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PUZZLER
This sky diver is falling at more than
50 m/s (120 mi/h), but once her parachute opens, her downward velocity will
be greatly reduced. Why does she slow
down rapidly when her chute opens, enabling her to fall safely to the ground? If
the chute does not function properly, the
sky diver will almost certainly be seriously injured. What force exerted on
her limits her maximum speed?(Guy Savage/Photo Researchers, Inc.)

chapter

Circular Motion and Other
Applications of Newton’s Laws
Chapter Outline
6.1 Newton’s Second Law Applied to
Uniform Circular Motion

6.2 Nonuniform Circular Motion
6.3 (Optional) Motion in Accelerated

6.4 (Optional) Motion in the Presence
of Resistive Forces

6.5 (Optional) Numerical Modeling in
Particle Dynamics

Frames

151152

CHAPTER 6

Circular Motion and Other Applications of Newton’s Laws

I

n the preceding chapter we introduced Newton’s laws of motion and applied
them to situations involving linear motion. Now we discuss motion that is
slightly more complicated. For example, we shall apply Newton’s laws to objects
traveling in circular paths. Also, we shall discuss motion observed from anaccelerating frame of reference and motion in a viscous medium. For the most part, this
chapter is a series of examples selected to illustrate the application of Newton’s
laws to a wide variety of circumstances.

6.1

NEWTON’S SECOND LAW APPLIED TO
UNIFORM CIRCULAR MOTION

In Section 4.4 we found that a particle moving with uniform speed v in a circular
path of radius r experiences an accelerationar that has a magnitude
v2
r

ar

4.7

The acceleration is called the centripetal acceleration because ar is directed toward
the center of the circle. Furthermore, ar is always perpendicular to v. (If there
were a component of acceleration parallel to v, the particle’s speed would be
changing.)
Consider a ball of mass m that is tied to a string of length r and is being
whirled atconstant speed in a horizontal circular path, as illustrated in Figure 6.1.
Its weight is supported by a low-friction table. Why does the ball move in a circle?
Because of its inertia, the tendency of the ball is to move in a straight line; however, the string prevents motion along a straight line by exerting on the ball a
force that makes it follow the circular path. This force is directed alongthe string
toward the center of the circle, as shown in Figure 6.1. This force can be any one
of our familiar forces causing an object to follow a circular path.
If we apply Newton’s second law along the radial direction, we find that the
value of the net force causing the centripetal acceleration can be evaluated:

Force causing centripetal
acceleration

Fr

mar

m

v2
r

(6.1)m
Fr
r

Fr

Figure 6.1 Overhead view of a ball moving
in a circular path in a horizontal plane. A
force Fr directed toward the center of the circle keeps the ball moving in its circular path.

6.1

Newton’s Second Law Applied to Uniform Circular Motion

153

Figure 6.2 When the string breaks, the
ball moves in the direction tangent to the
circle.
r

A force causing acentripetal acceleration acts toward the center of the circular
path and causes a change in the direction of the velocity vector. If that force
should vanish, the object would no longer move in its circular path; instead, it
would move along a straight-line path tangent to the circle. This idea is illustrated
in Figure 6.2 for the ball whirling at the end of a string. If the string breaks at
someinstant, the ball moves along the straight-line path tangent to the circle at
the point where the string broke.

Quick Quiz 6.1
Is it possible for a car to move in a circular path in such a way that it has a tangential acceleration but no centripetal acceleration?

CONCEPTUAL EXAMPLE 6.1

An athlete in the process of throwing the hammer at the 1996
Olympic Games in Atlanta, Georgia. The...