With 1. speed v = 11200 m/s, we find
K= 1 2 1 mv = (2.9 105 ) (11200) 2 = 18 1013 J. . 2 2
2. (a) The change in kinetic energy for the meteorite would be
1 1 K = K f - K i = - K i = - mi vi2 = - 4 106 kg 15 103 m/s 2 2
(
)(
)
2
= -5 1014 J ,
or | K |= 5 1014 J . The negative sign indicates that kinetic energy is lost. (b) The energy loss in units of megatons of TNT would be
-K = ( 5 1014 J )
1 megaton TNT 4.2 1015 J
= 0.1megaton TNT.
(c) The number of bombs N that the meteorite impact would correspond to is found by noting that megaton = 1000 kilotons and setting up the ratio:
N= 0.1 1000 kiloton TNT = 8. 13kiloton TNT
2 3. (a) From Table 2-1, we have v 2 = v0 + 2ax . Thus,
2 v = v0 + 2ax =
( 2.4 10 )
7 2
+ 2 ( 3.6 1015 ) ( 0.035) = 2.9 107 m/s.
(b) The initial kinetic energy is
Ki =
2 1 2 1 mv0 = (1.67 10-27 kg )( 2.4 107 m/s ) = 4.8 10-13 J. 2 2
The final kinetic energy is
Kf =
2 1 2 1 mv = (1.67 10-27 kg )( 2.9 107 m/s ) = 6.9 10-13 J. 2 2
The change in kinetic energy is K = (6.9 1013 4.8 1013) J = 2.1 1013 J.
4. We apply the equation x(t ) = x0 + v0t + 1 at 2 , found in Table 2-1. Since at t = 0 s, x0 = 0 2 and v0 = 12 m/s , the equation becomes (in unit of meters)
x(t ) = 12t + 1 at 2 . 2
With x = 10 m when t = 1.0 s , the acceleration is found to be a = - 4.0 m/s 2 . The fact that a < 0 implies that the bead is decelerating. Thus, the position is described by x(t ) = 12t - 2.0t 2 . Differentiating x with respect to t then yields
v(t ) = dx = 12 - 4.0t . dt
Indeed at t =3.0 s, v(t = 3.0) = 0 and the bead stops momentarily. The speed at t = 10 s is v(t = 10) = - 28 m/s , and the corresponding kinetic energy is
K= 1 2 1 mv = (1.8 10- 2 kg)( - 28 m/s) 2 = 7.1 J. 2 2
5. We denote the mass of the father as m and his initial speed vi. The initial kinetic energy of the father is
Ki = 1 Kson 2
and his final kinetic energy (when his speed is vf = vi + 1.0 m/s) is K f = Kson . We use these relations along with Eq. 7-1 in our solution.
1 (a) We see from the above that Ki = 2 K f which (with SI units understood) leads to
1 2 1 mvi = 2 2
1 2 m ( vi + 1.0 ) . 2
The mass cancels and we find a second-degree equation for vi :
1 2 1 vi - vi - = 0. 2 2
The positive root (from the quadratic formula) yields vi = 2.4 m/s.
1 (b) From the first relation above Ki = 2 Kson , we have
b
1 2 mvi 2
g 1 F 1 F mI I = G G Jv J K 2 H2 H 2K
2 son
and (after canceling m and one factor of 1/2) are led to vson = 2vi = 4.8 m s.
6. By the work-kinetic energy theorem,
W = K = 1 2 1 2 1 mv f - mvi = (2.0 kg) (6.0 m/s) 2 - (4.0 m/s) 2 = 20 J. 2 2 2
(
)
We note that the directions of v f and vi play no role in the calculation.
7. Eq. 7-8 readily yields (with SI units understood)
W = Fx x + Fy y = 2cos(100)(3.0) + 2sin(100)(4.0) = 6.8 J.
8. Using Eq. 7-8 (and Eq. 3-23), we find the work done by the water on the ice block:
W = F d = 210 ^ - 150 ^ 15 ^ - 12^ = (210) (15) + (-150) (-12) = 5.0 103 J. i j i j
(
)(
)
9. Since this involves constant-acceleration motion, we can apply the equations of Table 1 2-1, such as x = v0t + 2 at 2 (where x0 = 0 ). We choose to analyze the third and fifth points, obtaining
1 0.2 m = v0 (1.0 s) + a (1.0 s) 2 2 1 0.8m = v0 (2.0 s) + a (2.0 s) 2 2 Simultaneous solution of the equations leads to v0 = 0 and a = 0.40 m s2 . We now have two ways to finish the problem. One is to compute force from F = ma and then obtain the work from Eq. 7-7. The other is to find K as a way of computing W (in accordance with Eq. 7-10). In this latter approach, we find the velocity at t = 2.0 s from v = v0 + at (so v = 0.80 m s) . Thus,
W = K =
1 (3.0 kg) (0.80 m/s) 2 = 0.96 J. 2
10. The change in kinetic energy can be written as
K =
1 1 m(v 2 - vi2 ) = m(2a x) = ma x f 2 2
where we have used v 2 = vi2 + 2a x from Table 2-1. From Fig. 7-27, we see that f K = (0 - 30) J = - 30 J when x = + 5 m . The acceleration can then be obtained as a= K (- 30 J) = = - 0.75 m/s 2 . m x (8.0 kg)(5.0 m)
The negative sign indicates that the mass is decelerating. From the figure, we also see that when x = 5 m the kinetic energy becomes zero, implying that the mass comes to rest momentarily. Thus,
2 v0 = v 2 - 2a x = 0 - 2(- 0.75 m/s 2 )(5.0 m) = 7.5 m 2 /s 2 ,
or v0 = 2.7 m/s . The speed of the object when x = -3.0 m is
2 v = v0 + 2a x = 7.5 + 2(- 0.75)(- 3.0) = 12 = 3.5 m/s .
11. We choose +x as the direction of motion (so a and F are negative-valued). (a) Newton's second law readily yields F = (85 kg) ( - 2.0 m/s 2 ) so that
F =| F | = 1.7 102 N .
(b) From Eq. 2-16 (with v = 0) we have
0 = v + 2ax
2 0
x = -
2 -2.0 m/s
(
( 37 m/s )
2 2
)
= 3.4 102 m .
