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4 CHAPTER FORCES AND NEWTON'S LAWS
OF MOTION
ANSWERS TO FOCUS ON CONCEPTS QUESTIONS
1. (b) If only one force acts on the object, it is the net force; thus, the net force must be nonzero.
Consequently, the velocity would change, according to Newtons first law, and could not be
constant.
2. (d) This situation violates the first law, which predicts that the rabbits foot tends to remain in
place where it was when the car begins accelerating. The car would leave the rabbits foot
behind. That is, the rabbits foot would swing away from, not toward, the windshield.
3. (e) Newtons first law states that an object continues in a state of rest or in a state of motion
at a constant speed along a straight line, unless compelled to change that state by a net force.
All three statements are consistent with the first law.
4. (a) Newtons second law with a net force of 7560 N 7340 N = 220 N due north gives the
answer directly.
5. (c) Newtons second law gives the answer directly, provided the net force is calculated by
vector addition of the two given forces. The direction of the net force gives the direction of
the acceleration.
6. (e) Newtons second law gives the answer directly. One method is to determine the total
acceleration by vector addition of the two given components. The net force has the same
direction as the acceleration.
7. (e) Answers a and b are false, according to the third law, which states that whenever one
body exerts a force on a second body, the second body exerts an oppositely directed force of
equal magnitude on the first body. It does not matter whether one of the bodies is stationary
or whether it collapses. Answer c is false, because according to the third law, Sam and his
sister experience forces of equal magnitudes during the push-off. Since Sam has the greater
mass, he flies off with the smaller acceleration, according to the second law. Answer d is
false, because in catching and throwing the ball each astronaut applies a force to it, and,
according to the third law, the ball applies an oppositely directed force of equal magnitude to
each astronaut. These reaction forces accelerate the astronauts away from each other, so that
the distance between them increases.
8. (b) Newtons third law indicates that Paul and Tom apply forces of equal magnitude to each
other. According to Newtons second law, the magnitude of each of these forces is the mass
160 FORCES AND NEWTON'S LAWS OF MOTION
times the magnitude of the acceleration. Thus, we have mPaulaPaul = mTomaTom, or
mPaul/mTom = aTom/aPaul.
9. (e) Newtons law of gravitation gives the answer directly. According to this law the weight
is directly proportional to the mass of the planet, so twice the mass means twice the weight.
However, this law also indicates that the weight is inversely proportional to the square of the
planets radius, so three times the radius means one ninth the weight. Together, these two
factors mean that the weight on the planet is 2/9 or 0.222 times your earth-weight.
10. (c) Newtons law of gravitation gives the answer, provided that the distance between the
centers of the spheres is used for r (r = 0.50 m + 1.20 m + 0.80 m), rather than the distance
between the surfaces of the spheres.
11. (a) The answer follows directly from the fact that weight W is given by W = mg, where m is
the mass and g is the acceleration due to the earths gravity. Thus, m = (784 N)/(9.80 m/s2) =
80.0 kg. The mass is the same on Mercury as on Earth, because mass is an intrinsic property
of matter.
12. (d) What matters is the direction of the elevators acceleration. When the acceleration is
upward, the apparent weight is greater than the true weight. When the acceleration is
downward, the apparent weight is less than the true weight. In both possibilities the
acceleration points upward.
13. (b) According to Newtons third law, the pusher and the wall exert forces of equal
magnitude but opposite directions on each other. The normal force is the component of the
walls force that is perpendicular to the wall. Thus, it has the same magnitude as the
component of the pushers force that is perpendicular to the wall. As a result, the normal
forces are ranked in the same order as the perpendicular components of the pushers forces.
The smallest perpendicular component is in B, and the largest is in C.
14. (a) The static frictional force is balancing the component of the blocks weight that points
down the slope of the incline. This component is smallest in B and greatest in A.
15. (b) The static frictional force that blocks A and B exert on each other has a magnitude f.
The force that B exerts on A is directed to the right (the positive direction), while the force
that A exerts on B is directed to the left. Blocks B and C also exert static frictional forces on
each other, but these forces have a magnitude 2f, because the normal force pressing B and C
together is twice the normal force pressing A and B together. The force that C exerts on B is
directed to the right, while the force that B exerts on C is directed to the left. In summary,
then, block A experiences a single frictional force +f, which is the net frictional force; block
B experiences two frictional forces, and +2f, the net frictional force being +2f = +f;
f
f
block C experiences a single frictional force +2f, which is the net frictional force. It follows
that fs, A = fs, B = fs, C/2.
16. (c) The magnitude of the kinetic frictional force is proportional to the magnitude of the
normal force. The normal force is smallest in B, because the vertical component of F
Chapter 4 Answers to Focus on Concepts Questions
161
compensates for part of the blocks weight. In contrast, the normal force is greatest in C,
because the vertical component of F adds to the weight of the block.
17. (d) Acceleration is inversely proportional to mass, according to Newtons second law. This
law also indicates that acceleration is directly proportional to the net force. The frictional
force is the net force acting on a block, and its magnitude is directly proportional to the
magnitude of the normal force. However, in each of the pictures the normal force is directly
proportional to the weight and, thus, the mass of a block. The inverse proportionality of the
acceleration to mass and the direct proportionality of the net force to mass offset each other.
The result is that the deceleration is the same in each case.
18. (e) In B the tension T is the smallest, because three rope segments support the weight W of
the block, with the result that 3T = W, or T = W/3. In A the tension is the greatest, because
only one rope segment supports the weight of the block, with the result that T = W.
19. (c) Since the engines are shut down and since nothing is nearby to exert a force, the net
force acting on the probe is zero, and its acceleration must be zero, according to Newtons
second law. With zero acceleration the probe is in equilibrium.
20. (a) The hallmark of an object in equilibrium is that it has no acceleration. Therefore, an
object in equilibrium need not be at rest. It can be moving with a constant velocity.
21. (b) Since the object is not in equilibrium, it must be accelerating. Newtons second law, in
turn, implies that a net force must be present to cause the acceleration. Whether the net
forces arises from a single force, two perpendicular forces, or three forces is not important,
because only the net force appears in the second law.
22. (d) The block is at rest and, therefore, in equilibrium. According to Newtons second law,
then, the net force acting on the block in a direction parallel to the inclined surface of the
incline must be zero. This means that the force of static friction directed up the incline must
balance the component of the blocks weight directed down the incline [(8.0 kg)
(9.8 m/s2) sin 22 = 29 N].
23. (b) Since the boxes move at a constant velocity, they have no acceleration and are,
therefore, in equilibrium. According to Newtons second law, the net force acting on each
box must be zero. Thus, Newtons second law applied to each box gives two equations in
two unknowns, the magnitude of the tension in the rope between the boxes and the kinetic
frictional force that acts on each box. Note that the frictional forces acting on the boxes are
identical, because the boxes are identical. Solving these two equations shows that the tension
is one-half of the applied force.
24. 31 kgm/s2
25. 517 N
162 FORCES AND NEWTON'S LAWS OF MOTION
CHAPTER 4 FORCES AND NEWTON'S LAWS
OF MOTION
PROBLEMS
_____________________________________________________________________________
_
1.
REASONING AND SOLUTION According to Newtons second law, the acceleration is
a = F/m. Since the pilot and the plane have the same acceleration, we can write
F
F
=
m PILOT m PLANE
F
m PLANE
( F ) PILOT = mPILOT
or
Therefore, we find
3.7 104 N
= 93 N
4
3.1 10 kg
_____________________________________________________________________________
_
( F ) PILOT = ( 78 kg )
2.
REASONING Suppose the bobsled is moving along the + x direction. There are two forces
acting on it that are parallel to its motion; a force + Fx propelling it forward and a force of
450 N that is resisting its motion. The net force is the sum of these two forces. According
to Newtons second law, Equation 4.2a, the net force is equal to the mass of the bobsled
times its acceleration. Since the mass and acceleration are known, we can use the second
law to determine the magnitude of the propelling force.
SOLUTION
a. Newtons second law states that
+ Fx 450 N = max
1 44 2 4 43
Fx
(4.2a)
Solving this equation for Fx gives
(
)
Fx = max + 450 N = ( 270 kg ) 2.4 m/s 2 + 450 N = 1100 N
b. The magnitude of the net force that acts on the bobsled is
(
)
Fx = max = ( 270 kg ) 2.4 m/s 2 = 650 N
(4.2a)
____________________________________________________________________________________________
Chapter 4 Problems
3.
163
REASONING In each case, we will apply Newtons second law. Remember that it is the
net force that appears in the second law. The net force is the vector sum of both forces.
SOLUTION
a. We will use Newtons second law, Fx = max, to find the force F2. Taking the positive x
direction to be to the right, we have
F1 + F2
14 2 43 = ma
x
Fx
so
F2 = max F1
F2 = (3.0 kg)(+5.0 m/s2) (+9.0 N) = +6 N
b. Applying Newtons second law again gives
F2 = max F1 = (3.0 kg)(5.0 m/s2) (+9.0 N) = 24 N
c. An application of Newtons second law gives
F2 = max F1 = (3.0 kg)(0 m/s2) (+9.0 N) = 9.0 N
_____________________________________________________________________________
_
4.
REASONING According to Newtons second law, Equation 4.1, the average net force F
is equal to the product of the objects mass m and the average acceleration a . The average
acceleration is equal to the change in velocity divided by the elapsed time (Equation 2.4),
where the change in velocity is the final velocity v minus the initial velocity v0.
SOLUTION The average net force exerted on the car and riders is
v v0
(
)
45 m/s 0 m/s
= 5.5 103 kg
= 3.5 104 N
t t0
7.0 s
_____________________________________________________________________________
_
F = ma = m
5.
SSM REASONING The net force acting on the ball can be calculated using Newton's
second law. Before we can use Newton's second law, however, we must use Equation 2.9
from the equations of kinematics to determine the acceleration of the ball.
SOLUTION According to Equation 2.9, the acceleration of the ball is given by
164 FORCES AND NEWTON'S LAWS OF MOTION
a=
2
v 2 v0
2x
Thus, the magnitude of the net force on the ball is given by
2
v 2 v0
(45 m/s) 2 (0 m/s) 2
F = ma = m
= (0.058 kg)
= 130 N
2x
2(0.44 m)
_____________________________________________________________________________
_
6.
REASONING AND SOLUTION The acceleration is obtained from
x = v0t +
1
2
at2
where v0 = 0 m/s. So
a = 2x/t2
Newtons second law gives
2 ( 18 m )
2x
= 2900 N
F = ma = m 2 = ( 72 kg )
( 0.95 s ) 2
t
_____________________________________________________________________________
_
7.
SSM REASONING AND SOLUTION The acceleration required is
a=
2
v 2 v0
2x
2
( 15.0 m/s )
=
= 2.25 m/s 2
2 ( 50.0 m )
Newton's second law then gives the magnitude of the net force as
F = ma = (1580 kg)(2.25 m/s2) = 3560 N
____________________________________________________________________________________________
8.
REASONING We do not have sufficient information to calculate the average net force
applied to the fist by taking the vector sum of the individually applied forces. However, we
have the mass m of the fist, as well as its initial velocity (v0 = 0 m/s, since the fist starts from
rest), final velocity (v = 8.0 m/s), and the elapsed time (t = 0.15 s). Therefore we can use
v v v0
=
Equation 2.4 a =
to determine the average acceleration a of the fist and then
t
t
use Equation 4.1 (Newtons second law, F = ma ) to find the average net force F
applied to the fist.
Chapter 4 Problems
SOLUTION Inserting the relation a =
average net force applied to the fist:
165
v v v0
=
into Newtons second law yields the
t
t
v v0
8.0 m/s 0 m/s
F = ma = m
= ( 0.70 kg )
= 37 N
0.15 s
t
9.
SSM WWW REASONING Let due east be chosen as the positive direction. Then,
when both forces point due east, Newton's second law gives
FA + F = ma1
14 2 4B
3
F
(1)
where a1 = 0.50 m/s 2 . When FA points due east and FB points due west, Newton's second
law gives
FA F = ma2
14 2 4B
3
F
(2)
where a2 = 0.40 m/s 2 . These two equations can be used to find the magnitude of each
force.
SOLUTION
a. Adding Equations 1 and 2 gives
FA =
m ( a1 + a2 )
=
2
( 8.0 kg ) ( 0.50 m / s 2 + 0.40 m / s2 )
2
= 3.6 N
b. Subtracting Equation 2 from Equation 1 gives
FB =
m ( a1 a2 )
2
=
( 8.0 kg ) ( 0.50 m / s2 0.40 m / s 2 )
2
= 0.40 N
____________________________________________________________________________________________
166 FORCES AND NEWTON'S LAWS OF MOTION
10. REASONING From Newtons second law, we know that the net force F acting on the
electron is directly proportional to its acceleration, so in part a we will first find the
electrons acceleration. The problem text gives the electrons initial velocity
(v0 = +5.40105 m/s) and final velocity (v = +2.10106 m/s), as well as its displacement
(x = +0.038 m) during the interval of acceleration. The elapsed time is not known, so we
2
2
will use Equation 2.9 v = v0 + 2ax to calculate the electrons acceleration. Then we will
(
)
find the net force acting on the electron from Equation 4.1 ( F = ma ) and the electrons
mass. Because F1 points in the +x direction and F2 points in the x direction, the net force
acting on the electron is F = F1 F2 . In part b of the problem, we will rearrange this
expression to obtain the magnitude of the second electric force.
SOLUTION
a. Solving Equation 2.9 for the electrons acceleration, we find that
a=
2
v 2 v0
2x
( +2.10 106 m/s ) ( +5.40 105 m/s )
=
2
2 ( +0.038 m )
2
= +5.42 1013 m/s2
Newtons 2nd law of motion then gives the net force causing the acceleration of the electron:
F = ma = ( 9.11 1031 kg ) ( +5.42 1013 m/s 2 ) = +4.94 10 17 N
b. The net force acting on the electron is F = F1 F2 , so the magnitude of the second
electric force is F2 = F1 F , where F is the net force found in part a:
F2 = F1 F = 7.50 1017 N 4.94 1017 N = 2.56 10 17 N
11. REASONING Newtons second law gives the acceleration as a = (F)/m. Since we seek
only the horizontal acceleration, it is the x component of this equation that we will use;
ax = (Fx)/m. For completeness, however, the free-body diagram will include the vertical
forces also (the normal force FN and the weight W).
Chapter 4 Problems
SOLUTION
The free-body diagram is
shown at the right, where
167
+y
F1 = 59.0 N
F2 = 33.0 N
FN
= 70.0
F2
When F1 is replaced by its x and y
components, we obtain the free body
diagram in the following drawing.
+x
F1
W
Choosing right to be the positive direction, we have
F
F cos F2
ax = x = 1
m
m
ax
( 59.0 N ) cos 70.0 ( 33.0 N )
=
7.00 kg
= 1.83 m/s
+y
FN
2
F1cos
F2
The minus sign indicates that the horizontal acceleration
points to the left .
+x
F1sin
W
12. REASONING The net force F has a horizontal component Fx and a vertical component
Fy. Since these components are perpendicular, the Pythagorean theorem applies (Equation
1.7), and the magnitude of the net force is F =
( Fx )
2
(
+ Fy
)
2
. Newtons second law
allows us to express the components of the net force acting on the ball in terms of its mass
and the horizontal and vertical components of its acceleration: Fx = ma x , Fy = ma y
(Equations 4.2a and 4.2b).
SOLUTION Combining the Pythagorean theorem with Newtons second law, we obtain the
magnitude of the net force acting on the ball:
F =
( Fx )
2
(
+ Fy
= ( 0.430 kg )
)
2
=
( max )
2
(
+ ma y
)
2
( 810 m/s2 ) + ( 1100 m/s 2 )
2
2
2
= m ax + a 2
y
= 590 N
168 FORCES AND NEWTON'S LAWS OF MOTION
13.
SSM REASONING To determine the acceleration we will use Newtons second law
F = ma. Two forces act on the rocket, the thrust T and the rockets weight W, which is mg
= (4.50 105 kg)(9.80 m/s2) = 4.41 106 N. Both of these forces must be considered when
determining the net force F. The direction of the acceleration is the same as the direction
of the net force.
SOLUTION In constructing the free-body diagram for the rocket we choose upward and to
the right as the positive directions. The free-body diagram is as follows:
+y
T
The x component of the net force is
Fx = T cos 55.0
= 7.50 106 N cos 55.0 = 4.30 106 N
(
Ty
)
55.0
The y component of the net force is
W
(
Tx
= T sin 55.0
+x
= T cos 55.0
)
Fy = T sin 55.0 W = 7.50 106 N sin 55.0 4.41 106 N = 1.73 106 N
The magnitudes of the net force and of the acceleration are
F =
a=
( Fx )
2
(
+ Fy
)
2
m
( Fx ) 2 + ( Fy )
2
( 4.30 106 N ) + ( 1.73 106 N )
2
=
2
5
4.50 10 kg
= 10.3 m/s 2
The direction of the acceleration is the same as the direction of the net force. Thus, it is
directed above the horizontal at an angle of
Fy
= tan 1
F
x
6
1 1.73 10 N
= tan
4.30 106 N = 21.9
14. REASONING The acceleration of the sky diver can be obtained directly from Newtons
second law as the net force divided by the sky divers mass. The net force is the vector sum
of the sky divers weight and the drag force.
