Chapter 5
Forces and Motion II
5.1
The Important Stuff
5.1.1
Friction Forces
Forces which are known collectively as “friction forces” are all around us in daily life. In
elementary physics we discuss the friction force as it occurs between two objects whose
surfaces are in contact and which slide against one another.
If in such a situation, a body is
not moving
while an applied force
F
acts on it, then
static friction
forces are opposing the applied force, resulting in zero net force. Empirically,
one finds that this force can have a
maximum
value given by:
f
max
s
=
μ
s
N
(5.1)
where
μ
s
is the
coefficient of static friction
for the two surfaces and
N
is the normal
(perpendicular) force between the two surfaces.
If one object is in motion relative to the other one (i.e. it is sliding on the surface) then
there is a force of
kinetic friction
between the two objects. The direction of this force is
such as to oppose the sliding motion and its magnitude is given by
f
k
=
μ
k
N
(5.2)
where again
N
is the normal force between the two objects and
μ
k
is the
coefficient of
kinetic friction
for the two surfaces.
5.1.2
Uniform Circular Motion Revisited
Recall the result given in Chapter 3: When an object is in uniform circular motion, moving
in a circle of radius
r
with speed
v
, the acceleration is directed toward the center of the circle
and has magnitude
a
cent
=
v
2
r
.
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CHAPTER 5.
FORCES AND MOTION II
Therefore, by Newton’s Second Law of Motion, the
net force
on this object must also be
directed toward the center of the circle and have magnitude
F
cent
=
mv
2
r
.
(5.3)
Such a force is called a
centripetal force
, as indicated in this equation.
5.1.3
Newton’s Law of Gravity (Optional for Calculus–Based)
The force of gravity is one of the fundamental forces in nature. Although in our first physics
examples we only dealt with the fact that the
earth
pulls downward on all masses, in fact all
masses exert an attractive gravitational force on each other, but for most objects the force
is so small that we can ignore it.
Newton’s Law of Gravity
says that for two masses
m
1
and
m
2
separated by a distance
r
, the magnitude of the (attractive) gravitational force is
F
=
G
m
1
m
2
r
2
where
G
= 6
.
67
×
10

11 N
·
m
2
kg
2
(5.4)
While the law as given really applies to
point
(i.e. small) masses, it can be used for
spherical
masses as long as we take
r
to be the distance between the centers of the two masses.
5.2
Worked Examples
5.2.1
Friction Forces
1. An ice skater moving at
12
m
s
coasts to a halt in
95 m
on an ice surface. What
is the coefficient of (kinetic) friction between ice and skates?
[Ser4 551]
The information which we are given about the skater’s (onedimensional) motion is shown
in Fig. 5.1(a). We know that the skater’s notion is one of constant acceleration so we can use
the results in Chapter 2. In particular we know the initial and final velocities of the skater:
v
0
= 12
m
s
v
= 0
and we know the displacement for this interval:
x

x
0
= 95 m
we can use 2.8 to find the (constant) acceleration
a
. We find:
v
2
x
=
v
2
0
x
+ 2
a
x
(
x

x
0
)
=
⇒
a
x
=
(
v
2
x

v
2
0
x
)
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 Spring '11
 Chiu
 Physics, Force, Friction

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