Alternatively, this can be worked using the work-energy theorem. (c) Since F is opposite to the direction of motion (so the angle between F and
d = x is 180) then Eq. 7-7 gives the work done as W = - F x = -5.8 104 J .
(d) In this case, Newton's second law yields F = ( 85kg ) ( -4.0 m/s2 ) so that
F =| F | = 3.4 10 2 N .
(e) From Eq. 2-16, we now have x = -
2 ( -4.0 m/s
( 37 m/s )
2 2
)
= 1.7 102 m.
(f) The force F is again opposite to the direction of motion (so the angle is again 180) so that Eq. 7-7 leads to W = - F x = -5.8 104 J. The fact that this agrees with the result of part (c) provides insight into the concept of work.
12. (a) From Eq. 7-6, F = W/x = 3.00 N (this is the slope of the graph). (b) Eq. 7-10 yields K = Ki + W = 3.00 J + 6.00 J = 9.00 J.
13. (a) The forces are constant, so the work done by any one of them is given by W = F d , where d is the displacement. Force F1 is in the direction of the displacement, so
W1 = F1d cos 1 = (5.00 N) (3.00 m) cos 0 = 15.0 J. Force F2 makes an angle of 120 with the displacement, so W2 = F2 d cos 2 = (9.00 N) (3.00 m) cos120 = -13.5 J. Force F3 is perpendicular to the displacement, so W3 = F3d cos 3 = 0 since cos 90 = 0. The net work done by the three forces is W = W1 + W2 + W3 = 15.0 J - 13.5 J + 0 = +1.50 J. (b) If no other forces do work on the box, its kinetic energy increases by 1.50 J during the displacement.
14. The forces are all constant, so the total work done by them is given by W = Fnet x , where Fnet is the magnitude of the net force and x is the magnitude of the displacement. We add the three vectors, finding the x and y components of the net force:
Fnet x = - F1 - F2 sin 50.0 + F3 cos 35.0 = -3.00 N - (4.00 N)sin 35.0 + (10.0 N) cos 35.0 = 2.13 N Fnet y = - F2 cos 50.0 + F3 sin 35.0 = -(4.00 N) cos50.0 + (10.0 N) sin 35.0 = 3.17 N.
The magnitude of the net force is
2 2 Fnet = Fnet x + Fnet y = (2.13) 2 + (3.17) 2 = 3.82 N.
The work done by the net force is W = Fnet d = (3.82 N) (4.00 m) = 15.3 J where we have used the fact that d || Fnet (which follows from the fact that the canister started from rest and moved horizontally under the action of horizontal forces -- the resultant effect of which is expressed by Fnet ).
15. Using the work-kinetic energy theorem, we have
K = W = F d = Fd cos
In addition, F = 12 N and d = (2.00) 2 + (- 4.00) 2 + (3.00)2 = 5.39 m . (a) If K = + 30.0 J , then
= cos - 1
(b) K = - 30.0 J , then
K 30.0 = cos - 1 = 62.3 . Fd (12.0)(5.39)
= cos - 1
K - 30.0 = cos - 1 = 118 Fd (12.0)(5.39)
16. In both cases, there is no acceleration, so the lifting force is equal to the weight of the object. (a) Eq. 7-8 leads to W = F d = (360 kN) (0.10 m) = 36 kJ. (b) In this case, we find W = (4000 N)(0.050 m) = 2.0 102 J .
17. (a) We use F to denote the upward force exerted by the cable on the astronaut. The force of the cable is upward and the force of gravity is mg downward. Furthermore, the acceleration of the astronaut is g/10 upward. According to Newton's second law, F mg = mg/10, so F = 11 mg/10. Since the force F and the displacement d are in the same direction, the work done by F is
2 11mgd 11 (72 kg) 9.8 m / s (15 m) WF = Fd = = = 1164 104 J . 10 10
c
h
which (with respect to significant figures) should be quoted as 1.2 104 J. (b) The force of gravity has magnitude mg and is opposite in direction to the displacement. Thus, using Eq. 7-7, the work done by gravity is
Wg = - mgd = - (72 kg) 9.8 m / s2 (15 m) = - 1058 104 J .
c
h
which should be quoted as 1.1 104 J. (c) The total work done is W = 1164 104 J - 1.058 104 J = 1.06 103 J . Since the . astronaut started from rest, the work-kinetic energy theorem tells us that this (which we round to 1.1 103 J ) is her final kinetic energy.
1 (d) Since K = 2 mv 2 , her final speed is
. 2K 2(106 103 J) v= = = 5.4 m / s. 72 kg m
18. (a) Using notation common to many vector capable calculators, we have (from Eq. 78) W = dot([20.0,0] + [0, -(3.00)(9.8)], [0.500 30.0]) = +1.31 J. (b) Eq. 7-10 (along with Eq. 7-1) then leads to v = 2(1.31 J)/(3.00 kg) = 0.935 m/s.
19. (a) We use F to denote the magnitude of the force of the cord on the block. This force is upward, opposite to the force of gravity (which has magnitude Mg). The acceleration is a = g / 4 downward. Taking the downward direction to be positive, then Newton's second law yields
Fnet = ma Mg - F = M
FG g IJ H 4K
so F = 3Mg/4. The displacement is downward, so the work done by the cord's force is, using Eq. 7-7, WF = Fd = 3Mgd/4. (b) The force of gravity is in the same direction as the displacement, so it does work Wg = Mgd . (c) The total work done on the block is -3 M gd 4 + M gd = M gd 4 . Since the block starts from rest, we use Eq. 7-15 to conclude that this M gd 4 is the block's kinetic energy K at the moment it has descended the distance d. (d) Since K = 1 Mv 2 , the speed is 2
v= 2K 2( Mgd / 4) = = M M gd 2
b
g
at the moment the block has descended the distance d.
20. The fact that the applied force Fa causes the box to move up a frictionless ramp at a constant speed implies that there is no net change in the kinetic energy: K = 0 . Thus, the work done by Fa must be equal to the negative work done by gravity: Wa = -Wg . Since the box is displaced vertically upward by h = 0.150 m , we have Wa = + mgh = (3.00)(9.80)(0.150) = 4.41 J
21. Eq. 7-15 applies, but the wording of the problem suggests that it is only necessary to examine the contribution from the rope (which would be the "Wa" term in Eq. 7-15): Wa = -(50 N)(0.50 m) = -25 J (the minus sign arises from the fact that the pull from the rope is anti-parallel to the direction of motion of the block). Thus, the kinetic energy would have been 25 J greater if the rope had not been attached (given the same displacement).