SOLUTION From Newtons second law, F = ma (Equation
4.1), the sky divers acceleration is
a=
f
F
m
+
W
Free-body diagram
Chapter 4 Problems
169
The free-body diagram shows the two forces acting on the sky diver, his weight W and the
drag force f. The net force is F = f W . Thus, the acceleration can be written as
a=
f W
m
The acceleration of the sky diver is
a=
f W 1027 N 915 N
=
= +1.20 m/s 2
m
93.4 kg
Note that the acceleration is positive, indicating that it points upward .
____________________________________________________________________________________________
15. REASONING Newtons second law, F = ma, states that a net force of F must act on an
object of mass m in order to impart an acceleration a to the object. In the impact shock test
the box is subjected to a large deceleration and, hence, a correspondingly large net force.
To determine the net force we will determine the deceleration in a kinematics calculation
and use it in Newtons second law.
SOLUTION According to Newtons second law, the net force is F = ma, where the
acceleration can be determined with the aid of Equation 2.4 (v = v0 + at). According to this
equation
v v0
a=
t
Substituting this result for the acceleration into the second law gives
v v0
F = ma = m
t
Since the initial velocity (v0 = +220 m/s), final velocity ( v = 0 m/s), and the duration of the
collision (t = 6.5 10 3 s) are known, we find
v v0
0 m/s 220 m/s
6
F = m
= ( 41 kg )
= 1.39 10 N
3
t
6.5 10 s
The minus sign indicates that the net force points opposite to the direction in which the box
is thrown, which has been assumed to be the positive direction. The magnitude of the net
force is 1.39 106 N , which is over three hundred thousand pounds.
170 FORCES AND NEWTON'S LAWS OF MOTION
16. REASONING Since there is only one force acting on the man in the horizontal direction, it
is the net force. According to Newtons second law, Equation 4.1, the man must accelerate
under the action of this force. The factors that determine this acceleration are (1) the
magnitude and (2) the direction of the force exerted on the man, and (3) the mass of the
man.
When the woman exerts a force on the man, the man exerts a force of equal magnitude, but
opposite direction, on the woman (Newtons third law). It is the only force acting on the
woman in the horizontal direction, so, as is the case with the man, she must accelerate. The
factors that determine her acceleration are (1) the magnitude and (2) the direction of the
force exerted on her, and (3) the her mass.
SOLUTION
a. The acceleration of the man is, according to Equation 4.1, equal to the net force acting on him
divided by his mass.
aman =
F 45 N
=
= 0.55 m / s 2 (due east)
m 82 kg
b. The acceleration of the woman is equal to the net force acting on her divided by her mass.
awoman =
F 45 N
=
= 0.94 m / s 2 (due west)
m 48 kg
______________________________________________________________________________________
17. REASONING Equations 3.5a
( x = v0 xt + 1 axt 2 )
2
and 3.5b
( y = v0 yt + 12 a yt 2 )
give the
displacements of an object under the influence of constant accelerations ax and ay. We can
add these displacements as vectors to find the magnitude and direction of the resultant
displacement. To use Equations 3.5a and 3.5b, however, we must have values for ax and ay.
We can obtain these values from Newtons second law, provided that we combine the given
forces to calculate the x and y components of the net force acting on the duck, and it is here
that our solution begins.
SOLUTION Let the directions due east and due north, respectively, be the +x and +y
directions. Then, the components of the net force are
Fx = 0.10 N + ( 0.20 N ) cos 52 = 0.2231 N
Fy = ( 0.20 N ) sin 52 = 0.1576 N
According to Newtons second law, the components of the acceleration are
Chapter 4 Problems
ax =
ay =
Fx
m
Fy
m
=
0.2231 N
= 0.08924 m/s 2
2.5 kg
=
171
0.1576 N
= 0.06304 m/s 2
2.5 kg
From Equations 3.5a and 3.5b, we now obtain the displacements in the x and y directions:
(
)
x = v0 xt + 1 axt 2 = ( 0.11 m/s ) ( 3.0 s ) + 1 0.08924 m/s 2 ( 3.0 s ) = 0.7316 m
2
2
(
2
)
y = v0 y t + 1 a y t 2 = ( 0 m/s ) ( 3.0 s ) + 1 0.06304 m/s 2 ( 3.0 s ) = 0.2837 m
2
2
2
The magnitude of the resultant displacement is
r = x2 + y2 =
( 0.7316 m ) 2 + ( 0.2837 m ) 2 =
0.78 m
The direction of the resultant displacement is
0.2837 m
= tan 1
= 21 south of east
0.7316 m
____________________________________________________________________________________________
(
)
2
18. REASONING For both the tug and the asteroid, Equation 2.8 x = v0t + 1 at applies with
2
v0 = 0 m/s, since both are initially at rest. In applying this equation, we must be careful and
use the proper acceleration for each object. Newtons second law indicates that the
acceleration is given by a = F/m. In this expression, we note that the magnitudes of the net
forces acting on the tug and the asteroid are the same, according to Newtons action-reaction
law. The masses of the tug and the asteroid are different, however. Thus, the distance
traveled for either object is given by, where we use for F only the magnitude of the pulling
force
F 2
x = v0t + 1 at 2 = 1
t
2
2 m
SOLUTION Let L be the initial distance between the tug and the asteroid. When the two
objects meet, the distances that each has traveled must add up to equal L. Therefore,
L = xT + x A = 1 aT t 2 + 1 a At 2
2
2
F 2 1 F 2 1
1
12
L= 1
+
t + 2
t = 2 F
t
2m
m
m
T
A
T mA
172 FORCES AND NEWTON'S LAWS OF MOTION
Solving for the time t gives
t=
2L
1
1
F
m +m
T
A
=
2 ( 450 m )
1
1
+
( 490 N )
3500 kg 6200 kg
= 64 s
____________________________________________________________________________________________
19.
SSM WWW REASONING We first determine the acceleration of the boat. Then,
using Newton's second law, we can find the net force F that acts on the boat. Since two
of the three forces are known, we can solve for the unknown force FW once the net force
F is known.
SOLUTION Let the direction due east be the positive x direction and the direction due
north be the positive y direction. The x and y components of the initial velocity of the boat
are then
v0 x = (2.00 m/s) cos 15.0 = 1.93 m/s
v0 y = (2.00 m/s) sin 15.0 = 0.518 m/s
Thirty seconds later, the x and y velocity components of the boat are
vx = (4.00 m/s) cos 35.0 = 3.28 m/s
v y = (4.00 m/s) sin 35.0 = 2.29 m/s
Therefore, according to Equations 3.3a and 3.3b, the x and y components of the acceleration
of the boat are
v v
3.28 m/s 1.93 m/s
ax = x 0 x =
= 4.50 102 m/s 2
t
30.0 s
ay =
v y v0 y
t
=
2.29 m/s 0.518 m/s
= 5.91 102 m/s 2
30.0 s
Thus, the x and y components of the net force that act on the boat are
Fx = ma x = (325 kg) (4.50 10 2 m/s 2 ) = 14.6 N
Fy = ma y = (325 kg) (5.9110 2 m/s 2 ) = 19.2 N
Chapter 4 Problems
173
The following table gives the x and y components of the net force F and the two known
forces that act on the boat. The fourth row of that table gives the components of the
unknown force FW .
Force
x-Component
y-Component
F
14.6 N
19.2 N
F1
(31.0 N) cos 15.0 = 29.9 N
(31.0 N) sin 15.0 = 8.02 N
F2
(23.0 N ) cos 15.0 = 22.2 N
(23.0 N) sin 15.0 = 5.95 N
FW = F F1 F2
14.6 N 29.9 N + 22.2 N = 6.9 N
19.2 N 8.02 N + 5.95 N = 17.1 N
The magnitude of FW is given by the Pythagorean theorem as
FW = (6.9 N) 2 + (17.1N ) 2 = 18.4 N
The angle that FW makes with the x axis is
17.1 N
= tan 1
= 68
6.9 N
17.1 N
6.9 N
Therefore, the direction of FW is 68, north of east .
____________________________________________________________________________________________
20. REASONING The gravitational force acting on each object is specified by Newtons law of
universal gravitation. The acceleration of each object when released can be determined with
the aid of Newtons second law. We recognize that the gravitational force is the only force
acting on either object, so that it is the net force to use when applying the second law.
SOLUTION
a. The magnitude of the gravitational force exerted on the rock by the earth is given by
Equation 4.3 as
Gmearth mrock
Frock =
=
(
2
rearth
6.67 1011 N m 2 / kg 2 5.98 1024 kg ( 5.0 kg )
)(
( 6.38 10 m )
6
)
2
= 49 N
174 FORCES AND NEWTON'S LAWS OF MOTION
The magnitude of the gravitational force exerted on the pebble by the earth is
Gmearth mpebble
Fpebble =
=
(
2
rearth
6.67 1011 N m 2 / kg 2 5.98 1024 kg 3.0 10 4 kg
)(
( 6.38 106 m )
)(
2
)=
2.9 10 3 N
b. According to the second law, the magnitude of the acceleration of the rock is equal to the
gravitational force exerted on the rock divided by its mass.
arock =
Frock
mrock
Gmearth
=
=
(
2
rearth
6.67 1011 N m 2 / kg 2 5.98 1024 kg
)(
( 6.38 10 m )
6
2
)=
9.80 m /s 2
According to the second law, the magnitude of the acceleration of the pebble is equal to the
gravitational force exerted on the pebble divided by its mass.
apebble =
Fpebble
mpebble
=
Gmearth
2
rearth
( 6.67 1011 N m2 / kg 2 ) ( 5.98 1024 kg ) =
=
2
( 6.38 106 m )
9.80 m /s 2
_____________________________________________________________________________
_
21. REASONING AND SOLUTION
a. According to Equation 4.5, the weight of the space traveler of mass
m = 115 kg
on
earth is
W = mg = (115 kg) (9.80 m/s 2 ) = 1.13 103 N
b. In interplanetary space where there are no nearby planetary objects, the gravitational
force exerted on the space traveler is zero and g = 0 m/s2. Therefore, the weight is
W = 0 N . Since the mass of an object is an intrinsic property of the object and is
independent of its location in the universe, the mass of the space traveler is still
m = 115 kg .
Chapter 4 Problems
175
B
+
mg
Free-body diagram
of jumper
____________________________________________________________________________________________
22. REASONING With air resistance neglected, only two forces act on the bungee jumper at
this instant (see the free-body diagram): the bungee cord pulls up on her with a force B, and
the earth pulls down on her with a gravitational force mg. Because we know the jumpers
mass and acceleration, we can apply Newtons second law to this free-body diagram and
solve for B.
SOLUTION We will take the direction of the jumpers acceleration (downward) as
negative. Then, the net force acting on the bungee jumper is F = B mg. With Newtons
second law (F = ma), this becomes ma = B mg. We now solve for B:
(
)
B = ma + mg = m ( a + g ) = ( 55 kg ) 7.6 m/s 2 + 9.80 m/s 2 = 120 N
As indicated in the free-body diagram, the direction of the force applied by the bungee cord
is upward .
23. REASONING The earth exerts a gravitational force on the raindrop, and simultaneously the
raindrop exerts a gravitational force on the earth. This gravitational force is equal in
magnitude to the gravitational force that the earth exerts on the raindrop. The forces that the
raindrop and the earth exert on each other are Newtons third law (actionreaction) forces.
Newtons law of universal gravitation specifies the magnitude of both forces.
SOLUTION
a. The magnitude of the gravitational force exerted on the raindrop by the earth is given by
Equation 4.3:
Gmearth mraindrop
Fraindrop =
=
(
2
rearth
6.67 1011 N m 2 / kg 2 5.98 10 24 kg 5.2 10 7 kg
)(
( 6.38 10 m )
6
)(
2
)=
5.1 106 N
176 FORCES AND NEWTON'S LAWS OF MOTION
b. The magnitude of the gravitational force exerted on the earth by the raindrop is
Gmearth mraindrop
Fearth =
=
(
2
rearth
6.67 10 11 N m 2 / kg 2 5.98 1024 kg 5.2 10 7 kg
)(
( 6.38 106 m )
)(
2
)=
5.1 106 N
_____________________________________________________________________________
_
24. REASONING Newtons law of universal gravitation indicates that the gravitational force
that each uniform sphere exerts on the other has a magnitude that is inversely proportional
to the square of the distance between the centers of the spheres. Therefore, the maximum
gravitational force between two uniform spheres occurs when the centers of the spheres are
as close together as possible, and this occurs when the surfaces of the spheres are touching.
Then, the distance between the centers of the spheres is the sum of the two radii.
SOLUTION When the bowling ball and the billiard ball are touching, the distance between
their centers is r = rBowling + rBilliard. Using this expression in Newtons law of universal
gravitation gives
F=
GmBowling mBilliard
GmBowling mBilliard
=
( rBowling + rBilliard )
( 6.67 1011 N m 2 / kg 2 ) ( 7.2 kg ) ( 0.38 kg ) = 9.6 109 N
=
r2
2
( 0.11 m + 0.028 m ) 2
25.
SSM REASONING AND SOLUTION
a. Combining Equations 4.4 and 4.5, we see that the acceleration due to gravity on the
surface of Saturn can be calculated as follows:
gSaturn = G
M Saturn
2
rSaturn
(
= 6.67 10
11
2
N m /kg
2
( 5.67 1026 kg ) =
) (6.00 107 m)2
10.5 m/s 2
b. The ratio of the persons weight on Saturn to that on earth is
WSaturn
Wearth
=
mgSaturn
mgearth
=
gSaturn
gearth
=
10.5 m/s 2
= 1.07
9.80 m/s 2
____________________________________________________________________________________________
Chapter 4 Problems
177
26. REASONING As discussed in Conceptual Example 7, the same net force is required on the
moon as on the earth. This net force is given by Newtons second law as F = ma, where
the mass m is the same in both places. Thus, from the given mass and acceleration, we can
calculate the net force. On the moon, the net force comes about due to the drive force and
the opposing frictional force. Since the drive force is given, we can find the frictional force.
SOLUTION Newtons second law, with the direction of motion taken as positive, gives
F = ma
( 1430 N )
or
(
)(
f = 5.90 103 kg 0.220 m/s 2
)
Solving for the frictional force f , we find
(
)(
)
f = ( 1430 N ) 5.90 103 kg 0.220 m/s 2 = 130N
____________________________________________________________________________________________
27.
SSM REASONING AND SOLUTION According to Equations 4.4 and 4.5, the weight
of an object of mass m at a distance r from the center of the earth is
mg =
GM E m
r2
In a circular orbit that is 3.59 107 m above the surface of the earth ( radius = 6.38 106 m ,
mass = 5.98 1024 kg ), the total distance from the center of the earth is
r = 3.59 107 m + 6.38 106 m . Thus the acceleration g due to gravity is
g=
GM E
r
2
=
(6.67 1011N m 2 /kg 2 )(5.98 1024 kg)
=
(3.59 107 m + 6.38 106 m) 2
0.223 m/s 2
____________________________________________________________________________________________
28. REASONING AND SOLUTION
The magnitude of the net force acting on the
moon is found by the Pythagorean theorem to
be
F
SM
Sun
F=
2
FSM
F
Moon
F
EM
2
+ FEM
Earth
Newton's law of gravitation applied to the sunmoon (the units have been suppressed)
178 FORCES AND NEWTON'S LAWS OF MOTION
FSM =
GmS mM
2
rSM
=
( 6.67 1011 ) ( 1.99 1030 ) ( 7.35 1022 )
( 1.50 1011 ) 2
= 4.34 1020 N
A similar application to the earth-moon gives
FEM =
GmE mM
2
rEM
=
( 6.67 1011 ) ( 5.98 1024 ) ( 7.35 1022 )
( 3.85 108 ) 2
= 1.98 10 20 N
The net force on the moon is then
F=
( 4.34 1020 N ) + ( 1.98 1020 N )
2
2
= 4.77 10 20 N
____________________________________________________________________________________________
29. REASONING The magnitude of the gravitational force exerted on the satellite by the
earth is given by Equation 4.3 as F = Gmsatellite mearth / r 2 , where r is the distance between
the satellite and the center of the earth. This expression also gives the magnitude of the
gravitational force exerted on the earth by the satellite. According to Newtons second law,
the magnitude of the earths acceleration is equal to the magnitude of the gravitational force
exerted on it divided by its mass. Similarly, the magnitude of the satellites acceleration is
equal to the magnitude of the gravitational force exerted on it divided by its mass.
SOLUTION
a. The magnitude of the gravitational force exerted on the satellite when it is a distance of
two earth radii from the center of the earth is
F=
Gmsatellite mearth
r2
=
( 6.67 1011 N m2 / kg 2 ) ( 425 kg ) ( 5.98 1024 kg )
2
( 2 ) ( 6.38 106 m )
= 1.04 103 N
b. The magnitude of the gravitational force exerted on the earth when it is a distance of two
earth radii from the center of the satellite is
F=
Gmsatellite mearth
r2
=
( 6.67 1011 N m2 / kg 2 ) ( 425 kg ) ( 5.98 1024 kg )
2
( 2 ) ( 6.38 106 m )
= 1.04 103 N
c. The acceleration of the satellite can be obtained from Newtons second law.
asatellite =
F
msatellite
=
1.04 103 N
=
425 kg
2.45 m / s 2
Chapter 4 Problems
179
d. The acceleration of the earth can also be obtained from Newtons second law.
aearth =
F
mearth
1.04 103 N
= 1.74 1022 m / s 2
24
5.98 10 kg
=
____________________________________________________________________________________________
30. REASONING The weight of a person on the earth is the gravitational force Fearth that it
exerts on the person. The magnitude of this force is given by Equation 4.3 as
Fearth = G
mearth mperson
2
rearth
where rearth is the distance from the center of the earth to the person. In a similar fashion, the
weight of the person on another planet is
Fplanet = G
mplanet mperson
2
rplanet
We will use these two expressions to obtain the weight of the traveler on the planet.