22. We use d to denote the magnitude of the spelunker's displacement during each stage. The mass of the spelunker is m = 80.0 kg. The work done by the lifting force is denoted Wi where i = 1, 2, 3 for the three stages. We apply the work-energy theorem, Eq. 17-15.
1 (a) For stage 1, W1 - mgd = K1 = 2 mv12 , where v1 = 5.00 m / s . This gives
1 1 W1 = mgd + mv12 = (80.0) (9.80) (10.0) + (8.00)(5.00)2 = 8.84 103 J. 2 2
(b) For stage 2, W2 mgd = K2 = 0, which leads to
W2 = mgd = (80.0 kg) ( 9.80 m/s 2 ) (10.0 m) = 7.84 103 J.
1 (c) For stage 3, W3 - mgd = K3 = - 2 mv12 . We obtain
1 1 W3 = mgd - mv12 = (80.0) (9.80) (10.0) - (80.0)(5.00)2 = 6.84 103 J. 2 2
23. (a) The net upward force is given by F + FN - (m + M ) g = (m + M )a where m = 0.250 kg is the mass of the cheese, M = 900 kg is the mass of the elevator cab, F is the force from the cable, and FN = 3.00 N is the normal force on the cheese. On the cheese alone, we have
FN - mg = ma a= 3.00 - (0.250)(9.80) = 2.20 m/s 2 . 0.250
Thus the force from the cable is F = (m + M )(a + g ) - FN = 1.08 10 4 N , and the work done by the cable on the cab is
W = Fd1 = (1.80 104 )(2.40) = 2.59 104 J.
(b) If W = 92.61 kJ and d 2 = 10.5 m , the magnitude of the normal force is FN = (m + M ) g - W 9.261 104 = (0.250 + 900)(9.80) - = 2.45 N. d2 10.5
24. The spring constant is k = 100 N/m and the maximum elongation is xi = 5.00 m. Using Eq. 7-25 with xf = 0, the work is found to be
W= 1 2 1 kxi = (100)(5.00) 2 = 125 103 J. . 2 2
25. We make use of Eq. 7-25 and Eq. 7-28 since the block is stationary before and after the displacement. The work done by the applied force can be written as
Wa = -Ws = 1 k ( x 2 - xi2 ) . f 2
The spring constant is k = (80 N) /(2.0 cm)=4.0 103 N/m. With Wa = 4.0 J , and xi = - 2.0 cm , we have
xf = 2Wa 2(4.0 J) + xi2 = + (- 0.020 m) 2 = 0.049 m = 4.9 cm. k (4.0 103 N/m)
26. From Eq. 7-25, we see that the work done by the spring force is given by
Ws = 1 k ( xi2 - x 2 ) . f 2
The fact that 360 N of force must be applied to pull the block to x = + 4.0 cm implies that the spring constant is
k= 360 N = 90 N/cm=9.0 103 N/m . 4.0 cm
(a) When the block moves from xi = + 5.0 cm to x = + 3.0 cm , we have
Ws = 1 (9.0 103 N/m)[(0.050 m)2 - (0.030 m)2 ] = 7.2 J. 2
(b) Moving from xi = + 5.0 cm to x = - 3.0 cm , we have
1 Ws = (9.0 103 N/m)[(0.050 m) 2 - (- 0.030 m) 2 ] = 7.2 J. 2
(c) Moving from xi = + 5.0 cm to x = - 5.0 cm , we have
Ws = 1 (9.0 103 N/m)[(0.050 m)2 - (- 0.050 m)2 ] = 0 J. 2
(d) Moving from xi = + 5.0 cm to x = - 9.0 cm , we have
1 Ws = (9.0 103 N/m)[(0.050 m) 2 - (- 0.090 m) 2 ] = - 25 J. 2
27. The work done by the spring force is given by Eq. 7-25:
Ws = 1 k ( xi2 - x 2 ) . f 2
Since Fx = - kx , the slope in Fig. 7-35 corresponds to the spring constant k. Its value is given by k = 80 N/cm=8.0 103 N/m . (a) When the block moves from xi = + 8.0 cm to x = + 5.0 cm , we have
Ws = 1 (8.0 103 N/m)[(0.080 m) 2 - (0.050 m) 2 ] = 15.6 J 16 J. 2
(b) Moving from xi = + 8.0 cm to x = - 5.0 cm , we have
1 Ws = (8.0 103 N/m)[(0.080 m) 2 - (- 0.050 m)2 ] = 15.6 J 16 J. 2
(c) Moving from xi = + 8.0 cm to x = - 8.0 cm , we have
Ws = 1 (8.0 103 N/m)[(0.080 m) 2 - (- 0.080 m) 2 ] = 0 J. 2
(d) Moving from xi = + 8.0 cm to x = - 10.0 cm , we have
Ws = 1 (8.0 103 N/m)[(0.080 m) 2 - (- 0.10 m) 2 ] = - 14.4 J - 14 J. 2
28. The work done by the spring force is given by Eq. 7-25: Ws =
1 k ( xi2 - x 2 ) . f 2
The spring constant k can be deduced from Fig. 7-36 which shows the amount of work done to pull the block from 0 to x = 3.0 cm. The parabola Wa = kx 2 / 2 contains (0,0), (2.0 cm, 0.40 J) and (3.0 cm, 0.90 J). Thus, we may infer from the data that k = 2.0 103 N/m . (a) When the block moves from xi = + 5.0 cm to x = + 4.0 cm , we have
Ws = 1 (2.0 103 N/m)[(0.050 m) 2 - (0.040 m) 2 ] = 0.90 J. 2
(b) Moving from xi = + 5.0 cm to x = - 2.0 cm , we have
1 Ws = (2.0 103 N/m)[(0.050 m)2 - (- 0.020 m)2 ] = 2.1 J. 2
(c) Moving from xi = + 5.0 cm to x = - 5.0 cm , we have
1 Ws = (2.0 103 N/m)[(0.050 m)2 - (- 0.050 m)2 ] = 0 J. 2
29. (a) As the body moves along the x axis from xi = 3.0 m to xf = 4.0 m the work done by the force is
W=
xf xi
Fx dx =
xf xi
-6 x dx = -3( x 2 - xi2 ) = -3 (4.02 - 3.02 ) = -21 J. f
According to the work-kinetic energy theorem, this gives the change in the kinetic energy:
W = K = 1 m v 2 - vi2 f 2
d
i
where vi is the initial velocity (at xi) and vf is the final velocity (at xf). The theorem yields vf = 2W 2(-21) + vi2 = + (8.0) 2 = 6.6 m/s. m 2.0
(b) The velocity of the particle is vf = 5.0 m/s when it is at x = xf. The work-kinetic energy theorem is used to solve for xf. The net work done on the particle is W = -3 ( x 2 - xi2 ) , so f the theorem leads to