SOLUTION Dividing Fplanet by Fearth we have
Fplanet
Fearth
G
=
G
mplanet mperson
2
rplanet
mearth mperson
2
rearth
mplanet
=
m
earth
rearth
r
planet
2
or
Fplanet
Since we are given that
mplanet
mearth
mplanet
= Fearth
m
earth
= 3 and
rearth
rplanet
=
rearth
r
planet
2
1
, the weight of the space traveler on the
2
planet is
2
Fplanet
1
= ( 540.0 N ) ( 3) = 405.0 N
2
180 FORCES AND NEWTON'S LAWS OF MOTION
_____________________________________________________________________________
_
31.
SSM REASONING According to Equation 4.4, the weights of an object of mass m on
the surfaces of planet A (mass = MA, radius = R ) and planet B (mass = MB , radius = R ) are
WA =
GM A m
and
R2
WB =
GM Bm
R2
The difference between these weights is given in the problem.
SOLUTION The difference in weights is
WA WB =
GM A m
R
2
GM Bm
R
2
=
Gm
( MA MB )
R2
Rearranging this result, we find
MA MB
( WA WB ) R 2 =
=
Gm
( 3620 N ) ( 1.33 107 m )
(6.67 10 11N m 2 /kg 2 ) ( 5450 kg )
2
= 1.76 10 24 kg
____________________________________________________________________________________________
32. REASONING AND SOLUTION The figure at the right
shows the three spheres with sphere 3 being the sphere
of unknown mass. Sphere 3 feels a force F31 due to the
presence of sphere 1, and a force F32 due to the
presence of sphere 2. The net force on sphere 3 is the
resultant of F31 and F32.
Note that since the spheres form an equilateral triangle,
each interior angle is 60. Therefore, both F31 and F32
make a 30 angle with the vertical line as shown.
3
F
F
31
32
1.20 m
30
1
2
Furthermore, F31 and F32 have the same magnitude
given by
GMm3
F=
r2
where M is the mass of either sphere 1 or 2 and m3 is the mass of sphere 3. The components
of the two forces are shown in the following drawings:
Chapter 4 Problems
F
31
F cos
30.0
F cos
F sin
181
F
30.0
32
F sin
Clearly, the horizontal components of the two forces add to zero. Thus, the net force on
sphere 3 is the resultant of the vertical components of F31 and F32:
F3 = 2 F cos = 2
GMm3
r2
cos
The acceleration of sphere 3 is given by Newton's second law:
(6.67 10
GM
a3 =
= 2 2 cos = 2
m3
r
F3
11
N m 2 /kg 2 ) ( 2.80 kg )
( 1.20 m ) 2
cos 30.0
= 2.25 10 10 m/s 2
____________________________________________________________________________________________
33.
SSM WWW REASONING AND SOLUTION There are two forces that act on the
balloon; they are, the combined weight of the balloon and its load, Mg, and the upward
buoyant force FB . If we take upward as the positive direction, then, initially when the
balloon is motionless, Newton's second law gives FB Mg = 0 . If an amount of mass m is
dropped overboard so that the balloon has an upward acceleration, Newton's second law for
this situation is
FB ( M m) g = ( M m) a
But FB = mg , so that
Mg ( M m ) g = mg = ( M m ) a
Solving for the mass m that should be dropped overboard, we obtain
Ma
(310 kg )(0.15 m/s 2 )
=
= 4.7 kg
g + a 9.80 m/s 2 + 0.15 m/s 2
_____________________________________________________________________________
_
m=
34. REASONING AND SOLUTION
acceleration, it follows that
Since both motions are characterized by constant
182 FORCES AND NEWTON'S LAWS OF MOTION
yJ
yE
=
1 a t2
2 JJ
1 a t2
2 EE
where the subscripts designate those quantities that pertain to Jupiter and Earth. Since both
objects fall the same distance (yJ = yE), the above ratio is equal to unity. Solving for the
ratio of the times yields
tJ
tE
=
aE
aJ
=
2
GM E / RE
2
GM J / RJ
=
RJ
ME
RE
MJ
= ( 11.2 )
1
= 0.628
318
____________________________________________________________________________________________
35. REASONING The gravitational force that the sun exerts on a person standing on the earth
2
is given by Equation 4.3 as Fsun = GM sun m / rsun-earth , where Msun is the mass of the sun, m
is the mass of the person, and rsun-earth is the distance from the sun to the earth. Likewise,
the gravitational force that the moon exerts on a person standing on the earth is given by
2
Fmoon = GM moon m / rmoon-earth , where Mmoon is the mass of the moon and rmoon-earth is the
distance from the moon to the earth. These relations will allow us to determine whether the
sun or the moon exerts the greater gravitational force on the person.
SOLUTION Taking the ratio of Fsun to Fmoon, and using the mass and distance data from
the inside of the texts front cover, we find
GM sun m
Fsun
Fmoon
2
M
rsun-earth
=
= sun
M
GM moon m
moon
2
rmoon-earth
rmoon-earth
r
sun-earth
2
2
1.99 1030 kg 3.85 108 m
=
7.35 1022 kg 1.50 1011 m = 178
Therefore, the sun exerts the greater gravitational force.
____________________________________________________________________________________________
Chapter 4 Problems
183
36. REASONING The gravitational attraction between the planet and the moon is governed by
Newtons law of gravitation F = GMm/r2 (Equation 4.3), where M is the planets mass and
m is the moons mass. Because the magnitude of this attractive force varies inversely with
the square of the distance r between the center of the moon and the center of the planet, the
maximum force Fmax occurs at the minimum distance rmin, and the minimum force Fmin at
the maximum distance rmax. The problem text states that Fmax exceeds Fmin by 11%, or
Fmax = 1.11 Fmin. This expression can be rearranged to give the ratio of the
forces: Fmax/ Fmin = 1.11. We will use Equation 4.3 to compute the desired distance ratio in
terms of this force ratio.
SOLUTION From Equation 4.3, the ratio of the maximum gravitational force to the
minimum gravitational force is
Fmax
Fmin
=
GMm
2
rmin
GMm
2
rmax
1
=
2
2
rmin rmax
=2
1
rmin
2
rmax
Taking the square root of both sides of this expression and substituting the ratio of forces
Fmax/ Fmin = 1.11 yields the ratio of distances:
rmax
rmin
=
Fmax
Fmin
= 1.11 = 1.05
The moons maximum distance from the center of the planet is therefore about 5% larger
than its minimum distance.
37. REASONING We place the third particle (mass = m3) as shown in the following drawing:
L
D
m3
m
2m
The magnitude of the gravitational force that one particle exerts on another is given by
Newtons law of gravitation as F = Gm1m2/r2. Before the third particle is in place, this law
indicates that the force on each particle has a magnitude Fbefore = Gm2m/L2. After the third
particle is in place, each of the first two particles experiences a greater net force, because the
third particle also exerts a gravitational force on them.
SOLUTION For the particle of mass m, we have
184 FORCES AND NEWTON'S LAWS OF MOTION
Fafter
=
Fbefore
Gmm3
Gm 2m
2
D
L2 = L m3 + 1
Gm2m
2mD 2
L2
2
+
For the particle of mass 2m, we have
G 2mm3
Fafter
Fbefore
=
( L D)
2
+
Gm2m
L2
Gm2m
L2
=
L2 m3
m( L D)
2
+1
Since Fafter/Fbefore = 2 for both particles, we have
L2 m3
2mD
2
+1 =
L2 m3
m ( L D)
2
+1
or
2D2 = ( L D )
2
Expanding and rearranging this result gives D 2 + 2 LD L2 = 0 , which can be solved for D
using the quadratic formula:
D=
2 L
( 2 L ) 2 4 ( 1) ( L2 )
2 ( 1)
= 0.414 L or
2.414 L
The negative solution is discarded because the third particle lies on the + x axis between m
and 2m. Thus, D = 0.414 L .
____________________________________________________________________________________________
38. REASONING In each case the object is in equilibrium. According to Equation 4.9b,
Fy = 0, the net force acting in the y (vertical) direction must be zero. The net force is
composed of the weight of the object(s) and the normal force exerted on them.
SOLUTION
a. There are three vertical forces acting on the crate: an upward normal force + FN that the
floor exerts, the weight m1g of the crate, and the weight m2g of the person standing on the
crate. Since the weights act downward, they are assigned negative numbers. Setting the sum
of these forces equal to zero gives
Chapter 4 Problems
185
F + ( m g ) + ( m2 g ) = 0
1 N 4 4 4 12 4 4 4 4 4
4
4
3
Fy
The magnitude of the normal force is
FN = m1g + m2g = (35 kg + 65 kg)(9.80 m/s2) = 980 N
b. There are only two vertical forces acting on the person: an upward normal force + FN that
the crate exerts and the weight m2g of the person. Setting the sum of these forces equal to
zero gives
F + ( m2 g ) = 0
1 N 4 2 4 43
4
Fy
The magnitude of the normal force is
FN = m2g = (65 kg)(9.80 m/s2) = 640 N
____________________________________________________________________________________________
39.
SSM REASONING The book is kept from falling as long as the total static frictional
force balances the weight of the book. The forces that act on the book are shown in the
following free-body diagram, where P is the pressing force applied by each hand.
fsMAX
fsMAX
P
P
W
In this diagram, note that there are two pressing forces, one from each hand. Each hand also
applies a static frictional force, and, therefore, two static frictional forces are shown. The
maximum static frictional force is related in the usual way to a normal force FN, but in this
problem the normal force is provided by the pressing force, so that FN = P.
SOLUTION Since the frictional forces balance the weight, we have
2 f sMAX = 2 ( s FN ) = 2 ( s P ) = W
Solving for P, we find that
186 FORCES AND NEWTON'S LAWS OF MOTION
P=
W
31 N
=
= 39 N
2 s 2 ( 0.40 )
____________________________________________________________________________________________
40. REASONING AND SOLUTION
a. The apparent weight of the person is given by Equation 4.6 as
F N = mg + ma
= (95.0 kg)(9.80 m/s2 + 1.80 m/s2) = 1.10 103 N
b.
FN = (95.0 kg)(9.80 m/s2) = 931 N
c.
FN = (95.0 kg)(9.80 m/s2 1.30 m/s2) = 808 N
____________________________________________________________________________________________
41. REASONING As the drawing shows, the normal
FN
force FN points perpendicular to the hill, while the weight
W points vertically down. Since the car does not leave the
surface of the hill, there is no acceleration in this
perpendicular direction. Therefore, the magnitude of the
perpendicular component of the weight W cos must equal
W cos
the magnitude of the normal force, FN = W cos . Thus, the
W
magnitude of the normal force is less than the magnitude of
the weight. As the hill becomes steeper, increases, and cos decreases. Consequently, the
normal force decreases as the hill becomes steeper. The magnitude of the normal force does
not depend on whether the car is traveling up or down the hill.
SOLUTION
a. From the REASONING, we have that FN = W cos . The ratio of the magnitude of the
normal force to the magnitude W of the weight is
FN
W
=
W cos
= cos 15 = 0.97
W
b. When the angle is 35, the ratio is
W cos
= cos 35 = 0.82
W
W
_____________________________________________________________________________
_
FN
=
Chapter 4 Problems
187
42. REASONING The reading on the bathroom scale is
proportional to the normal force it exerts on the man.
When he simply stands on the scale, his acceleration
FN1
is zero, so the normal force pushing up on him
balances the downward pull of gravity: FN1 = mg (see
P FN2
the free-body diagram). Thus, the first reading on the
scale is his actual mass m, the ratio of the normal
force the scale exerts on him to the acceleration due to
gravity: First reading = m = FN1/g = 92.6 kg. With the
mg
mg
chin-up bar helping to support him, the normal force
exerted on him by the scale decreases, and the second Standing
Pulli
reading is the ratio of the reduced normal force FN2 to
the acceleration due to gravity: Second reading = FN2/g = 75.1 kg. Lastly, we note that the
magnitude P of the force the chin-up bar exerts on the man is exactly equal to the magnitude
P of the force that the man exerts on the chin-up bar. This prediction is due to Newtons
third law. Therefore, it is a value for P that we seek.
SOLUTION When the man is pulling down on the chin-up bar, there are two upward forces
acting on him (see the second part of the drawing), and he is still at rest, so the sum of these
two forces balances the downward pull of gravity: FN2 + P = mg, or P = mg FN 2 . Since
the second reading on the scale is equal to FN2/g, the normal force the scale exerts on him is
FN2 = (Second reading)g. Thus we obtain the magnitude P of the force the man exerts on
the chin-up bar:
P = mg FN2 = mg (Second reading) g = ( m Second reading ) g
(
)
= ( 92.6 kg 75.1 kg ) 9.80 m/s 2 = 172 N
43. REASONING As shown in the free-body diagram below, three forces act on the car: the
static frictional force fs (directed up the hill), the normal force FN (directed perpendicular to
the road), and its weight mg. As it sits on the hill, the car has an acceleration of zero
(ax = ay = 0 m/s2). Therefore, the net force acting on the car in the x direction must be zero
( Fx = 0 )
and the net force in the y direction must be zero ( Fy = 0 ) . These two relations
will allow us to find the normal force and the static frictional force.
SOLUTION
a. Applying Newtons second
( Fy = 0 ) yields
law to the y direction
Fy = + FN mg cos15 = 0
(4.2b)
188 FORCES AND NEWTON'S LAWS OF MOTION
where the term mg cos 15 is the y component of the cars weight (negative, because this
component points along the negative y axis). Solving for the magnitude FN of the normal
force, we obtain
FN = mg cos15 = ( 1700 kg ) ( 9.80 m/s 2 ) cos15 = 1.6 104 N
b. Applying Newtons second law to the x direction ( Fx = 0 ) gives
Fx = + mg sin15 fs = 0
(4.2a)
where the term mg sin 15 is the x component of the cars weight. Solving this expression for
the static frictional force gives
fs = mg sin15 = ( 1700 kg ) ( 9.80 m/s 2 ) sin15 = 4.3 103 N
44. REASONING
a. Since the refrigerator does not move, the static frictional force must be equal in
magnitude, but opposite in direction, to the horizontal pushing force that the person exerts
on the refrigerator.
b. The magnitude of the maximum static frictional force is given by Equation 4.7 as
fsMAX = s FN . This is also the largest possible force that the person can exert on the
refrigerator before it begins to move. Thus, the factors that determine this force magnitude
are the coefficient of static friction s and the magnitude FN of the normal force (which is
equal to the weight of the refrigerator in this case).
SOLUTION
a. Since the refrigerator does not move, it is in equilibrium, and the magnitude of the static
frictional force must be equal to the magnitude of the horizontal pushing force. Thus, the
magnitude of the static frictional force is 267 N . The direction of this force must be
opposite to that of the pushing force, so the static frictional force is in the +x direction .
b. The magnitude of the largest pushing force is given by Equation 4.7 as
fsMAX = s FN = s mg = (0.65)(57 kg)(9.80 m/s 2 ) = 360 N
____________________________________________________________________________________________
45.
SSM REASONING AND SOLUTION Four forces act on the sled. They are the pulling
force P, the force of kinetic friction fk , the weight mg of the sled, and the normal force FN
Chapter 4 Problems
189
exerted on the sled by the surface on which it slides. The following figures show free-body
diagrams for the sled. In the diagram on the right, the forces have been resolved into their x
and y components.
y
P sin
y
FN
P
fk
FN
fk
P cos
x
mg
x
mg
Since the sled is pulled at constant velocity, its acceleration is zero, and Newton's second
law in the direction of motion is (with right chosen as the positive direction)
Fx = P cos f k = ma x = 0
From Equation 4.8, we know that f k = k FN , so that the above expression becomes
P cos k FN = 0
(1)
In the vertical direction,
Fy = P sin + FN mg = ma y = 0
(2)
Solving Equation (2) for the normal force, and substituting into Equation (1), we obtain
P cos k ( mg P sin ) = 0
Solving for k , the coefficient of kinetic friction, we find
k =
P cos
(80.0 N) cos 30.0
=
= 0.444
mg P sin (20.0 kg) (9.80 m/s 2 ) (80.0 N) sin 30.0
____________________________________________________________________________________________
46. REASONING In each of the three cases under consideration the kinetic frictional force is
given by fk = kFN. However, the normal force FN varies from case to case. To determine
the normal force, we use Equation 4.6 ( FN = mg + ma) and thereby take into account the
190 FORCES AND NEWTON'S LAWS OF MOTION
acceleration of the elevator. The normal force is greatest when the elevator accelerates
upward (a positive) and smallest when the elevator accelerates downward (a negative).