-3 x 2 - xi2 = f 1 m v 2 - vi2 . f 2
d
i
d
i
Thus, xf = - m 2 2 ( v f - vi ) + xi2 = - 2.0 kg ( (5.0 m/s)2 - (8.0 m/s)2 ) + (3.0 m)2 = 4.7 m. 6 6 N/m
30. (a) This is a situation where Eq. 7-28 applies, so we have Fx =
1 2 kx 2
(3.0 N) x = 2 (50 N/m)x2
1
which (other than the trivial root) gives x = (3.0/25) m = 0.12 m. (b) The work done by the applied force is Wa = Fx = (3.0 N)(0.12 m) = 0.36 J. (c) Eq. 7-28 immediately gives Ws = Wa = 0.36 J. (d) With Kf = K considered variable and Ki = 0, Eq. 7-27 gives K = Fx
1 2 kx . 2
We take
the derivative of K with respect to x and set the resulting expression equal to zero, in order to find the position xc which corresponds to a maximum value of K: xc = k = (3.0/50) m = 0.060 m. We note that xc is also the point where the applied and spring forces "balance." (e) At xc we find K = Kmax = 0.090 J.
F
31. According to the graph the acceleration a varies linearly with the coordinate x. We may write a = x, where is the slope of the graph. Numerically,
=
20 m / s2 = 2.5 s-2 . 8.0 m
The force on the brick is in the positive x direction and, according to Newton's second law, its magnitude is given by F = a m = m x. If xf is the final coordinate, the work done by the force is
b g
W=
xf 0
F dx =
m
xf 0
x dx =
2m
x2 = f
2.5 (8.0)2 = 8.0 102 J. 2(10)
32. From Eq. 7-32, we see that the "area" in the graph is equivalent to the work done. Finding that area (in terms of rectangular [length width] and triangular 1 [ 2 base height] areas) we obtain W = W0< x < 2 + W2< x < 4 + W4< x< 6 + W6< x <8 = (20 + 10 + 0 - 5) J = 25 J.
33. (a) The graph shows F as a function of x assuming x0 is positive. The work is negative as the object moves from x = 0 to x = x0 and positive as it moves from x = x0 to x = 2 x0 .
Since the area of a triangle is (base)(altitude)/2, the work done from x = 0 to x = x0 is -( x0 )( F0 ) / 2 and the work done from x = x0 to x = 2 x0 is (2 x0 - x0 )( F0 ) / 2 = ( x0 )( F0 ) / 2 The total work is the sum, which is zero. (b) The integral for the work is W =
z
2 x0
0
Fx F G Hx
0
0
I Fx - 1J dx = F G K H 2x
0
2
0
I - xJ K
2 x0
= 0.
0
34. Using Eq. 7-32, we find W =
z
1.25
0.25
e -4 x dx = 0.21 J
2
where the result has been obtained numerically. Many modern calculators have that capability, as well as most math software packages that a great many students have access to.
35. We choose to work this using Eq. 7-10 (the work-kinetic energy theorem). To find the initial and final kinetic energies, we need the speeds, so
v= dx = 3.0 - 8.0t + 3.0t 2 dt
in SI units. Thus, the initial speed is vi = 3.0 m/s and the speed at t = 4 s is vf = 19 m/s. The change in kinetic energy for the object of mass m = 3.0 kg is therefore
K = 1 m v 2 - vi2 = 528 J f 2
(
)
which we round off to two figures and (using the work-kinetic energy theorem) conclude that the work done is W = 5.3 102 J.
36. (a) Using the work-kinetic energy theorem
K f = Ki +
2.0 0
1 (2.5 - x 2 ) dx = 0 + (2.5)(2.0) - (2.0)3 = 2.3 J. 3
(b) For a variable end-point, we have Kf as a function of x, which could be differentiated to find the extremum value, but we recognize that this is equivalent to solving F = 0 for x:
F = 0 2.5 - x 2 = 0 .
Thus, K is extremized at x =
K f = Ki +
2.5 0
2.5 1.6 m and we obtain
1 ( 2.5)3 = 2.6 J. 3
(2.5 - x 2 )dx = 0 + (2.5)( 2.5) -
Recalling our answer for part (a), it is clear that this extreme value is a maximum.
37. (a) We first multiply the vertical axis by the mass, so that it becomes a graph of the applied force. Now, adding the triangular and rectangular "areas" in the graph (for 0 x 4) gives 42 J for the work done. (b) Counting the "areas" under the axis as negative contributions, we find (for 0 x 7) the work to be 30 J at x = 7.0 m. (c) And at x = 9.0 m, the work is 12 J. (d) Eq. 7-10 (along with Eq. 7-1) leads to speed v = 6.5 m/s at x = 4.0 m. Returning to the original graph (where a was plotted) we note that (since it started from rest) it has received acceleration(s) (up to this point) only in the +x direction and consequently must have a velocity vector pointing in the +x direction at x = 4.0 m. (e) Now, using the result of part (b) and Eq. 7-10 (along with Eq. 7-1) we find the speed is 5.5 m/s at x = 7.0 m. Although it has experienced some deceleration during the 0 x 7 interval, its velocity vector still points in the +x direction. (f) Finally, using the result of part (c) and Eq. 7-10 (along with Eq. 7-1) we find its speed v = 3.5 m/s at x = 9.0 m. It certainly has experienced a significant amount of deceleration during the 0 x 9 interval; nonetheless, its velocity vector still points in the +x direction.