SOLUTION
a. When the elevator is stationary, its acceleration is a = 0 m/s2. Using Equation 4.6, we
can express the kinetic frictional force as
f k = k FN = k ( mg + ma ) = k m ( g + a )
(
)(
)
= ( 0.360 ) ( 6.00 kg ) 9.80 m/s 2 + 0 m/s 2 = 21.2 N
b. When the elevator accelerates upward, a = +1.20 m/s2. Then,
f k = k FN = k ( mg + ma ) = k m ( g + a )
(
)(
)
= ( 0.360 ) ( 6.00 kg ) 9.80 m/s 2 + 1.20 m/s 2 = 23.8 N
c. When the elevator accelerates downward, a = 1.20 m/s2. Then,
f k = k FN = k ( mg + ma ) = k m ( g + a )
(
)(
)
= ( 0.360 ) ( 6.00 kg ) 9.80 m/s 2 + 1.20 m/s 2 = 18.6 N
____________________________________________________________________________________________
47. REASONING The magnitude of the kinetic frictional force is given by Equation 4.8 as the
coefficient of kinetic friction times the magnitude of the normal force. Since the slide into
second base is horizontal, the normal force is vertical. It can be evaluated by noting that
there is no acceleration in the vertical direction and, therefore, the normal force must
balance the weight.
To find the players initial velocity v0, we will use kinematics. The time interval for the
slide into second base is given as t = 1.6 s. Since the player comes to rest at the end of the
slide, his final velocity is v = 0 m/s. The players acceleration a can be obtained from
Newtons second law, since the net force is the kinetic frictional force, which is known from
part (a), and the mass is given. Since t, v, and a are known and we seek v0, the appropriate
kinematics equation is Equation 2.4 (v = v0 + at).
SOLUTION
a. Since the normal force FN balances the weight mg, we know that FN = mg. Using this
fact and Equation 4.8, we find that the magnitude of the kinetic frictional force is
(
)
f k = k FN = k mg = ( 0.49 ) ( 81 kg ) 9.8 m/s 2 = 390 N
Chapter 4 Problems
191
b. Solving Equation 2.4 ( v = v0 + at) for v0 gives v0 = v at. Taking the direction of the
players slide to be the positive direction, we use Newtons second law and Equation 4.8 for
the kinetic frictional force to write the acceleration a as follows:
a=
F k mg
=
= k g
m
m
The acceleration is negative, because it points opposite to the players velocity, since the
player slows down during the slide. Thus, we find for the initial velocity that
(
)
v0 = v ( k g ) t = 0 m/s ( 0.49 ) 9.8 m/s 2 ( 1.6 s ) = +7.7 m/s
48. REASONING AND SOLUTION The deceleration produced by the frictional force is
a=
fk
m
=
k mg
m
= k g
The speed of the automobile after 1.30 s have elapsed is given by Equation 2.4 as
v = v0 + at = v0 + ( k g ) t = 16.1 m/s ( 0.720 ) ( 9.80 m/s 2 ) ( 1.30 s ) = 6.9 m/s
____________________________________________________________________________________________
49.
SSM REASONING Let us assume that the skater is moving horizontally along the + x
axis. The time t it takes for the skater to reduce her velocity to vx = +2.8 m/s from
v0x = +6.3 m/s can be obtained from one of the equations of kinematics:
vx = v0 x + a x t
(3.3a)
The initial and final velocities are known, but the acceleration is not. We can obtain the
acceleration from Newtons second law ( Fx = max , Equation 4.2a ) in the following
manner. The kinetic frictional force is the only horizontal force that acts on the skater, and,
since it is a resistive force, it acts opposite to the direction of the motion. Thus, the net force
in the x direction is Fx = f k , where fk is the magnitude of the kinetic frictional force.
Therefore, the acceleration of the skater is a x = Fx /m = f k / m .
The magnitude of the frictional force is f k = k FN (Equation 4.8), where k is the
coefficient of kinetic friction between the ice and the skate blades and FN is the magnitude
of the normal force. There are two vertical forces acting on the skater: the upward-acting
192 FORCES AND NEWTON'S LAWS OF MOTION
normal force FN and the downward pull of gravity (her weight) mg. Since the skater has no
vertical acceleration, Newton's second law in the vertical direction gives (taking upward as
the positive direction) Fy = FN mg = 0 . Therefore, the magnitude of the normal force is
FN = mg and the magnitude of the acceleration is
ax =
f k k FN k m g
=
=
= k g
m
m
m
SOLUTION
Solving the equation v x = v0 x + a x t for the time and substituting the expression above for
the acceleration yields
t=
vx v0 x
ax
=
vx v0 x
k g
=
2.8 m/s 6.3 m/s
= 4.4 s
( 0.081) ( 9.80 m/s 2 )
____________________________________________________________________________________________
50. REASONING We assume the car accelerates in the
+x direction. The air resistance force fA opposes the
cars motion (see the free-body diagram). The
frictional force is static, because the tires do not slip,
and points in the direction of the cars acceleration.
The reason for this is that without friction the cars
wheels would simply spin in place, and the cars
acceleration would be severely limited. The frictional
force has its maximum value fsMAX because we seek
the maximum acceleration before slipping occurs.
Applying Newtons second law (Fx = max, Equation
4.2a) to the horizontal motion gives
fsMAX
fsMAX f A = ma x
(1)
where ax is the maximum acceleration we seek. The air resistance force fA is given, and we
(
)
MAX
= FN . Because
will find the maximum static frictional force from Equation 4.7 fs
the cars acceleration has no vertical component, the net vertical force acting on the car
must be zero, so the upward normal force FN must balance the two downward forces, the
cars weight W and the downforce D:
FN = W + D = mg + D
(2)
Chapter 4 Problems
SOLUTION According to Equation 4.7
193
( fsMAX = s FN ) and Equation (2), the maximum
static frictional force the track can exert on the car is
fsMAX = FN = ( mg + D )
(3)
Now solving Equation (1) for the cars acceleration ax and then substituting Equation (3) for
the static frictional force fsMAX , we obtain
ax =
=
fsMAX f A
m
=
s ( mg + D ) f A
m
( 0.87 ) ( 690 kg ) ( 9.80 m/s 2 ) + 4060 N 1190 N
690 kg
= 12 m/s 2
51. REASONING The diagram shows the two applied forces that act on the crate. These two
forces, plus the kinetic frictional force fk constitute the net force that acts on the crate. Once
the net force has been determined, Newtons second law, F = ma (Equation 4.1) can be
used to find the acceleration of the crate.
SOLUTION The sum of the applied forces is F = F1 + F2.
The x-component of this sum is Fx = F1 cos 55.0 + F2 =
(88.0 N) cos 55.0 + 54.0 N = 104 N. The y-component of F
is Fy = F1 sin 55.0 = (88.0 N) sin 55.0 = 72.1 N. The
magnitude of F is
F = Fx2 + Fy2 =
( 104 N ) 2 + ( 72.1 N ) 2
= 127 N
55.0
F2
Since the crate starts from rest, it moves along the direction of F. The kinetic frictional force
fk opposes the motion, so it points opposite to F. The net force acting on the crate is the
sum of F and fk. The magnitude a of the crates acceleration is equal to the magnitude F of
the net force divided by the mass m of the crate
a=
F f k + F
=
m
m
(4.1)
According to Equation 4.8, the magnitude fk of the kinetic frictional force is given by
f k = k FN , where FN is the magnitude of the normal force. In this situation, FN is equal to
194 FORCES AND NEWTON'S LAWS OF MOTION
the magnitude of the crates weight, so FN = mg. Thus, the x-component of the acceleration
is
(
)
2
k mg + F ( 0.350 ) ( 25.0 kg ) 9.80 m/s + 127 N
a=
=
= 1.65 m/s 2
m
25.0 kg
The crate moves along the direction of F, whose x and y components have been determined
previously. Therefore, the acceleration is also along F. The angle that F makes with the xaxis can be found using the inverse tangent function:
Fy
= tan 1
F
x
F1 sin 55.0
1
= tan
F cos 55.0 + F
2
1
( 88.0 N ) sin 55.0
= tan 1
= 34.6 above the x axis
( 88.0 N ) cos 55.0 + 54.0 N
____________________________________________________________________________________________
52. REASONING AND SOLUTION
horizontal directions gives
Newtons second law applied in the vertical and
L cos 21.0 W = 0
L sin 21.0 R = 0
(1)
(2)
a. Equation (1) gives
21.0
W
53 800 N
L=
=
= 57 600 N
cos 21.0 cos 21.0
b. Equation (2) gives
L
R
R = L sin 21.0 = ( 57 600 N ) sin 21.0 = 20 600 N
W
____________________________________________________________________________________________
53.
SSM REASONING In order for the object to move with constant velocity, the net force
on the object must be zero. Therefore, the north/south component of the third force must be
equal in magnitude and opposite in direction to the 80.0 N force, while the east/west
component of the third force must be equal in magnitude and opposite in direction to the
60.0 N force. Therefore, the third force has components: 80.0 N due south and 60.0 N due
east. We can use the Pythagorean theorem and trigonometry to find the magnitude and
direction of this third force.
Chapter 4 Problems
SOLUTION The magnitude of the third force is
195
N
F3 = (80.0 N)2 + (60.0 N) 2 = 1.00 10 2 N
E
The direction of F3 is specified by the angle where
80.0 N
80.0 N
= tan 1
= 53.1, south of east
60.0 N
F3
60.0 N
____________________________________________________________________________________________
54. REASONING AND SOLUTION
a. In the horizontal direction the thrust F is balanced by the resistive force fr of the water.
That is,
Fx = 0
or
fr = F = 7.40 105 N
b. In the vertical direction, the weight, mg, is balanced by the buoyant force, Fb. So
Fy = 0
gives
Fb = mg = (1.70 108 kg)(9.80 m/s2) = 1.67 109 N
____________________________________________________________________________________________
55. REASONING The drawing shows the two
forces, T and T , that act on the tooth. To
obtain the net force, we will add the two
16.0
forces using the method of components (see
T
Section 1.8).
+y
+x
16.0
T
SOLUTION The table lists the two vectors and their x and y components:
Vector
x component
y component
T
+T cos 16.0
T sin 16.0
T
T cos 16.0
T sin 16.0
T + T
+T cos 16.0 T cos 16.0
T sin 16.0 T sin 16.0
196 FORCES AND NEWTON'S LAWS OF MOTION
Since we are given that T = T = 21.0 N, the sum of the x components of the forces is
Fx = +T cos 16.0 T cos 16.0 = +(21 N) cos 16.0 (21 N) cos 16.0 = 0 N
The sum of the y components is
Fy = T sin16.0 T sin 16.0 = (21 N) sin 16.0 (21 N) sin 16.0 = 11.6 N
The magnitude F of the net force exerted on the tooth is
F=
( Fx ) 2 + ( Fy ) 2 =
( 0 N ) 2 + ( 11.6 N ) 2 = 11.6 N
_____________________________________________________________________________
_
56. REASONING At first glance there seems to be very little information given. However, it
is enough. In part a of the drawing the bucket is hanging stationary and, therefore, is in
equilibrium. The forces acting on it are its weight and the two tension forces from the rope.
There are two tension forces from the rope, because the rope is attached to the bucket handle
at two places. These three forces must balance, which will allow us to determine the weight
of the bucket. In part b of the drawing, the bucket is again in equilibrium, since it is
traveling at a constant velocity and, therefore, has no acceleration. The forces acting on the
bucket now are its weight and a single tension force from the rope, and they again must
balance. In part b, there is only a single tension force, because the rope is attached to the
bucket handle only at one place. This will allow us to determine the tension in part b, since
the weight is known.
SOLUTION Let W be the weight of the bucket, and let T be the tension in the rope as the
bucket is being pulled up at a constant velocity. The free-body diagrams for the bucket in
parts a and b of the drawing are as follows:
Chapter 4 Problems
197
T
92.0 N
92.0 N
W
W
Free-body diagram for
part a
Free-body diagram for
part b
Since the bucket in part a is in equilibrium, the net force acting on it is zero. Taking upward
to be the positive direction, we have
F = 92.0 N + 92.0 N W = 0
or
W = 184 N
Similarly, in part b we have
F = T W = 0
57.
or
T = W = 184 N
SSM REASONING AND SOLUTION The free body diagram for the plane is shown
below to the left. The figure at the right shows the forces resolved into components parallel
to and perpendicular to the line of motion of the plane.
L
L
T
R
T
R
W
W sin
W cos
If the plane is to continue at constant velocity, the resultant force must still be zero after the
fuel is jettisoned. Therefore (using the directions of T and L to define the positive
directions),
T R W(sin ) = 0
(1)
L W (cos ) = 0
(2)
From Example 13, before the fuel is jettisoned, the weight of the plane is 86 500 N, the
thrust is 103 000 N, and the lift is 74 900 N. The force of air resistance is the same before
and after the fuel is jettisoned and is given in Example 13 as R = 59 800 N.
After the fuel is jettisoned, W = 86 500 N 2800 N = 83 700 N
198 FORCES AND NEWTON'S LAWS OF MOTION
From Equation (1) above, the thrust after the fuel is jettisoned is
T = R + W (sin ) = [(59 800 N) + (83 700 N)(sin 30.0)] = 101 600 N
From Equation (2), the lift after the fuel is jettisoned is
L = W (cos ) = (83 700 N)(cos 30.0) = 72 500 N
a. The pilot must, therefore, reduce the thrust by
103 000 N 101 600 N = 1400 N
b. The pilot must reduce the lift by
74 900 N 72 500 N = 2400 N
____________________________________________________________________________________________
58. REASONING The worker is standing still. Therefore,
he is in equilibrium, and the net force acting on him is
zero. The static frictional force fs that prevents him from
slipping points upward along the roof, an angle of
degrees above the horizontal; we choose this as the x
direction (see the free-body diagram). The normal force
FN is perpendicular to the roof and thus has no x
component. But the gravitational force mg of the earth
on the worker points straight down, and thus has a
component parallel to the roof. We will use this freebody diagram and find the workers mass m by applying
Newtons second law with the acceleration equal to zero.
+y
FN
fs
mg sin
+x
mg
Free-body diagram of the worker
SOLUTION The static frictional force fs points in the x direction, and the x component of
the workers weight mg points in the +x direction. Because there are no other forces with x
components, and the workers acceleration is zero, these two forces must balance each
other. The x component of the workers weight is mg sin , therefore fs = mg sin . Solving
this relation for the workers mass, we obtain
m=
fs
g sin
=
390 N
( 9.80 m/s2 ) ( sin 36o )
= 68 kg
59. REASONING The sum of the angles the right and left surfaces
make with the horizontal and the angle between the two surface
+y
F1
F2
45.0
45.0
+x
W
Free-body diagram of
the wine bottle
Chapter 4 Problems
199
must be 180.0. Therefore, the angle that the left surface makes with respect to the
horizontal is 180.0 90.0 45.0 = 45.0. In the free-body diagram of the wine bottle, the
x axis is the horizontal. The force each surface exerts on the bottle is perpendicular to the
surfaces, so both forces are directed 45.0 above the horizontal. Letting the surface on the
right be surface 1, and the surface on the left be surface 2, the forces F1 and F2 are as shown
in the free-body diagram. The third force acting on the bottle is W, its weight or the
gravitational force exerted on it by the earth. We will apply Newtons second law and
analyze the vertical forces in this free-body diagram to determine the magnitude F of the
forces F1 and F2, using the fact that the bottle is in equilibrium.
SOLUTION The vertical components of the forces exerted by the surfaces are
F1y = F1 sin 45.0 and F2y = F2 sin 45.0. But the forces F1 and F2 have the same magnitude
F, so the two vertical components become F1y = F2y = F sin 45.0. Because the bottle is in
equilibrium, the upward forces must balance the downward force:
F1 y + F2 y = W
or
F sin 45.0o + F sin 45.0o = W
or
2 F sin 45.0o = W
Therefore,
F=
W
2sin 45.0o
=
mg
2sin 45.0o
=
( 1.40 kg ) ( 9.80 m/s 2 )
2 sin 45.0o
60. REASONING The free-body diagram in the drawing at the right
shows the forces that act on the clown (weight = W). In this
drawing, note that P denotes the pulling force. Since the rope
passes around three pulleys, forces of magnitude P are applied
both to the clowns hands and his feet. The normal force due to
the floor is FN , and the maximum static frictional force is fsMAX.
At the instant just before the clowns feet move, the net vertical
and net horizontal forces are zero, according to Newtons second
law, since there is no acceleration at this instant.
= 9.70 N
FN
P
fsMAX
P
W
SOLUTION According to Newtons second law, with upward and to the right chosen as
the positive directions, we have
F + P W = 0
1N4 4 2 4 43
Vertical forces
and
f MAX P = 0
1s 4 2 4 3
Horizontal forces
200 FORCES AND NEWTON'S LAWS OF MOTION
From the horizontal-force equation we find P = fsMAX. But fsMAX = sFN . From the
vertical-force equation, the normal force is FN = W P . With these substitutions, it follows
that
P = f sMAX = s FN = s ( W P )
Solving for P gives
P=
sW ( 0.53) ( 890 N )
=
= 310 N
1 + s
1 + 0.53
____________________________________________________________________________________________
61. REASONING Since the boxes are at rest, they are in equilibrium. According to Equation
4.9b, the net force in the vertical, or y, direction is zero, Fy = 0. There are two unknowns
in this problem, the normal force that the exerts table on box 1 and the tension in the rope
that connects boxes 2 and 3. To determine these unknowns we will apply the relation
Fy = 0 twice, once to the boxes on the left of the pulley and once to the box on the right.
SOLUTION There are four forces acting on the two boxes on
the left. The boxes are in equilibrium, so that the net force must
be zero. Choosing the +y direction as being the upward direction,
we have that
W W2 + FN + T = 0
1 414 4 2 4 4 43
(1)
Fy
where W1 and W2 are the magnitudes of the weights of the boxes,
FN is the magnitude of the normal force that the table exerts on
box 1, and T is the magnitude of the tension in the rope. We
know the weights. To find the unknown tension, note that the box
3 is also in equilibrium, so that the net force acting on it must be
zero.
W3 + T = 0
1 4 2 43
Fy
so that
T
3
T
FN
W3
2
1
T = W3
Substituting this expression for T into Equation (1) and solving
for the normal force gives
W2
W1
FN = W1 + W2 W3 = 55 N + 35 N 28 N = 62 N
____________________________________________________________________________________________
62. REASONING Since the mountain climber is at rest, she is in equilibrium and the net force
acting on her must be zero. Three forces comprise the net force, her weight, and the tension
forces from the left and right sides of the rope. We will resolve the forces into components
and set the sum of the x components and the sum of the y components separately equal to
Chapter 4 Problems
201
zero. In so doing we will obtain two equations containing the unknown quantities, the
tension TL in the left side of the rope and the tension TR in the right side. These two
equations will be solved simultaneously to give values for the two unknowns.
SOLUTION Using W to denote the weight of the mountain climber and choosing right and
upward to be the positive directions, we have the following free-body diagram for the
climber:
For the x components of the forces we have
Fx = TR sin 80.0 TL sin 65.0 = 0
+y
TL
65.0 80.0
For the y components of the forces we have
Fy = TR cos80.0 + TL cos 65.0 W = 0
TR
+x
W
Solving the first of these equations for TR, we find that
TR = TL
sin 65.0
sin 80.0
Substituting this result into the second equation gives
TL
sin 65.0
cos80.0 + TL cos 65.0 W = 0
sin 80.0
or
TL = 1.717 W
Using this result in the expression for TR reveals that
TR = TL
sin 65.0
sin 65.0
= ( 1.717W )
= 1.580 W
sin 80.0
sin 80.0
Since the weight of the climber is W = 535 N, we find that
TL = 1.717 W = 1.717 ( 535 N ) = 919 N
TR = 1.580 W = 1.580 ( 535 N ) = 845 N
63.
SSM REASONING There are four forces that act on the chandelier; they are the forces
of tension T in each of the three wires, and the downward force of gravity mg. Under the
influence of these forces, the chandelier is at rest and, therefore, in equilibrium.
Consequently, the sum of the x components as well as the sum of the y components of the
forces must each be zero. The figure below shows a quasi-free-body diagram for the
202 FORCES AND NEWTON'S LAWS OF MOTION
chandelier and the force components for a suitable system of x, y axes. Note that the
diagram only shows one of the forces of tension; the second and third tension forces are not
shown in the interest of clarity. The triangle at the right shows the geometry of one of the
cords, where l is the length of the cord, and d is the distance from the ceiling.
y
T
d
x
l
mg
We can use the forces in the y direction to find the magnitude T of the tension in any one
wire.
SOLUTION
that
Remembering that there are three tension forces, we see from the diagram
3T sin = mg
or
T=
mg
mg
mg l
=
=
3 sin 3(d / l) 3d
Therefore, the magnitude of the tension in any one of the cords is
T=
(44 kg)(9.80 m/s 2 )(2.0 m)
= 1.9 102 N
3(1.5 m)
____________________________________________________________________________________________
64. REASONING The diagram at the right shows the force F that
F
the ground exerts on the end of a crutch. This force, as
mentioned in the statement of the problem, acts along the crutch
and, therefore, makes an angle with respect to the vertical.
The horizontal and vertical components of this force are also
shown. The horizontal component, F sin , is the static F sin
frictional force that prevents the crutch from slipping on the
floor, so fs = F sin . The largest value that the static frictional
F cos
force can have before the crutch begins to slip is then given by fsMAX = F sin MAX . We
also know from Section 4.9 (see Equation 4.7) that the maximum static frictional force is
related to the magnitude FN of the normal force by fsMAX = s FN , where s is the
coefficient of static friction. These two relations will allow us to find MAX .
SOLUTION The magnitude of the maximum static frictional force is given by
fsMAX = s FN . But, as mentioned in the REASONING section, fsMAX is also the
Chapter 4 Problems
203
horizontal component of the force F, so fsMAX = F sin MAX . The vertical component of F
is F cos MAX and is the magnitude FN of the normal force that the ground exerts on the
crutch. Thus, we have
f MAX = s FN
{
1s 2 4
43
F cos MAX
F sin MAX
The force F can be algebraically eliminated from this equation, leaving
sin MAX
= s
cos MAX
or
tan MAX = s
The maximum angle that a crutch can have is
MAX = tan 1 ( s ) = tan 1 ( 0.90 ) = 42
_____________________________________________________________________________
_
65.
REASONING The toboggan has a constant velocity, so it has no acceleration and is in
equilibrium. Therefore, the forces acting on the toboggan must balance, that is, the net force
acting on the toboggan must be zero. There are three forces present, the kinetic frictional
force, the normal force from the inclined surface, and the weight mg of the toboggan. Using
Newtons second law with the acceleration equal to zero, we will obtain the kinetic friction
coefficient.
SOLUTION
In drawing the free-body
diagram for the toboggan we choose the + x
axis to be parallel to the hill surface and
downward, the +y direction being
perpendicular to the hill surface. We also
use fk to symbolize the frictional force.
Since the toboggan is in equilibrium, the zero
net force components in the x and y
directions are
+y
FN
fk
8.00
+x
mg
8.00
Fx = mg sin 8.00 k FN = 0
Fy = FN mg cos8.00 = 0
In the first of these expressions we have used Equation 4.8 for fk to express the kinetic
frictional force. Solving the second equation for the normal force FN and substituting into
the first equation gives
204 FORCES AND NEWTON'S LAWS OF MOTION
mg sin 8.00 k mg cos8.00 = 0
1 4 2 44
4
3
F
or
k =
sin 8.00
= tan 8.00 = 0.141
cos8.00
N
66. REASONING
The block is in equilibrium in each case. Since the block moves at a constant velocity in
each case, it is not accelerating. A zero acceleration is the hallmark of equilibrium. At
equilibrium, the net force is zero (i.e., the forces balance to zero), and we will obtain the
magnitude of the pushing force by utilizing this fact as it pertains to the vertical or y
(
)
direction. We will use Equation 4.9b Fy = 0 for this purpose.
+y
+y
It is important to note, however, that the
direction of the kinetic frictional force is
+x
+x
not the same in each case.
The
frictional force always opposes the
fk
relative motion between the surface of
fk
the block and the wall. Therefore, when
P
P
the block slides upward, the frictional
W
W
force points downward. When the block
Free-body diagram for
Free-body diagram for
slides downward, the frictional force upward motion of the
downward motion of
points upward. These directions are
block
the block
shown in the free-body diagrams (not to
scale) for the two cases. In these drawings W is the weight of the block and fk is the kinetic
frictional force.
In each case the magnitude of the frictional force is the same. It is given by Equation 4.8 as
fk = kFN, where k is the coefficient of kinetic friction and FN is the magnitude of the
normal force. The coefficient of kinetic friction does not depend on the direction of the
motion. Furthermore, the magnitude of the normal force in each case is the component of
the pushing force that is perpendicular to the wall, or FN = P sin .
SOLUTION Using Equations 4.9b to describe the balance of forces that act on the block in
the y direction and referring to the free-body diagrams, we have
Upward motion
Fy = P cos W f k = 0
Downward motion
Fy = P cos W + f k = 0
According to Equation 4.8, the magnitude of the kinetic frictional force is fk = kFN, where
we have pointed out in the REASONING that the magnitude of the normal force is
FN = P sin . Substituting into the equations for Fy in the two cases, we obtain
Chapter 4 Problems
Upward motion
Fy = P cos W k P sin = 0
Downward motion
205
Fy = P cos W + k P sin = 0
Solving each case for P, we find that
a. Upward motion
P=
W
39.0 N
=
= 52.6 N
cos k sin cos 30.0 ( 0.250 ) sin 30.0
b. Downward motion
P=
67.
W
39.0 N
=
= 39.4 N
cos + k sin cos 30.0 + ( 0.250 ) sin 30.0
SSM REASONING When the bicycle is coasting straight down the hill, the forces that
act on it are the normal force FN exerted by the surface of the hill, the force of gravity mg,
and the force of air resistance R. When the bicycle climbs the hill, there is one additional
force; it is the applied force that is required for the bicyclist to climb the hill at constant
speed. We can use our knowledge of the motion of the bicycle down the hill to find R.
Once R is known, we can analyze the motion of the bicycle as it climbs the hill.
SOLUTION The figure to the left below shows the free-body diagram for the forces
during the downhill motion. The hill is inclined at an angle above the horizontal. The
figure to the right shows these forces resolved into components parallel to and perpendicular
to the line of motion.
+y
+x
FN
FN
R
R
mg sin
mg
mg cos
Since the bicyclist is traveling at a constant velocity, his acceleration is zero. Therefore,
according to Newton's second law, we have Fx = 0 and Fy = 0 . Taking the direction up
the hill as positive, we have Fx = R mg sin = 0 , or
R = mg sin = (80.0 kg)(9.80 m / s 2 ) sin 15.0 = 203 N
206 FORCES AND NEWTON'S LAWS OF MOTION
When the bicyclist climbs the same hill at
constant speed, an applied force P must push the
system up the hill. Since the speed is the same,
the magnitude of the force of air resistance will
remain 203 N. However, the air resistance will
oppose the motion by pointing down the hill.
The figure at the right shows the resolved forces
that act on the system during the uphill motion.
F
N
P
mg sin
mg cos
R
Using the same sign convention as above, we have Fx = P mg sin R = 0 , or
P = R + mg sin = 203 N + 203 N =
406 N
____________________________________________________________________________________________
68. REASONING Because the kite line is straight, the distance between the kite and the person
holding the line is L = 43 m, as shown in the right part of the drawing. In order to find the
height h of the kite, we need to know the angle that the kite line makes with the horizontal
(see the drawing). Once that is known, h = L sin will give the kites height relative to the
person. The key to finding the angle is the realization that the tension force T exerted on
the kite by the line is parallel to the line itself. Therefore, the tension T is directed at an
angle below the horizontal (see the free-body diagram below). To find the angle , it is
sufficient, then, to find one of the components of the tension force, either Tx or Ty, because
the magnitude T of the tension is known. With values for Tx or Ty and T, we can use either
the sine or cosine function to determine .
Both the winds force f and the tension T have horizontal and vertical components, while
the weight force W is purely vertical. Therefore, there are only two horizontal forces acting
on the kite (Tx, fx), but three vertical forces (Ty, fy, W), so it will be easier to calculate Tx, the
horizontal component of the tension. Because the kite is stationary, the horizontal
component of the tension must balance the horizontal component of the force exerted on the
kite by the wind:
Tx = f x
(1)
Chapter 4 Problems
207
f +y
fy
L
Tx
56
fx
+x
W
T
h
Ty
Free-body diagram
of the kite
Height of the kite
SOLUTION From the free-body diagram of the kite, we can see that the x components of
the tension and air resistance forces are Tx = T cos and fx = f cos 56. Substituting these
expressions into Equation (1), we find that
Tx = f x
or
T cos = f cos 56o
or
cos =
f cos 56o
T
o
f cos 56o
1 ( 19 N ) cos 56
o
= cos 1
= 48
= cos
T
16 N
Therefore, the kites height relative to the person is
h = L sin = ( 43 m ) sin 48o = 32 m
69. REASONING The weight of the part of the washcloth off the table is moff g. At the instant
just before the washcloth begins to slide, this weight is supported by a force that has
magnitude equal to fsMAX, which is the static frictional force that the table surface applies to
the part of the washcloth on the table. This force is transmitted around the bend in the
washcloth hanging over the edge by the tension forces between the molecules of the
washcloth, in much the same way that a force applied to one end of a rope is transmitted
along the rope as it passes around a pulley.
SOLUTION Since the static frictional supports the weight of the washcloth off the table,
we have fsMAX = moff g. The static frictional force is fsMAX = sFN . The normal force FN is
applied by the table to the part of the washcloth on the table and has a magnitude equal to
the weight of that part of the washcloth. This is so, because the table is assumed to be
208 FORCES AND NEWTON'S LAWS OF MOTION
horizontal and the part of the washcloth on it does not accelerate in the vertical direction.
Thus, we have
f sMAX = s FN = s mon g = moff g
The magnitude g of the acceleration due to gravity can be eliminated algebraically from this
result, giving smon = moff . Dividing both sides by mon + moff gives
mon
s
mon + moff
moff
=
mon + moff
or
s f on = f off
where we have used fon and foff to denote the fractions of the washcloth on and off the table,
respectively. Since fon + foff = 1, we can write the above equation on the left as
(
)
s 1 f off = f off
or
f off =
s
0.40
=
= 0.29
1 + s 1 + 0.40
____________________________________________________________________________________________
70. REASONING In addition to the upward buoyant force B and the
downward resistive force R, a downward gravitational force mg acts
on the submarine (see the free-body diagram), where m denotes the
mass of the submarine. Because the submarine is not in contact with
any rigid surface, no normal force is exerted on it. We will calculate
the acceleration of the submarine from Newtons second law, using the
free-body diagram as a guide.
+y
B
R
SOLUTION Choosing up as the positive direction, we sum the forces
mg
acting on the submarine to find the net force F. According to
Free-body diagram
Newtons second law, the acceleration a is
of the submarine
(
)
2
F B R mg 16 140 N 1030 N ( 1450 kg ) 9.80 m/s
a=
=
=
= +0.62 m/s 2
m
m
1450 kg
where the positive value indicates that the direction is upward.
71. REASONING According to Newtons second law, the acceleration has the same direction
as the net force and a magnitude given by a = F/m.
SOLUTION Since the two forces are perpendicular, the magnitude of the net force is given
by the Pythagorean theorem as F =
( 40.0 N ) 2 + ( 60.0 N ) 2 . Thus, according to Newtons
second law, the magnitude of the acceleration is
Chapter 4 Problems
F
a=
=
m
( 40.0 N ) 2 + ( 60.0 N ) 2
4.00 kg
209
= 18.0 m/s 2
The direction of the acceleration vector is given by
60.0 N
= tan 1
= 56.3 above the +x axis
40.0 N
_____________________________________________________________________________
_
72. REASONING
a. Since the fish is being pulled up at a constant speed, it has no acceleration. According to
Newtons second law, the net force acting on the fish must be zero. We will use this fact to
determine the weight of the heaviest fish that can be pulled up.
b. When the fish has an upward acceleration, we can still use Newtons second law to find
the weight of the heaviest fish. However, because the fish has an acceleration, we will see
that the maximum weight is less than that in part (a).
SOLUTION
a. There are two forces acting on the fish (taking the upward vertical direction to be the + y
direction): the maximum force of +45 N due to the line, and the weight W of the fish
(negative, because the weight points down). Newtons second law ( Fy = 0, Equation 4.9b )
gives
Fy = +45 N W = 0
b. Since
the
fish
has
( Fy = ma y , Equation 4.2b )
an
upward
or
W = 45 N
acceleration
ay,
Newtons
second
law
becomes
Fy = +45 N W = ma y
Where W is the weight of the heaviest fish that can be pulled up with an acceleration.
Solving this equation for W gives
W = +45 N ma y
The mass m of the fish is the magnitude W of its weight divided by the magnitude g of the
acceleration due to gravity (see Equation 4.5), or m = W /g. Substituting this relation for m
into the previous equation gives
210 FORCES AND NEWTON'S LAWS OF MOTION
W
W = +45 N
a y
g
Solving this equation for W yields
45 N
45 N
=
= 37 N
ay
2.0 m/s 2
1+
1+
g
9.80 m/s 2
______________________________________________________________________________
W =
73.
SSM REASONING If we assume that the acceleration is constant, we can use Equation
2.4 ( v = v0 + at ) to find the acceleration of the car. Once the acceleration is known,
Newton's second law ( F = ma ) can be used to find the magnitude and direction of the net
force that produces the deceleration of the car.
SOLUTION The average acceleration of the car is, according to Equation 2.4,
a=
v v0
t
=
17.0 m/s 27.0 m/s
= 1.25 m/s 2
8.00 s
where the minus sign indicates that the direction of the acceleration is opposite to the
direction of motion; therefore, the acceleration points due west.
According to Newton's Second law, the net force on the car is
F = ma = (1380 kg)(1.25 m/s 2 ) = 1730 N
The magnitude of the net force is 1730 N . From Newton's second law, we know that the
direction of the force is the same as the direction of the acceleration, so the force also points
due west .
____________________________________________________________________________________________
74. REASONING In the absence of air resistance, the two forces
acting on the sensor are its weight W and the tension T in the
towing cable (see the free-body diagram). We see that Tx is the
only horizontal force acting on the sensor, and therefore
Newtons second law Fx = max (Equation 4.2a) gives Tx = max.