38. As the body moves along the x axis from xi = 0 m to xf = 3.00 m the work done by the force is
W=
xf xi
Fx dx =
xf xi
(cx - 3.00 x 2 )dx =
c 2 x - x3 2
3
=
0
c (3.00) 2 - (3.00)3 2
= 4.50c - 27.0.
However, W = K = (11.0 - 20.0) = - 9.00 J from the work-kinetic energy theorem. Thus,
4.50c - 27.0 = - 9.00
or c = 4.00 N/m .
39. We solve the problem using the work-kinetic energy theorem which states that the change in kinetic energy is equal to the work done by the applied force, K = W . In our problem, the work done is W = Fd , where F is the tension in the cord and d is the length of the cord pulled as the cart slides from x1 to x2. From Fig. 7-40, we have
2 d = x12 + h 2 - x2 + h 2 = (3.00) 2 + (1.20) 2 - (1.00)2 + (1.20) 2 = 3.23 m - 1.56 m = 1.67 m
which yields K = Fd = (25.0 N)(1.67 m) = 41.7 J.
40. that Recognizing the force in the cable must equal the total weight (since there is no acceleration), we employ Eq. 7-47:
P = Fv cos = mg
FG x IJ H t K
where we have used the fact that = 0 (both the force of the cable and the elevator's motion are upward). Thus,
P = (3.0 103 kg) (9.8 m / s2 )
FG 210 mIJ H 23 s K
= 2.7 105 W.
41. The power associated with force F is given by P = F v , where v is the velocity of the object on which the force acts. Thus,
P = F v = Fv cos = (122 N)(5.0 m/s)cos37 = 4.9 102 W.
42. (a) Since constant speed implies K = 0, we require Wa = -Wg , by Eq. 7-15. Since Wg is the same in both cases (same weight and same path), then Wa = 9.0 10 2 J just as it was in the first case. (b) Since the speed of 1.0 m/s is constant, then 8.0 meters is traveled in 8.0 seconds. Using Eq. 7-42, and noting that average power is the power when the work is being done at a steady rate, we have
P = W 900 J = = 1.1102 W. t 8.0 s
(c) Since the speed of 2.0 m/s is constant, 8.0 meters is traveled in 4.0 seconds. Using Eq. 7-42, with average power replaced by power, we have
P= W 900 J = 225 W 2.3 102 W . = t 4.0 s
43. (a) The power is given by P = Fv and the work done by F from time t1 to time t 2 is given by
W =
z
t2
t1
P dt =
z
t2
t1
Fv dt .
Since F is the net force, the magnitude of the acceleration is a = F/m, and, since the initial velocity is v0 = 0 , the velocity as a function of time is given by v = v0 + at = ( F m)t . Thus
W=
t2 t1
1 2 ( F 2 / m)t dt = (F 2 / m)(t2 - t12 ). 2
For t1 = 0 and t2 = 1.0s, W= (b) For t1 = 1.0s, and t2 = 2.0s, W= 1 (5.0 N) 2 [(2.0 s) 2 - (1.0 s) 2 ] = 2.5 J. 2 15 kg 1 (5.0 N)2 (1.0 s) 2 = 0.83 J. 2 15 kg
(c) For t1 = 2.0s and t2 = 3.0s, W= 1 (5.0 N) 2 [(3.0 s) 2 - (2.0 s) 2 ] = 4.2 J. 2 15 kg
(d) Substituting v = (F/m)t into P = Fv we obtain P = F 2 t m for the power at any time t. At the end of the third second
F (5.0 N) (3.0 s) IJ P = G H 15 kg K
2
= 5.0 W.
44. (a) Using Eq. 7-48 and Eq. 3-23, we obtain
P = F v = (4.0 N)( - 2.0 m/s) + (9.0 N)(4.0 m/s) = 28 W.
(b) We again use Eq. 7-48 and Eq. 3-23, but with a one-component velocity: v = vj. P = Fv -12 W = ( -2.0 N)v which yields v = 6 m/s.
45. The total work is the sum of the work done by gravity on the elevator, the work done by gravity on the counterweight, and the work done by the motor on the system: WT = We + Wc + Ws. Since the elevator moves at constant velocity, its kinetic energy does not change and according to the work-kinetic energy theorem the total work done is zero. This means We + Wc + Ws = 0. The elevator moves upward through 54 m, so the work done by gravity on it is
We = - me gd = -(1200 kg)(9.80 m/s 2 )(54 m) = -6.35 105 J.
The counterweight moves downward the same distance, so the work done by gravity on it is
Wc = mc gd = (950 kg) (9.80 m/s 2 ) (54 m) = 5.03 105 J.
Since WT = 0, the work done by the motor on the system is
Ws = -We - Wc = 6.35 105 J - 5.03 105 J = 1.32 105 J.
This work is done in a time interval of t = 3.0 min = 180 s, so the power supplied by the motor to lift the elevator is Ws 1.32 105 J P= = = 7.4 10 2 W. 180 s t
46. (a) Since the force exerted by the spring on the mass is zero when the mass passes through the equilibrium position of the spring, the rate at which the spring is doing work on the mass at this instant is also zero. (b) The rate is given by P = F v = - Fv , where the minus sign corresponds to the fact that F and v are anti-parallel to each other. The magnitude of the force is given by F = kx = (500 N/m)(0.10 m) = 50 N, while v is obtained from conservation of energy for the spring-mass system:
E = K + U = 10 J = 1 1 1 1 mv 2 + kx 2 = (0.30 kg) v 2 + (500 N/m)(0.10 m) 2 2 2 2 2
which gives v = 7.1 m/s. Thus P = - Fv = -(50 N)(7.1 m/s) = -3.5 102 W.
47. (a) Eq. 7-8 yields (with SI units understood) W = Fx x + Fy y + Fz z = (2.0)(7.5 0.50) + (4.0)(12.0 0.75) + (6.0)(7.2 0.20) =101 J 1.0 102 J. (b) Dividing this result by 12 s (see Eq. 7-42) yields P = 8.4 W.