Because the vertical component of the sensors acceleration is
zero, the vertical component of the cables tension T must
balance the sensors weight: Ty = W = mg. We thus have
sufficient information to calculate the horizontal and vertical
+y
T
Ty
+x
T x = m ax
W
Free-body diagram
of the sensor
Chapter 4 Problems
211
components of the tension force T, and therefore to calculate its magnitude T from the
2
2
2
Pythagorean theorem: T = Tx + Ty .
SOLUTION Given that Tx = max and that Ty = mg, the Pythagorean theorem yields the
magnitude T of the tension in the cable:
T = Tx2 + Ty2 =
= (129 kg)
75.
( ma ) 2 + ( mg ) 2
= m2 a 2 + m2 g 2 = m a 2 + g 2
( 2.84 m/s2 ) + ( 9.80 m/s2 )
2
2
= 1320 N
SSM REASONING AND SOLUTION
a. Each cart has the same mass and acceleration; therefore, the net force acting on any one
of the carts is, according to Newton's second law
F = ma = (26 kg)(0.050 m/s 2 ) = 1.3 N
b. The fifth cart must essentially push the sixth, seventh, eight, ninth and tenth cart. In
other words, it must exert on the sixth cart a total force of
F = ma = 5(26 kg)(0.050 m/s 2 )= 6.5 N
____________________________________________________________________________________________
76. REASONING AND SOLUTION
Newton's second law applied to
Object 1
object 1 (422 N) gives
Object 2
N1
T = m1a1
T
T
Similarly, for object 2 (185 N)
W1
T m2g = m2a2
W
2
If the string is not to break or go
slack, both objects must have
accelerations of the same magnitude.
Then a1 = a and a2 = a. The above equations become
T = m1a
T m2g = m2a
(1)
(2)
212 FORCES AND NEWTON'S LAWS OF MOTION
a. Substituting Equation (1) into Equation (2) and solving for a yields
a=
m2 g
m1 + m2
The masses of objects 1 and 2 are
m1 = W1 / g = ( 422 N ) / ( 9.80 m/s 2 ) = 43.1 kg
m2 = W2 / g = ( 185 N ) / ( 9.80 m/s 2 ) = 18.9 kg
The acceleration is
a=
m2 g
( 18.9 kg ) ( 9.80 m/s 2 )
=
= 2.99 m/s 2
m1 + m2
43.1 kg + 18.9 kg
b. Using this value in Equation (1) gives
T = m1a = ( 43.1 kg ) ( 2.99 m/s 2 ) = 129 N
____________________________________________________________________________________________
77.
F
N
SSM WWW REASONING The speed of the skateboarder
at the bottom of the ramp can be found by solving Equation 2.9
2
( v 2 = v0 + 2ax , where x is the distance that the skater moves
mgsin
mgcos
down the ramp) for v. The figure at the right shows the freebody diagram for the skateboarder. The net force F, which
accelerates the skateboarder down the ramp, is the component of
the weight that is parallel to the incline: F = mg sin . Therefore, we know from
Newton's second law that the acceleration of the skateboarder down the ramp is
a=
F mg sin
=
= g sin
m
m
SOLUTION Thus, the speed of the skateboarder at the bottom of the ramp is
2
2
v = v0 + 2ax = v0 + 2 gx sin = (2.6 m/s) 2 + 2(9.80 m/s 2 )(6.0 m) sin18 = 6.6 m/s
_____________________________________________________________________________
_
Chapter 4 Problems
213
78. REASONING AND SOLUTION From Newton's second law and the equation: v = v0 + at,
we have
v v0
F = ma = m
t
a. When the skier accelerates from rest (v0 = 0 m/s) to a speed of 11 m/s in 8.0 s, the
required net force is
F =m
v v0
t
= (73 kg)
(11 m/s) 0 m/s
= 1.0 102 N
8.0 s
b. When the skier lets go of the tow rope and glides to a halt ( v = 0 m/s) in 21 s, the net
force acting on the skier is
F =m
v v0
t
= (73 kg)
0 m/s (11 m/s)
= 38 N
21 s
The magnitude of the net force is 38 N .
____________________________________________________________________________________________
79. REASONING The only horizontal force acting on the boat and trailer is the tension in the
hitch; therefore, it is the net force. According to Newtons second law, the tension (or the net
force) equals the mass times the acceleration. The mass is known, and the acceleration can
be found by applying an appropriate equation of kinematics from Chapter 3.
SOLUTION Assume that the boat and trailer are moving in the + x direction. Newtons
second law is Fx = ma x (see Equation 4.2a), where the net force is just the tension + T in
the hitch, so Fx = T . Thus,
T = ma x
(1)
Since the initial and final velocities, v0x and vx, and the time t are known, we may use
Equation 3.3a from the equations of kinematics to relate these variables to the acceleration:
vx = v0 x + ax t
(3.3a)
Solving Equation (3.3a) for ax and substituting the result into Equation (1), we find that
v v
T = ma x = m x 0 x
t
11 m/s 0 m/s
= ( 410 kg )
= 160 N
28 s
____________________________________________________________________________________________
214 FORCES AND NEWTON'S LAWS OF MOTION
80. REASONING Since we assume that there is no
+y
frictional force resisting the airplanes motion, the only
horizontal force acting on the airplane arises because of
the tension (magnitude = T) in the cable. From Newtons
FN
second law F = Ma (Equation 4.1), we conclude that the
airplanes acceleration is given by a = F/M = T/M,
fsMAX T
where M is the mass of the airplane. The harder the man
+x
pulls on the cable, the greater the tension T, and the
greater the airplanes acceleration. According to
Newtons third law, however, the cable also exerts an
mg
opposing horizontal force of magnitude T on the man.
Thus, if he is to keep his footing, T cannot exceed the
Free-body diagram
maximum force of static friction fsMAX the runway
of the man
exerts on him. Therefore, the airplanes acceleration is
greatest when these two forces have equal magnitudes:
T = fsMAX (see the free-body diagram of the man). The maximum static frictional force the
runway can exert is determined by the relation fsMAX = s FN (Equation 4.7). Because the
man has no acceleration in the vertical direction, the normal force must balance the
downward pull of gravity: FN = mg.
SOLUTION
Combining FN = mg and fsMAX = s FN (Equation 4.7), we obtain the
maximum tension in the cable:
T = fsMAX = s FN = s mg
(1)
We can now substitute Equation (1) for the tension into Newtons second law ( a = T/M),
and calculate the maximum possible acceleration of the airplane:
(
)
2
T s mg ( 0.77 ) ( 85 kg ) 9.80 m/s
a=
=
=
= 5.9 103 m/s 2
M
M
109 000 kg
81. REASONING AND SOLUTION If the +x axis is taken to be parallel to and up the ramp,
then Fx = max gives
T fk mg sin 30.0 = max
where fk = kFN . Hence,
T = max + kFN + mg sin 30.0
Also, Fy = may gives
FN mg cos 3 0.0 = 0
(1)
Chapter 4 Problems
215
since no acceleration occurs in this direction. Then
FN = mg cos 3 0.0
(2)
Substitution of Equation (2) into Equation (1) yields
T = max + kmg cos 30.0 + mg sin 30.0
T = (205 kg)(0.800 m/s2) + (0.900)(205 kg)(9.80 m/s2)cos 30.0
+ (205 kg)(9.80 m/s2)sin 30.0 = 2730 N
____________________________________________________________________________________________
82. REASONING To determine the mans upward acceleration by
+
means of Newtons second law, we first need to identify all of the
forces exerted on him and then construct a free-body diagram. The
earth pulls down on the man with a gravitational force W = mg. Once
T
T
he begins accelerating upward, he is no longer in contact with the
ground, so there is no normal force acting on him. The pulling force P
that he exerts on the rope does not appear in his free-body diagram
because it is not a force exerted on him. Each end of the rope exerts a
W
tension force on him. If we assume that the rope is massless, and
ignore friction between the rope and the branch, then the magnitude
of the tension T is the same everywhere in the rope. Because the man Free-body diagram
of the man
pulls down on the free end intentionally and on the other end
inadvertently (because it is tied around his waist), Newtons third law
predicts that both ends of the rope pull upward on him. The third law predicts that the free
end of the rope pulls up on the man with a force exactly equal in magnitude to that of the
358-N pulling force. Thus, in addition to the downward gravitational force, there are two
upward tension forces with magnitudes T = 358 N acting on the man, as illustrated in the
free-body diagram.
SOLUTION Taking up as the positive direction and applying Newtons second law to the
mans free-body diagram yields
F = 2T W = ma
Solving for the acceleration a, we find
a=
2 ( 358 N )
2T W 2T mg 2T
=
=
g =
9.80 m/s 2 = 0.14 m/s 2
m
m
m
72.0 kg
216 FORCES AND NEWTON'S LAWS OF MOTION
83.
SSM REASONING The free-body diagrams for
Robin (mass = m) and for the chandelier (mass = M) are
given at the right. The tension T in the rope applies an Robin
upward force to both. Robin accelerates upward, while
the chandelier accelerates downward, each acceleration
having the same magnitude. Our solution is based on
separate applications of Newtons second law to Robin
and the chandelier.
T
T
Chandelier
mg
Mg
SOLUTION Applying Newtons second law, we find
T mg = ma
1 4 2 44
4
3
T Mg = Ma
1 44 2 4 43
and
Robin Hood
Chandelier
In these applications we have taken upward as the positive direction, so that Robins
acceleration is a, while the chandeliers acceleration is a. Solving the Robin-Hood
equation for T gives
T = mg + ma
Substituting this expression for T into the Chandelier equation gives
mg + ma Mg = Ma
M m
a =
g
M +m
or
a. Robins acceleration is
( 195 kg ) ( 77.0 kg )
M m
2
2
a=
9.80 m/s = 4.25 m/s
g =
M +m
( 195 kg ) + ( 77.0 kg )
(
)
b. Substituting the value of a into the expression for T gives
(
)
T = mg + ma = ( 77.0 kg ) 9.80 m/s 2 + 4.25 m/s 2 = 1080 N
____________________________________________________________________________________________
84. REASONING Newtons second law, Equation 4.2a, can be used to find the tension in the
coupling between the cars, since the mass and acceleration are known. The tension in the
coupling between the 30th and 31st cars is responsible for providing the acceleration for the
20 cars from the 31st to the 50th car. The tension in the coupling between the 49th and 50th
cars is responsible only for pulling one car, the 50th.
SOLUTION
a. The tension T between the 30th and 31st cars is
Chapter 4 Problems
Tx = (Mass of 20 cars) ax
217
(4.2a)
(
)(
)
= ( 20 cars ) 6.8 103 kg/car 8.0 102 m / s 2 = 1.1 10 4 N
b. The tension T between the 49th and 50th cars is
Tx = (Mass of 1 car)ax
(
(4.2a)
)(
)
= ( 1 car ) 6.8 103 kg/car 8.0 10 2 m / s 2 = 5.4 102 N
____________________________________________________________________________________________
85. REASONING The box comes to a halt because the kinetic
F
N
frictional force and the component of its weight parallel to the
incline oppose the motion and cause the box to slow down. The
distance that the box travels up the incline can be can be found
mg sin
2
by solving Equation 2.9 ( v 2 = v0 + 2ax ) for x. Before we use
this approach, however, we must first determine the acceleration
fk
of the box as it travels along the incline.
mg cos
SOLUTION The figure above shows the free-body diagram for the box. It shows the
resolved components of the forces that act on the box. If we take the direction up the incline
as the positive x direction, then, Newton's second law gives
Fx = mg sin f k = max
mg sin k FN = ma x
or
where we have used Equation 4.8, f k = k FN . In the y direction we have
Fy = FN mg cos = 0
or
FN = mg cos
since there is no acceleration in the y direction. Therefore, the equation for the motion in
the x direction becomes
mg sin k mg cos = ma x
or
a x = g (sin + k cos )
According to Equation 2.9, with this value for the acceleration and the fact that v = 0 m/s,
the distance that the box slides up the incline is
x=
2
v0
2a
=
2
v0
2 g (sin + k cos )
=
(1.50 m/s) 2
= 0.265 m
2(9.80 m/s 2 )[sin 15.0 + (0.180)cos 15.0]
____________________________________________________________________________________________
218 FORCES AND NEWTON'S LAWS OF MOTION
86. REASONING Since we assume that there is no frictional
force resisting the airplanes motion, the only horizontal
force acting on the airplane arises because of the tension
(magnitude = T) in the cable. The airplane (mass = M)
undergoes a horizontal acceleration caused by the
horizontal component Tx = T cos of the tension force,
where is the angle that the cable makes with the
horizontal. From Newtons second law, the acceleration of
the airplane is
ax =
Fx
M
=
T cos
M
+y
FN
fsMAX
T
Tx
+x
Ty
mg
(1)
Free-body diagram
of the man
The maximum tension in the cable is limited by the
condition that the mans feet must not slip. When the man
pulls as hard as possible without slipping, the horizontal component of the tension acting on
him matches the maximum static frictional force: Tx = T cos = fsMAX (see the free-body
diagram of the man). The maximum static frictional force itself is given by fsMAX = s FN
(Equation 4.7). Together, these two relations yield
T cos = s FN
(2)
To evaluate the magnitude FN of the normal force that acts on the man, we must consider
Newtons third law. This law indicates that when the man (mass = m) pulls up on the cable,
the cable pulls down on him (see the free-body diagram of the man). This additional
downward force increases the upward normal force FN the runway exerts on him. Applying
Newtons second law to the vertical direction in this diagram, with zero acceleration, we see
that Fy = FN Ty mg = FN T sin mg = 0 . Solving for FN yields
FN = T sin + mg
(3)
Substituting Equation (3) into Equation (2) yields an expression in which the tension T is the
only unknown quantity:
T cos = s ( T sin + mg )
(4)
We now solve Equation (4) for the tension in the cable:
T cos = sT sin + s mg
or
T cos sT sin = s mg
T=
s mg
cos s sin
or
T ( cos s sin ) = s mg
(5)
Chapter 4 Problems
219
Equation (5) may be substituted into Equation (1) for the airplanes acceleration:
ax =
s mg cos
T cos
=
M
M ( cos s sin )
(6)
SOLUTION We apply Equation (6) to calculate the acceleration of the airplane:
(
)
( 0.77 ) ( 85 kg ) 9.80 m/s 2 cos 9.0o
s mg cos
ax =
=
= 6.7 103 m/s 2
o
o
M ( cos s sin ) ( 109 000 kg ) cos 9.0 0.77 sin 9.0
(
)
87. REASONING As the free-body diagram shows, there are two forces
acting on the fireman as he slides down the pole: his weight W and the
kinetic frictional force fk. The kinetic frictional force opposes the
motion of the fireman, so it points upward in the +y direction. In accord
with Newtons second law, the net force, which is the sum of these two
forces, is equal to the firemans mass times his acceleration. His mass,
and therefore his weight, is known, but his acceleration is not. We will
turn to one of the equations of kinematics from Chapter 3 to determine
the acceleration.
SOLUTION Newtons second law
( Fy = ma y , Equation 4.2b )
can
+y
fk
W
Free-body diagram for
the fireman
be applied to this situation:
Fy = + f k W = ma y
The magnitude W of the firemans weight can be expressed in terms of his mass as W = mg
(Equation 4.5), where g is magnitude of the acceleration due to gravity. Solving the equation
above for the magnitude of the kinetic frictional force, and using W = mg, gives
f k = ma y + W = ma y + mg
(1)
Since the initial and final velocities, v0y and vy, and the displacement y are known, we will
use Equation 3.6b from the equations of kinematics to relate these variables to the
2
2
acceleration: v y = v0 y + 2a y y . Solving this equation for ay and substituting the result into
Equation 1 gives
2
v 2 v0 y
y
fk = m
2y
+ mg
220 FORCES AND NEWTON'S LAWS OF MOTION
We note that the fireman slides down the pole, so his displacement is negative, or
y = 4.0 m. The magnitude of the kinetic frictional force is, then,
2
v 2 v0 y
y
fk = m
2y
+ mg
( 1.4 m/s ) 2 ( 0 m/s ) 2
= ( 86 kg )
+ ( 86 kg ) ( 9.80 m/s 2 ) = 820 N
( 4.0 m )
2
_____________________________________________________________________________
_
88. REASONING
Consider the forces that act on each block. Only one
+
force contributes to the horizontal net force acting on
block 1, as shown in the free-body diagram. This is the
P
force P with which block 2 pushes on block 1. The
minus sign in the free-body diagram indicates the
Free-body diagram for
direction of the force is to the left. This force is part of
block 1
the action-reaction pair of forces that is consistent with
Newtons third law. Block 1 pushes forward and to the
right against block 2, and block 2 pushes backward and to the left against block 1 with an
oppositely directed force of equal magnitude.
Two forces contribute to the horizontal net force acting
+
on block 2, as shown in the free-body diagram. One is
the force P with which block 1 pushes on block 2.
P
fk
According to Newtons third law, this force has the
same magnitude but the opposite direction as the force
Free-body diagram for
with which block 2 pushes on block 1. The other force
block 2
is the kinetic frictional force fk, which points to the left,
in opposition to the relative motion between the block and the surface on which it slides.