48. (a) With SI units understood, the object's displacement is ^ i j d = d f - di = -8.00 ^ + 6.00 ^ + 2.00 k . Thus, Eq. 7-8 gives W = F d = (3.00)(-8.00) + (7.00)(6.00) + (7.00)(2.00) = 32.0 J. (b) The average power is given by Eq. 7-42:
Pavg = W 32.0 = = 8.00 W. t 4.00
(c)
The
distance
2
from
2
the
2
coordinate
origin
to
the
initial
position
is
d i = (3.00) + (-2.00) + (5.00) = 6.16 m , and the magnitude of the distance from the
coordinate origin to the final position is d f = (-5.00)2 + (4.00) 2 + (7.00) 2 = 9.49 m . Their scalar (dot) product is di d f = (3.00)(-5.00) + (-2.00)(4.00) + (5.00)(7.00) = 12.0 m 2 . Thus, the angle between the two vectors is
= cos - 1
di d f di d f
= cos - 1
12.0 = 78.2 . (6.16)(9.49)
49. From Eq. 7-32, we see that the "area" in the graph is equivalent to the work done. We 1 find the area in terms of rectangular [length width] and triangular [ 2 base height] areas and use the work-kinetic energy theorem appropriately. The initial point is taken to be x = 0, where v0 = 4.0 m/s.
2 1 (a) With Ki = 2 mv0 = 16 J, we have
K 3 - K 0 = W0< x <1 + W1< x < 2 + W2< x <3 = -4.0 J so that K3 (the kinetic energy when x = 3.0 m) is found to equal 12 J. (b) With SI units understood, we write W3< x < x f as Fx x = (-4.0)( x f - 3.0) and apply the work-kinetic energy theorem:
K x f - K3 = W3< x < x f K x f - 12 = ( -4)( x f - 3.0)
so that the requirement Kx f = 8.0 J leads to x f = 4.0 m. (c) As long as the work is positive, the kinetic energy grows. The graph shows this situation to hold until x = 1.0 m. At that location, the kinetic energy is K1 = K 0 + W0< x <1 = 16 J + 2.0 J = 18 J.
50. (a) The compression of the spring is d = 0.12 m. The work done by the force of gravity (acting on the block) is, by Eq. 7-12,
W1 = mgd = (0.25 kg) 9.8 m / s2 (0.12 m) = 0.29 J.
c
h
(b) The work done by the spring is, by Eq. 7-26,
1 1 W2 = - kd 2 = - (250 N / m) (0.12 m) 2 = -18 J. . 2 2
(c) The speed vi of the block just before it hits the spring is found from the work-kinetic energy theorem (Eq. 7-15).
1 K = 0 - mvi2 = W1 + W2 2
which yields vi = ( -2)(W1 + W2 ) ( -2)(0.29 - 18) . = = 35 m / s. . m 0.25
(d) If we instead had vi' = 7 m/s , we reverse the above steps and solve for d . Recalling the theorem used in part (c), we have
1 1 0 - mvi2 = W1 + W2 = mgd - kd 2 2 2
which (choosing the positive root) leads to d = mg + m2 g 2 + mkvi 2 k
which yields d = 0.23 m. In order to obtain this result, we have used more digits in our intermediate results than are shown above (so vi = 12.048 = 3.471 m / s and vi' = 6.942 m/s).
51. (a) The component of the force of gravity exerted on the ice block (of mass m) along . the incline is mg sin , where = sin -1 0.91 15 gives the angle of inclination for the inclined plane. Since the ice block slides down with uniform velocity, the worker must exert a force F "uphill" with a magnitude equal to mg sin . Consequently,
b
g
F = mg sin = (45 kg) 9.8 m/s 2
(
)
0.91m = 2.7 10 2 N. 1.5 m
(b) Since the "downhill" displacement is opposite to F , the work done by the worker is
W1 = - 2.7 102 N (1.5 m) = -4.0 102 J.
c
h
(c) Since the displacement has a vertically downward component of magnitude 0.91 m (in the same direction as the force of gravity), we find the work done by gravity to be
W2 = (45 kg) 9.8 m / s2 (0.91 m) = 4.0 102 J.
c
h
(d) Since FN is perpendicular to the direction of motion of the block, and cos 90 = 0, work done by the normal force is W3 = 0 by Eq. 7-7. (e) The resultant force Fnet is zero since there is no acceleration. Thus, its work is zero, as can be checked by adding the above results W1 + W2 + W3 = 0 .
52. (a) The force of the worker on the crate is constant, so the work it does is given by WF = F d = Fd cos , where F is the force, d is the displacement of the crate, and is the angle between the force and the displacement. Here F = 210 N, d = 3.0 m, and = 20. Thus WF = (210 N) (3.0 m) cos 20 = 590 J. (b) The force of gravity is downward, perpendicular to the displacement of the crate. The angle between this force and the displacement is 90 and cos 90 = 0, so the work done by the force of gravity is zero. (c) The normal force of the floor on the crate is also perpendicular to the displacement, so the work done by this force is also zero. (d) These are the only forces acting on the crate, so the total work done on it is 590 J.
53. The work done by the applied force Fa is given by W = Fa d = Fa d cos . From Fig. 7-43, we see that W = 25 J when = 0 and d = 5.0 cm . This yields the magnitude of Fa :
Fa = W 25 J = = 5.0 102 N . d 0.050 m
(a) For = 64 , we have W = Fa d cos = (5.0 102 N)(0.050 m) cos 64 = 11 J. (b) For = 147 , we have W = Fa d cos = (5.0 102 N)(0.050 m) cos147 = - 21 J.
54. (a) Eq. 7-10 (along with Eq. 7-1 and Eq. 7-7) leads to vf = (2 m F cos )1/2= (cos )1/2 with SI units understood. (b) With vi = 1, those same steps lead to vf = (1 + cos )1/2. (c) Replacing with 180 , and still using vi = 1, we find vf = [1 + cos(180 )]1/2 = (1 cos )1/2. (d) The graphs are shown below. Note that as is increased in parts (a) and (b) the force provides less and less of a positive acceleration, whereas in part (c) the force provides less and less of a deceleration (as its value increases). The highest curve (which slowly decreases from 1.4 to 1) is the curve for part (b); the other decreasing curve (starting at 1 and ending at 0) is for part (a). The rising curve is for part (c); it is equal to 1 where = 90.
d
55. (a) We can easily fit the curve to a concave-downward parabola: x = 10 t(10 t), from which (by taking two derivatives) we find the acceleration to be a = 0.20 m/s2. The (constant) force is therefore F = ma = 0.40 N, with a corresponding work given by W = 2 Fx = 50 t(t 10). It also follows from the x expression that vo = 1.0 m/s. This means that Ki = 2 mv2 = 1.0 J. Therefore, when t = 1.0 s, Eq. 7-10 gives K = Ki + W = 0.64 J 0.6 J , where the second significant figure is not to be taken too seriously. (b) At t = 5.0 s, the above method gives K = 0. (c) Evaluating the W = 50 t(t 10) expression at t = 5.0 s and t = 1.0 s, and subtracting, yields 0.6 J. This can also be inferred from the answers for parts (a) and (b).