Both blocks decelerate, the magnitude of the deceleration being the same for each block.
They have the same deceleration, because they are pressed together. Since the blocks are
moving to the right in the drawing, the acceleration vector points to the left, for it reflects
the slowing down of the motion. In Case A and in Case B we will apply Newtons second
law separately to each block in order to relate the net force to the acceleration.
SOLUTION Referring to the free-body diagram for block 1, we write Newtons second law
as follows:
P = m1 ( a )
{
(1)
Net force
on block 1
where a is the magnitude of the acceleration. The minus sign appears on the right side of
this equation because the acceleration, being a deceleration, points to the left, in the negative
Chapter 4 Problems
221
direction. Referring to the free-body diagram for block 2, we write Newtons second law as
follows:
P f = m2 ( a )
1 2 3k
(2)
Net force
on block 2
Solving Equation (1) for a gives a = P / m1 . Substituting this result into Equation (2) gives
P
P f k = m2
m
1
or
P=
m1 f k
m1 + m2
Substituting this result for P into a = P / m1 gives
a=
m1 f k
fk
P
=
=
m1 m1 ( m1 + m2 ) m1 + m2
We can now use these results to calculate P and a in both cases.
a. Case A
P=
m1 f k
( 3.0 kg ) ( 5.8 N )
=
= 2.9 N
m1 + m2 3.0 kg + 3.0 kg
Case B
P=
m1 f k
( 6.0 kg ) ( 5.8 N ) = 3.9 N
=
m1 + m2
6.0 kg + 3.0 kg
b. Case A
a=
fk
5.8 N
=
= 0.97 m/s 2
m1 + m2 3.0 kg + 3.0 kg
The magnitude of the acceleration is 0.97 m/s 2 .
Case B
a=
fk
5.8 N
=
= 0.64 m/s 2
m1 + m2 6.0 kg + 3.0 kg
The magnitude of the acceleration is 0.64 m/s 2 .
89. SSM REASONING The tension in each coupling bar is responsible for accelerating the
objects behind it. The masses of the cars are m1, m2, and m3. We can use Newtons second
law to express the tension in each coupling bar, since friction is negligible:
222 FORCES AND NEWTON'S LAWS OF MOTION
TA = ( m1 + m2 + m3 ) a
1 4 4 42 4 4 43
TB = ( m2 + m3 ) a
1 4 42 4 43
Coupling bar A
TC = m3a
14 2 4
3
Coupling bar C
Coupling bar B
In these expressions a = 0.12 m/s2 remains constant. Consequently, the tension in a given
bar will change only if the total mass of the objects accelerated by that bar changes as a
result of the luggage transfer. Using (Greek capital delta) to denote a change in the usual
fashion, we can express the changes in the above tensions as follows:
TA = ( m1 + m2 + m3 ) a
1 4 4 4 4 2 4 4 4 4 4
4
3
Coupling bar A
TB = ( m2 + m3 ) a
1 4 4 4 2 4 4 4 4
4
3
Coupling bar B
TC = ( m3 ) a
1 44 2 4 43
Coupling bar C
SOLUTION
a. Moving luggage from car 2 to car 1 does not change the total mass m1 + m2 + m3, so
(m1 + m2 + m3) = 0 kg and TA = 0 N .
The transfer from car 2 to car 1 causes the total mass m2 + m3 to decrease by 39 kg, so
(m2 + m3) = 39 kg and
(
)
TB = ( m2 + m3 ) a = ( 39 kg ) 0.12 m/s 2 = 4.7 N
The transfer from car 2 to car 1 does not change the mass m3, so m3 = 0 kg and
TC = 0 N .
b. Moving luggage from car 2 to car 3 does not change the total mass m1 + m2 + m3, so
(m1 + m2 + m3) = 0 kg and TA = 0 N .
The transfer from car 2 to car 3 does not change the total mass m2 + m3, so (m2 + m3) = 0
kg and TB = 0 N .
The transfer from car 2 to car 3 causes the mass m3 to increase by 39 kg, so m3 = +39 kg
and
TC = ( m3 ) a = ( +39 kg ) 0.12 m/s 2 = +4.7 N
(
)
____________________________________________________________________________________________
90. REASONING AND SOLUTION The distance required for the truck to stop is found from
x=
2
v 2 v0
2a
2
( 0 m/s ) 2 v0
=
2a
Chapter 4 Problems
223
The acceleration of the truck is needed. The frictional force decelerates the crate. The
maximum force that friction can supply is
fsMAX = sFN = smg
Newton's second law requires that
fsMAX = ma so a = sg
Now the stopping distance is
x=
2
v0
2 s g
=
( 25 m/s ) 2
2 ( 0.650 ) ( 9.80 m/s 2 )
= 49.1 m
____________________________________________________________________________________________
91. REASONING AND SOLUTION
a. Newton's second law for block 1 (10.0 kg) is
T = m1a
(1)
Block 2 (3.00 kg) has two ropes attached each carrying a tension T. Also, block 2 only
travels half the distance that block 1 travels in the same amount of time so its acceleration is
only half of block 1's acceleration. Newton's second law for block 2 is then
2T m2 g = 1 m2a
2
(2)
Solving Equation (1) for a, substituting into Equation (2), and rearranging gives
T=
1+
1
2
1
4
m2 g
( m2 / m1 )
= 13.7 N
b. Using this result in Equation (1) yields
a=
T
13.7 N
=
= 1.37 m/s 2
m1 10.0 kg
____________________________________________________________________________________________
92. REASONING AND SOLUTION
a. The force acting on the sphere which accelerates it is the horizontal component of the
tension in the string. Newton's second law for the horizontal motion of the sphere gives
224 FORCES AND NEWTON'S LAWS OF MOTION
T sin = ma
The vertical component of the tension in the string supports the weight of the sphere so
T cos = mg
Eliminating T from the above equations results in a = g tan .
b.
(
)
a = g tan = 9.80 m/s 2 tan 10.0 = 1.73 m/s 2
c. Rearranging the result of part a and setting a = 0 m/s2 gives
= tan 1 ( a / g ) = 0
____________________________________________________________________________________________
93.
SSM REASONING AND SOLUTION
a. The left mass (mass 1) has a tension T1 pulling it up. Newton's second law gives
T1 m1g = m1a
(1)
The right mass (mass 3) has a different tension, T3, trying to pull it up. Newton's second for
it is
T3 m3g = m3a
(2)
The middle mass (mass 2) has both tensions acting on it along with friction. Newton's
second law for its horizontal motion is
T3 T1 km2g = m2a
(3)
Solving Equation (1) and Equation (2) for T1 and T3, respectively, and substituting into
Equation (3) gives
( m m1 k m2 ) g
a= 3
m1 + m2 + m3
Hence,
(
)
25.0 kg 10.0 kg ( 0.100 ) ( 80.0 kg ) 9.80 m/s 2
a=
= 0.60 m/s 2
10.0 kg + 80.0 kg + 25.0 kg
b. From part a:
Chapter 4 Problems
225
(
)
T3 = m3(g a) = ( 25.0 kg ) ( 9.80 m/s 2 0.60 m/s 2 ) = 230 N
T1 = m1(g + a) = ( 10.0 kg ) 9.80 m/s 2 + 0.60 m/s 2 = 104 N
____________________________________________________________________________________________
94. REASONING AND SOLUTION
a. The static frictional force is responsible for accelerating the top block so that it does not
slip against the bottom one. The maximum force that can be supplied by friction is
fsMAX = sFN = sm1g
Newton's second law requires that fsMAX = m1a, so
a = sg
The force necessary to cause BOTH blocks to have this acceleration is
F = (m1 + m2)a = (m1 + m2) sg
2
F = (5.00 kg + 12.0 kg)(0.600)(9.80 m/s2) = 1.00 10 N
b. The maximum acceleration that the two block combination can have before slipping
occurs is
F
a=
17.0 kg
Newton's second law applied to the 5.00 kg block is
F sm1g = m1a = (5.00 kg)
F
17.0 kg
Hence
F = 41.6 N
____________________________________________________________________________________________
95.
SSM REASONING The magnitude of the gravitational force that each part exerts on the
other is given by Newtons law of gravitation as F = Gm1m2 / r 2 . To use this expression,
we need the masses m1 and m2 of the parts, whereas the problem statement gives the
weights W1 and W2. However, the weight is related to the mass by W = mg, so that for each
part we know that m = W/g.
SOLUTION The gravitational force that each part exerts on the other is
226 FORCES AND NEWTON'S LAWS OF MOTION
F=
Gm1m2
r2
=
G ( W1 / g ) ( W2 / g )
r2
m
( 6.67 1011 N 2 / kg2 ) ( 11 000 N ) ( 3400 N ) =
=
2
( 9.80 m/s2 ) ( 12 m ) 2
1.8 10 7 N
____________________________________________________________________________________________
96. REASONING The magnitudes of the initial (v0 = 0 m/s) and final (v = 805 m/s) velocities
are known. In addition, data is given for the mass and the thrust, so that Newtons second
law can be used to determine the acceleration of the probe. Therefore, kinematics Equation
2.4 (v = v0 + at) can be used to determine the time t.
SOLUTION Solving Equation 2.4 for the time gives
t=
v v0
a
Newtons second law gives the acceleration as a = (F)/m. Using this expression in
Equation 2.4 gives
t=
( v v0 )
( F ) / m
=
m ( v v0 )
F
=
( 474 kg ) ( 805 m/s 0 m/s )
56 103 N
= 6.8 106 s
Since one day contains 8.64 104 s, the time is
(
t = 6.8 106 s
97.
1 day
) 8.64 104 s = 79 days
SSM REASONING AND SOLUTION According to Equation 3.3b, the acceleration of
the astronaut is a y = (v y v0 y ) / t = v y / t . The apparent weight and the true weight of the
astronaut are related according to Equation 4.6. Direct substitution gives
vy
FN = { + ma y = m ( g + a y ) = m g +
mg
{
t
True
Apparent
weight
weight
45 m/s
2
= (57 kg) 9.80 m/s 2 +
= 7.3 10 N
15 s
____________________________________________________________________________________________
Chapter 4 Problems
227
98. REASONING According to Newton's second law ( F = ma ), the acceleration of the
object is given by a = F / m , where F is the net force that acts on the object. We must
first find the net force that acts on the object, and then determine the acceleration using
Newton's second law.
SOLUTION The following table gives the x and y components of the two forces that act
on the object. The third row of that table gives the components of the net force.
Force
x-Component
y-Component
F1
40.0 N
0N
F2
(60.0 N) cos 45.0 = 42.4 N
(60.0 N) sin 45.0 = 42.4 N
F = F1 + F2
82.4 N
42.4 N
The magnitude of F is given by the Pythagorean theorem as
F = (82.4 N) 2 + (42.4) 2 = 92.7 N
The angle that F makes with the +x axis is
1 42.4
N
= tan
= 27.2
82.4 N
F
42.4 N
82.4 N
According to Newton's second law, the magnitude of the acceleration of the object is
a=
F 92.7 N
=
= 30.9 m/s 2
m
3.00 kg
Since Newton's second law is a vector equation, we know that the direction of the right hand
side must be equal to the direction of the left hand side. In other words, the direction of the
acceleration a is the same as the direction of the net force F . Therefore, the direction of
the acceleration of the object is 27.2 above the +x axis .
____________________________________________________________________________________________
99.
SSM REASONING In order to start the crate moving, an external agent must supply a
force that is at least as large as the maximum value fsMAX = s FN , where s is the
coefficient of static friction (see Equation 4.7). Once the crate is moving, the magnitude of
228 FORCES AND NEWTON'S LAWS OF MOTION
the frictional force is very nearly constant at the value f k = k FN , where k is the
coefficient of kinetic friction (see Equation 4.8). In both cases described in the problem
statement, there are only two vertical forces that act on the crate; they are the upward
normal force FN, and the downward pull of gravity (the weight) mg. Furthermore, the crate
has no vertical acceleration in either case. Therefore, if we take upward as the positive
direction, Newton's second law in the vertical direction gives FN mg = 0 , and we see that,
in both cases, the magnitude of the normal force is FN = mg .
SOLUTION
a. Therefore, the applied force needed to start the crate moving is
fsMAX = s mg = (0.760)(60.0 kg)(9.80 m/s 2 ) = 447 N
b. When the crate moves in a straight line at constant speed, its velocity does not change,
and it has zero acceleration. Thus, Newton's second law in the horizontal direction becomes
P fk = 0, where P is the required pushing force. Thus, the applied force required to keep
the crate sliding across the dock at a constant speed is
P = f k = k mg = (0.410)(60.0 kg)(9.80 m/s 2 ) = 241 N
____________________________________________________________________________________________
100. REASONING Newtons second law of motion gives the relationship between the net force
F and the acceleration a that it causes for an object of mass m. The net force is the vector
sum of all the external forces that act on the object. Here the external forces are the drive
force, the force due to the wind, and the resistive force of the water.
SOLUTION We choose the direction of the drive force (due west) as the positive direction.
Solving Newtons second law ( F = ma ) for the acceleration gives
a=
F +4100 N 800 N 1200 N
=
= +0.31 m/s 2
m
6800 kg
The positive sign for the acceleration indicates that its direction is due west .
101. REASONING AND SOLUTION The acceleration needed so that the craft touches down
with zero velocity is
2
v 2 v0 ( 18.0 m/s ) 2
a=
=
= 0.982 m/s 2
2s
2 ( 165 m )
Newton's second law applied in the vertical direction gives
Chapter 4 Problems
229
F mg = ma
Then
F = m(a + g) = (1.14 104 kg)(0.982 m/s2 + 1.60 m/s2) = 29 400 N
____________________________________________________________________________________________
102. REASONING AND SOLUTION The apparent weight is
FN = mw(g + a)
We need to find the acceleration a. Let T represent the force applied by the hoisting cable.
Newton's second law applied to the elevator gives
T (mw + me)g = (mw + me)a
Solving for a gives
a=
T
9410 N
g =
9.80 m/s 2 = 0.954 m/s 2
mw + me
60.0 kg + 815 kg
Now the apparent weight is
FN = (60.0 kg)(9.80 m/s2 + 0.954 m/s2) = 645 N
____________________________________________________________________________________________
103. SSM REASONING We can use the appropriate equation of kinematics to find the
acceleration of the bullet. Then Newton's second law can be used to find the average net
force on the bullet.
SOLUTION According to Equation 2.4, the acceleration of the bullet is
a=
v v0
t
=
715 m/s 0 m/s
= 2.86 105 m/s 2
3
2.50 10 s
Therefore, the net average force on the bullet is
F = ma = (15 103 kg)(2.86 105 m/s 2 ) = 4290 N
____________________________________________________________________________________________
104. REASONING The magnitude F of the net force acting on the kayak is given by Newtons
second law as F = ma (Equation 4.1), where m is the combined mass of the person and
kayak, and a is their acceleration. Since the initial and final velocities, v0 and v, and the
displacement x are known, we can employ one of the equations of kinematics from
Chapter 2 to find the acceleration.
230 FORCES AND NEWTON'S LAWS OF MOTION
(
)
SOLUTION Solving Equation 2.9 v 2 = v0 2 + 2ax from the equations of kinematics for
the acceleration, we have
a=
v 2 v0 2
2x
Substituting this result into Newtons second law gives
( 0.60 m/s ) 2 ( 0 m/s ) 2
v 2 v0 2
F = ma = m
= 32 N
= ( 73 kg )
2 ( 0.41 m )
2x
_____________________________________________________________________________
_
105. REASONING AND SOLUTION
a. According to Equation 4.4, the weight of an object of mass m on the surface of Mars
would be given by
GM M m
W=
2
RM
where MM is the mass of Mars and RM is the radius of Mars. On the surface of Mars, the
weight of the object can be given as W = mg (see Equation 4.5), so
mg =
GM M m
2
RM
or
g=
GM M
2
RM
Substituting values, we have
g=
(6.67 1011N m 2 /kg 2 )(6.46 10 23 kg)
= 3.75 m/s 2
(3.39 106 m) 2
b. According to Equation 4.5,
W = mg = (65 kg)(3.75 m/s2) = 2.4 102 N
____________________________________________________________________________________________
106. REASONING Each particle experiences two gravitational forces, one due to each of the
remaining particles. To get the net gravitational force, we must add the two contributions,
taking into account the directions. The magnitude of the gravitational force that any one
particle exerts on another is given by Newtons law of gravitation as F = Gm1m2 / r 2 . Thus,
for particle A, we need to apply this law to its interaction with particle B and with particle
C. For particle B, we need to apply the law to its interaction with particle A and with
particle C. Lastly, for particle C, we must apply the law to its interaction with particle A
Chapter 4 Problems
231
and with particle B. In considering the directions, we remember that the gravitational force
between two particles is always a force of attraction.
SOLUTION We begin by calculating the magnitude of the gravitational force for each pair
of particles:
FAB =
FBC =
FAC =
GmA mB
m
( 6.67 1011 N 2 / kg2 ) ( 363 kg ) ( 517 kg ) = 5.007 105 N
=
GmB mC
m
( 6.67 1011 N 2 / kg2 ) ( 517 kg ) ( 154 kg ) = 8.497 105 N
=
r2
r2
Gm AmC
r2
( 0.500 m ) 2
( 0.500 m ) 2
m
( 6.67 1011 N 2 / kg2 ) ( 363 kg ) ( 154 kg ) = 6.629 106 N
=
( 0.500 m ) 2
In using these magnitudes we take the direction to the right as positive.
a. Both particles B and C attract particle A to the right, the net force being
FA = FAB + FAC = 5.007 10 5 N + 6.629 10 6 N = 5.67 10 5 N, right
b. Particle C attracts particle B to the right, while particle A attracts particle B to the left, the
net force being
FB = FBC FAB = 8.497 10 5 N 5.007 10 5 N = 3.49 10 5 N, right
c. Both particles A and B attract particle C to the left, the net force being
FC = FAC + FBC = 6.629 10 6 N + 8.497 10 5 N = 9.16 10 5 N, left
____________________________________________________________________________________________
y
107. SSM REASONING AND SOLUTION
The system is shown in the drawing. We
will let m1 = 21.0 kg , and m2 = 45.0 kg .
Then, m1 will move upward, and m2 will
move downward. There are two forces that
act on each object; they are the tension T in
the cord and the weight mg of the object.
The forces are shown in the free-body
diagrams at the far right.
y
T
m1
T
x
m2
m1g
m2g
232 FORCES AND NEWTON'S LAWS OF MOTION
We will take up as the positive direction.
acceleration of m2 must be a.
If the acceleration of m1 is a, then the
From Newton's second law, we have for m1
Fy = T m1g = m1a
(1)
Fy = T m2 g = m2 a
(2)
and for m2
a. Eliminating T between these two equations, we obtain
a=
m2 m1
45.0 kg 21.0 kg
2
2
g =
(9.80 m/s ) = 3.56 m/s
m2 + m1
45.0 kg + 21.0 kg
b. Eliminating a between Equations (1) and (2), we find
T=
2(21.0 kg)(45.0 kg)
2
g=
(9.80 m/s ) = 281 N
m1 + m2
21.0 kg + 45.0 kg
2m1m2
____________________________________________________________________________________________
108. REASONING Static friction determines the magnitude of the applied force at which either
the upper or lower block begins to slide. For the upper block the static frictional force is
applied only by the lower block. For the lower block, however, separate static frictional
forces are applied by the upper block and by the horizontal surface. The maximum
magnitude of any of the individual frictional forces is given by Equation 4.7 as the
coefficient of static friction times the magnitude of the normal force.
SOLUTION We begin by drawing the free-body diagram for the lower block.
max
fs, from A
FApplied
A
max
B
fs, from surface
Free-body diagram for
lower block B
This diagram shows that three horizontal forces act on the lower block, the applied force,
and the two maximum static frictional forces, one from the upper block and one from the
horizontal surface. At the instant that the lower block just begins to slide, the blocks are in
equilibrium and the applied force is balanced by the two frictional forces, with the result
that
Chapter 4 Problems
233
MAX
MAX
FApplied = fs, from A + fs, from surface
(1)
According to Equation 4.7, the magnitude of the maximum frictional force from the surface
is
MAX
fs, from surface = s FN = s 2mg
(2)
Here, we have recognized that the normal force FN from the horizontal surface must balance
the weight 2mg of both blocks.
It remains now to determine the
47.0 N
magnitude of the maximum frictional
max
fs, from B
MAX
force fs, from A from the upper block.
A
To this end, we draw the free-body
B
diagram for the upper block at the
Free-body diagram for
instant that it just begins to slip due to
upper block A
the 47.0-N applied force. At this
instant the block is in equilibrium,
so that the frictional force from the lower block B balances the 47.0-N force. Thus,
MAX
fs, from B = 47.0 N , and according to Equation 4.7, we have
MAX
fs, from B = s FN = s mg = 47.0 N
Here, we have recognized that the normal force FN from the lower block must balance the
weight mg of only the upper block. This result tells us that smg = 47.0 N. To determine
MAX
fs, from A we invoke Newtons third law to conclude that the magnitudes of the frictional
forces at the A-B interface are equal, since they are action-reaction forces.
MAX
fs, from A = s mg . Substituting this result and Equation (2) into Equation (1) gives
Thus,
MAX
MAX
FApplied = fs, from A + fs, from surface = s mg + s 2mg = 3 ( 47.0 N ) = 141 N
_____________________________________________________________________________
_
109. REASONING AND SOLUTION If the +x axis is taken in the direction of motion, Fx = 0
gives
F
N
F fk mg sin = 0
fk
where
fk = kFN
Then
F
W
2 5 .0
234 FORCES AND NEWTON'S LAWS OF MOTION
F kFN mg sin
=0
(1)
Also, Fy = 0 gives
FN mg cos = 0
so
FN = mg cos
(2)
Substituting Equation (2) into Equation (1) and solving for F yields
F = mg( sin + k cos )
F = (55.0 kg)(9.80 m/s2)[sin 25.0 + (0.120)cos 25.0] = 286 N
____________________________________________________________________________________________
110. REASONING Since the wire beneath the limb is at rest, it is in equilibrium and the net
force acting on it must be zero. Three forces comprise the net force, the 151-N force from
the limb, the 447-N tension force from the left section of the wire, and the tension force T
from the right section of the wire. We will resolve the forces into components and set the
sum of the x components and the sum of the y components separately equal to zero. In so
doing we will obtain two equations containing the unknown quantities, which are the
horizontal and vertical components of the tension force T. These two equations will be
solved simultaneously to give values for the two unknowns. Knowing the components of
the tension force, we can determine its magnitude and direction.
SOLUTION Let Tx and Ty be the horizontal and vertical components of the tension force.
The free-body diagram for the wire beneath the limb is as follows:
Taking upward and to the right as the positive
directions, we find for the x components of the
forces that
Fx = Tx ( 447 N ) cos 14.0 = 0
Tx = ( 447 N ) cos 14.0 = 434 N
+y
T
447 N
14.0
Tx
151 N
For the y components of the forces we have
Fy = Ty + ( 447 N ) sin14.0 151 N = 0
Ty = ( 447 N ) sin14.0 + 151 N = 43 N
The magnitude of the tension force is
Ty
+x
Chapter 4 Problems
T = Tx2 + Ty2 =
( 434 N ) 2 + ( 43 N ) 2
235
= 436 N
Since the components of the tension force and the angle are related by tan = Ty / Tx , we
find that
Ty
43 N
= tan 1 = tan 1
= 5.7
T
434 N
x
111. SSM REASONING
The shortest time to pull the person from the cave corresponds to
the maximum acceleration, a y , that the rope can withstand. We first determine this
2
acceleration and then use kinematic Equation 3.5b ( y = v0 y t + 1 a yt ) to find the time t.
2
SOLUTION As the person is being pulled from the cave, there are two forces that act on
him; they are the tension T in the rope that points vertically upward, and the weight of the
person mg that points vertically downward. Thus, if we take upward as the positive
direction, Newton's second law gives Fy = T mg = ma y . Solving for a y , we have
ay =
T
T
569 N
g=
g=
9.80 m/s 2 = 0.92 m/s 2
2
2
m
W /g
(5.20 10 N)/(9.80 m/s )
2
Therefore, from Equation 3.5b with v0 y = 0 m/s, we have y = 1 a y t . Solving for t, we find
2
t=
2y
2(35.1 m)
=
= 8.7 s
ay
0.92 m/s 2
____________________________________________________________________________________________
112. REASONING The free-body diagrams for the large cube (mass = M) and the small cube
(mass = m) are shown in the following drawings. In the case of the large cube, we have
omitted the weight and the normal force from the surface, since the play no role in the
solution (although they do balance).
fsMAX
FN
P
FN
mg
In these diagrams, note that the two blocks exert a normal force on each other; the large
block exerts the force FN on the smaller block, while the smaller block exerts the force FN
236 FORCES AND NEWTON'S LAWS OF MOTION
on the larger block. In accord with Newtons third law these forces have opposite directions
and equal magnitudes FN . Under the influence of the forces shown, the two blocks have the
same acceleration a. We begin our solution by applying Newtons second law to each one.
SOLUTION According to Newtons second law, we have
F = P FN = Ma
1 4 44 2 4 4 4
3
FN = ma
14 2 4
3
Large block
Small block
Substituting FN = ma into the large-block expression and solving for P gives
P = (M + m) a
For the smaller block to remain in place against the larger block, the static frictional force
must balance the weight of the smaller block, so that fsMAX = mg. But fsMAX is given by
fsMAX = sFN , where, from the Newtons second law, we know that FN = ma. Thus, we
have sma = mg or a = g/s . Using this result in the expression for P gives
P = ( M + m)
( M + m ) g = ( 25 kg + 4.0 kg ) ( 9.80 m/s
a=
s
2
0.71
)=
4.0 102 N
____________________________________________________________________________________________
113. REASONING According to Newtons second law, the acceleration of the probe is
a = F/m. Using this value for the acceleration in Equation 2.8 and noting that the probe
starts from rest (v0 = 0 m/s), we can write the distance traveled by the probe as
1 F 2
x = v0t + 1 at 2 =
t
2
2 m
This equation is the basis for our solution.
SOLUTION Since each engine produces the same amount of force or thrust T, the net force
is F = 2T when the engines apply their forces in the same direction and
F = T 2 + T 2 = 2T when they apply their forces perpendicularly. Thus, we write the
distances traveled in the two situations as follows:
1 2T
x=
2
1 44 m
24
2
t
3
4
Engines fired in
same direction
and
1 2T 2
x=
t
2 m
1 4 4 2 4 43
Engines fired
perpendicularly
Chapter 4 Problems
237
Since the distances are the same, we have
2 1 2T 2
t
t =
2 m
2 m
1 2T
2
2 t 2 = t
or
The firing time when the engines apply their forces perpendicularly is, then,
t =
( 4 2 ) t = ( 4 2 ) ( 28 s ) =
33 s
____________________________________________________________________________________________
114. REASONING The drawing shows the point between the earth and the moon where the
gravitational force exerted on the spacecraft by the earth balances that exerted by the moon.
The magnitude of the gravitational force exerted on the spacecraft by the earth is
Fearth = G
mearth mspacecraft
r2
while that exerted on the spacecraft by the moon is
Fmoon = G
mmoon mspacecraft
( rearth-moon r ) 2
By setting these two expressions equal to each other (since the gravitational forces balance),
we will be able to find the distance r.
Point where the
gravitational forces
b
Earth
Moon
SOLUTION Setting Fearth equal to Fmoon, we have
G
mearth mspacecraft
Solving this expression for r gives
r2
=G
mmoon mspacecraft
( rearth-moon r ) 2
238 FORCES AND NEWTON'S LAWS OF MOTION
mearth
mmoon
( rearth-moon )
r=
mearth
mmoon
1+
=
( 3.85 108 m )
81.4
1 + 81.4
= 3.47 108 m
_____________________________________________________________________________
_
115. SSM REASONING AND SOLUTION The free-body diagram is shown at
the right. The forces that act on the picture are the pressing force P, the
normal force FN exerted on the picture by the wall, the weight mg of the
picture, and the force of static friction fsMAX . The maximum magnitude for
MAX
fS
FN
P
mg
fsMAX
= s FN . The picture is in
the frictional force is given by Equation 4.7:
equilibrium, and, if we take the directions to the right and up as positive, we
have in the x direction
Fx = P FN = 0
or
P = FN
Fy = fsMAX mg = 0
or
fsMAX = mg
and in the y direction
Therefore,
fsMAX = s FN = mg
But since FN = P , we have
s P = mg
Solving for P, we have
P=
mg (1.10 kg)(9.80 m/s 2 )
=
= 16.3 N
s
0.660
____________________________________________________________________________________________
116. REASONING AND SOLUTION
a. The rope exerts a tension, T, acting upward on each block. Applying Newton's second
law to the lighter block (block 1) gives
T m1g = m1a
Similarly, for the heavier block (block 2)
T m2g = m2a
Chapter 4 Problems
239
Subtracting the second equation from the first and rearranging yields
m m1
a= 2
g = 3.68 m/s 2
m +m
1
2
b. The tension in the rope is now 908 N since the tension is the reaction to the applied force
exerted by the hand. Newton's second law applied to the block is
T m1g = m1a
Solving for a gives
a=
( 908 N ) 9.80 m/s 2 = 11.8 m/s 2
T
g=
m1
42.0 kg
c. In the first case, the inertia of BOTH blocks affects the acceleration whereas, in the
second case, only the lighter block's inertia remains.
____________________________________________________________________________________________
117. SSM REASONING AND SOLUTION The penguin comes to a halt on the horizontal
surface because the kinetic frictional force opposes the motion and causes it to slow down.
The time required for the penguin to slide to a halt ( v = 0 m/s) after entering the horizontal
patch of ice is, according to Equation 2.4,
t=
v v0
ax
=
v0
ax
We must, therefore, determine the acceleration of the penguin as it slides along the
horizontal patch (see the following drawing).
fk1
FN1
mg sin
mg cos
Free-body diagram A
fk2
FN2
mg
Free-body diagram B
For the penguin sliding on the horizontal patch of ice, we find from free-body diagram B
and Newton's second law in the x direction (motion to the right is taken as positive) that
240 FORCES AND NEWTON'S LAWS OF MOTION
Fx = f k2 = ma x
or
ax =
f k2 k FN2
=
m
m
In the y direction in free-body diagram B, we have Fy = FN2 mg = 0 , or FN2 = mg .
Therefore, the acceleration of the penguin is
ax =
k mg
= k g
m
(1)
Equation (1) indicates that, in order to find the acceleration ax, we must find the coefficient
of kinetic friction.
We are told in the problem statement that the coefficient of kinetic friction between the
penguin and the ice is the same for the incline as for the horizontal patch. Therefore, we can
use the motion of the penguin on the incline to determine the coefficient of friction and use
it in Equation (1).
For the penguin sliding down the incline, we find from free-body diagram A (see the
previous drawing) and Newton's second law (taking the direction of motion as positive) that
Fx = mg sin f k1 = ma x = 0
f k1 = mg sin
or
(2)
Here, we have used the fact that the penguin slides down the incline with a constant
velocity, so that it has zero acceleration. From Equation 4.8, we know that f k1 = k FN1 .
Applying Newton's second law in the direction perpendicular to the incline, we have
Fy = FN1 mg cos = 0
or
FN1 = mg cos
Therefore, f k1 = k mg cos , so that according to Equation (2), we find
f k1 = k mg cos = mg sin
Solving for the coefficient of kinetic friction, we have
k =
sin
= tan
cos
Finally, the time required for the penguin to slide to a halt after entering the horizontal patch
of ice is
v
v0
v0
1.4 m/s
t= 0 =
=
=
= 1.2 s
ax
k g g tan (9.80 m/s 2 ) tan 6.9
Chapter 4 Problems
241
____________________________________________________________________________________________
118. REASONING The following figure shows the crate on the incline and the free body
diagram for the crate. The diagram at the far right shows all the forces resolved into
components that are parallel and perpendicular to the surface of the incline. We can
analyze the motion of the crate using Newton's second law. The coefficient of friction can
be determined from the resulting equations.
y
FN
x
P sin
MAX
P
MAX
FN
fS
fS
P
P cos
mg sin
mg
mg cos
SOLUTION Since the crate is at rest, it is in equilibrium and its acceleration is zero in all
directions. If we take the direction down the incline as positive, Newton's second law
indicates that
Fx = P cos + mg sin f sMAX = 0
According to Equation 4.7, fsMAX = s FN . Therefore, we have
P cos + mg sin s FN = 0
(1)
The expression for the normal force can be found from analyzing the forces that are
perpendicular to the incline. Taking up to be positive, we have
Fy = P sin + FN mg cos = 0
or
FN = mg cos P sin
Equation (1) then becomes
P cos + mg sin s (mg cos P sin ) = 0
Solving for the coefficient of static friction, we find that
s =
P cos + mg sin (535 N) cos 20.0 + (225 kg)(9.80 m/s 2 ) sin 20.0
=
= 0.665
mg cos P sin (225 kg)(9.80 m/s 2 ) cos 20.0 (535 N) sin 20.0
____________________________________________________________________________________________
242 FORCES AND NEWTON'S LAWS OF MOTION
119. REASONING The free-body diagram for the box is shown in the following drawing on the
left. On the right the same drawing is repeated, except that the pushing force P is resolved
into its horizontal and vertical components.
P
P sin
FN
FN
P cos
fk
fk
mg
mg
Since the block is moving at a constant velocity, it has no acceleration, and Newtons
second law indicates that the net vertical and net horizontal forces must separately be zero.
SOLUTION Taking upward and to the right as the positive directions, we write the zero net
vertical and horizontal forces as follows:
F mg P sin = 0
1N4 4 4 2 4 4 4 3
Vertical
P cos f = 0
1 4 4 2 4 k43
Horizontal
From the equation for the horizontal forces, we have P cos = fk . But the kinetic frictional
force is fk = kFN . Furthermore, from the equation for the vertical forces, we have
FN = mg + P sin . With these substitutions, we obtain
P cos = f k = k FN = k ( mg + P sin )
Solving for P gives
P=
k mg
cos k sin
The necessary pushing force becomes infinitely large when the denominator in this
expression is zero. Hence, we find that cos k sin = 0 , which can be rearranged to
show that
sin
1
1
= tan =
or = tan 1
= 68
cos
k
0.41
____________________________________________________________________________________________