2 1
1
56. With SI units understood, Eq. 7-8 leads to W = (4.0)(3.0) c(2.0) = 12 2c. (a) If W = 0, then c = 6.0 N. (b) If W = 17 J, then c = 2.5 N. (c) If W = 18 J, then c = 15 N.
57. (a) Noting that the x component of the third force is F3x = (4.00 N)cos(60), we apply Eq. 7-8 to the problem: W = [5.00 1.00 + (4.00)cos 60](0.20 m) = 1.20 J. (b) Eq. 7-10 (along with Eq. 7-1) then yields v = 2W/m = 1.10 m/s.
58. (a) The plot of the function (with SI units understood) is shown below.
Estimating the area under the curve allows for a range of answers. Estimates from 11 J to 14 J are typical. (b) Evaluating the work analytically (using Eq. 7-32), we have W= 10 e-x/2 dx = -20 e-x/2|0 = 12.6 J 13 J .
2
59. (a) Eq. 7-6 gives Wa = Fd = (209 N)(1.50 m) 314 J. (b) Eq. 7-12 leads to Wg = (25.0 kg)(9.80 m/s2)(1.50 m)cos(115) 155 J. (c) The angle between the normal force and the direction of motion remains 90 at all times, so the work it does is zero. (d) The total work done on the crate is WT = 314 J 155 J =158 J.
60. (a) Eq. 7-8 gives W = (3.0)(5.0 3.0) + (7.0)[4.0 (2.0)] + (7.0)(7.0 5.0) = 32 J. (b) Eq. 7-42 gives P = W/t = 32/4.0 = 8.0 W. (c) Proceeding as in Sample Problem 3-6, we have
= cos-1
(-5.0)(3.0) + (4.0)(-2.0) + (7.0)(5.0) = 78 di df
38 and df = 90 (distances
where we used the Pythagorean theorem to find di =
understood to be in meters).
61. Hooke's law and the work done by a spring is discussed in the chapter. We apply work-kinetic energy theorem, in the form of K = Wa + Ws , to the points in Figure 7-48 at x = 1.0 m and x = 2.0 m, respectively. The "applied" work Wa is that due to the constant force P . 1 4 = P(10) - k (10) 2 . . 2 1 0 = P (2.0) - k (2.0) 2 2 (a) Simultaneous solution leads to P = 8.0 N, (b) and k = 8.0 N/m.
62. Using Eq. 7-8, we find
W = F d = ( F cos ^ sin ^ (x^ + y^ = Fx cos + Fy sin i+F j) i j)
where x = 2.0 m, y = 4.0 m, F = 10 N, and = 150 . Thus, we obtain W = 37 J. Note that the given mass value (2.0 kg) is not used in the computation.
63. There is no acceleration, so the lifting force is equal to the weight of the object. We note that the person's pull F is equal (in magnitude) to the tension in the cord. (a) As indicated in the hint, tension contributes twice to the lifting of the canister: 2T = mg. Since F = T , we find F = 98 N. (b) To rise 0.020 m, two segments of the cord (see Fig. 7-48) must shorten by that amount. Thus, the amount of string pulled down at the left end (this is the magnitude of d , the downward displacement of the hand) is d = 0.040 m. (c) Since (at the left end) both F and d are downward, then Eq. 7-7 leads to
W = F d = (98) (0.040) = 3.9 J.
(d) Since the force of gravity Fg (with magnitude mg) is opposite to the displacement
d c = 0.020 m (up) of the canister, Eq. 7-7 leads to
W = Fg dc = - (196) (0.020) = -3.9 J. This is consistent with Eq. 7-15 since there is no change in kinetic energy.
64. The acceleration is constant, so we may use the equations in Table 2-1. We choose the direction of motion as +x and note that the displacement is the same as the distance traveled, in this problem. We designate the force (assumed singular) along the x direction acting on the m = 2.0 kg object as F. (a) With v0 = 0, Eq. 2-11 leads to a = v/t. And Eq. 2-17 gives x = second law yields the force F = ma. Eq. 7-8, then, gives the work:
W = F x = m v t 1 1 vt = mv 2 2 2
1 2
vt . Newton's
as we expect from the work-kinetic energy theorem. With v = 10 m/s, this yields W = 1.0 102 J . (b) Instantaneous power is defined in Eq. 7-48. With t = 3.0 s, we find
P = Fv = m v v = 67 W. t
(c) The velocity at t = 1.5s is v ' = at ' = 5.0 m s . Thus, P = Fv = 33 W.
65. One approach is to assume a "path" from ri to rf and do the line-integral accordingly. Another approach is to simply use Eq. 7-36, which we demonstrate:
W =
xf xi
Fx dx +
yf yi
Fy dy =
-4
2
(2x)dx +
-3
3
(3) dy
with SI units understood. Thus, we obtain W = 12 18 = 6 J.
66. The total weight is (100)(660) = 6.6 104 N, and the words "raises ... at constant speed" imply zero acceleration, so the lift-force is equal to the total weight. Thus P = Fv = (6.6 104)(150/60) = 1.65 105 W.
67. (a) The force F of the incline is a combination of normal and friction force which is serving to "cancel" the tendency of the box to fall downward (due to its 19.6 N weight). Thus, F = mg upward. In this part of the problem, the angle between the belt and F is 80. From Eq. 7-47, we have
P = Fv cos = (19.6)(0.50) cos 80 = 1.7 W.
(b) Now the angle between the belt and F is 90, so that P = 0. (c) In this part, the angle between the belt and F is 100, so that P = (19.6)(0.50) cos 100 = 1.7 W.
68. Using Eq. 7-7, we have W = Fd cos = 1504 J . Then, by the work-kinetic energy theorem, we find the kinetic energy Kf = Ki + W = 0 + 1504 J. The answer is therefore 1.5 kJ .
69. (a) In the work-kinetic energy theorem, we include both the work due to an applied force Wa and work done by gravity Wg in order to find the latter quantity. K = Wa + Wg leading to Wg = 2.1 102 J . (b) The value of Wg obtained in part (a) still applies since the weight and the path of the child remain the same, so = Wg = 2.1 102 J . 30 = (100)(1.8) cos 180 + Wg
70. (a) To hold the crate at equilibrium in the final situation, F must have the same magnitude as the horizontal component of the rope's tension T sin , where is the angle between the rope (in the final position) and vertical:
= sin -1
FG 4.00IJ = 19.5 . H 12.0 K
But the vertical component of the tension supports against the weight: T cos = mg . Thus, the tension is T = (230)(9.80)/cos 19.5 = 2391 N and F = (2391) sin 19.5 = 797 N. An alternative approach based on drawing a vector triangle (of forces) in the final situation provides a quick solution. (b) Since there is no change in kinetic energy, the net work on it is zero. (c) The work done by gravity is Wg = Fg d = - mgh , where h = L(1 cos ) is the vertical component of the displacement. With L = 12.0 m, we obtain Wg = 1547 J which should be rounded to three figures: 1.55 kJ. (d) The tension vector is everywhere perpendicular to the direction of motion, so its work is zero (since cos 90 = 0). (e) The implication of the previous three parts is that the work due to F is Wg (so the net work turns out to be zero). Thus, WF = Wg = 1.55 kJ. (f) Since F does not have constant magnitude, we cannot expect Eq. 7-8 to apply.
71. (a) Hooke's law and the work done by a spring is discussed in the chapter. Taking absolute values, and writing that law in terms of differences F and x , we analyze the first two pictures as follows:
| F | = k | x | 240 N - 110 N = k (60 mm - 40 mm)
which yields k = 6.5 N/mm. Designating the relaxed position (as read by that scale) as xo we look again at the first picture:
110 N = k (40 mm - xo )
which (upon using the above result for k) yields xo = 23 mm. (b) Using the results from part (a) to analyze that last picture, we find
W = k (30 mm - xo ) = 45 N .
72. (a) Using Eq. 7-8 and SI units, we find
W = F d = (2 ^ - 4 ^ (8 ^ + c ^ = 16 - 4c i j) i j)
which, if equal zero, implies c = 16/4 = 4 m. (b) If W > 0 then 16 > 4c, which implies c < 4 m. (c) If W < 0 then 16 < 4c, which implies c > 4 m.
73. A convenient approach is provided by Eq. 7-48. P = F v = (1800 kg + 4500 kg)(9.8 m/s2)(3.80 m/s) = 235 kW. Note that we have set the applied force equal to the weight in order to maintain constant velocity (zero acceleration).
74. (a) To estimate the area under the curve between x = 1 m and x = 3 m (which should yield the value for the work done), one can try "counting squares" (or half-squares or thirds of squares) between the curve and the axis. Estimates between 5 J and 8 J are typical for this (crude) procedure. (b) Eq. 7-32 gives
3
a a a 2 dx = 1x 3 1= 6J
where a = 9 Nm2 is given in the problem statement.
75. (a) Using Eq. 7-32, the work becomes W = 2 x2 x3 (SI units understood). The plot is shown below:
9
(b) We see from the graph that its peak value occurs at x = 3.00 m. This can be verified by taking the derivative of W and setting equal to zero, or simply by noting that this is where the force vanishes. (c) The maximum value is W = 2 (3.00)2 (3.00)3 = 13.50 J. (d) We see from the graph (or from our analytic expression) that W = 0 at x = 4.50 m. (e) The case is at rest when v = 0 . Since W = K = mv 2 / 2 , the condition implies W = 0 . This happens at x = 4.50 m.
9
76. The problem indicates that SI units are understood, so the result (of Eq. 7-23) is in Joules. Done numerically, using features available on many modern calculators, the result is roughly 0.47 J. For the interested student it might be worthwhile to quote the "exact" answer (in terms of the "error function"):
1.2 .15
e-2x dx = 2 [erf(6 2 /5) erf(3 2 /20)] .
77. (a) In 10 min the cart moves
d = 6.0
mi h 5280 ft/mi 60 min/h (10 min) = 5280 ft
so that Eq. 7-7 yields W = F d cos = (40 lb) (5280 ft) cos 30 = 1.8 105 ft lb. (b) The average power is given by Eq. 7-42, and the conversion to horsepower (hp) can be found on the inside back cover. We note that 10 min is equivalent to 600 s. Pavg 1.8 105 ft lb = = 305 ft lb / s 600 s
which (upon dividing by 550) converts to Pavg = 0.55 hp.
78. (a) Estimating the initial speed from the slope of the graph near the origin is somewhat difficult, and it may be simpler to determine it from the constant-acceleration 1 equations from chapter 2: v = v0 + at and x = v0 + 2 at 2 , where x0 = 0 has been used. Applying these to the last point on the graph (where the slope is apparently zero) or applying just the x equation to any two points on the graph, leads to a pair of simultaneous equations from which a = -2.0 m s 2 and v0 = 10 m s can be found. Then,
K0 =
1 2 mv0 = 2.5 103 J = 2.5 kJ. 2
(b) The speed at t = 3.0 s is obtained by v = v0 + at = 10 + (-2.0)(3.0) = 4.0 m/s or by estimating the slope from the graph (not recommended). Then the work-kinetic energy theorem yields
1 W = K = (50 kg)(4.0 m/s) 2 - 2.5 103 J= - 2.1 kJ . 2
79. (a) We set up the ratio
50 km E = 1 km 1 megaton
FG H
IJ K
1/ 3
and find E = 503 1 105 megatons of TNT. (b) We note that 15 kilotons is equivalent to 0.015 megatons. Dividing the result from part (a) by 0.013 yields about ten million bombs.
80. After converting the speed to meters-per-second, we find K = 2 mv2 = 667 kJ.
1