Circular Motion and Other Applications of Newton's Laws

Some examples of forces acting during circular motion. (Left) As these speed skaters round a
curve, the force exerted by the ice on their skates provides the centripetal acceleration.
(Right) Passengers on a “corkscrew” roller coaster. What are the origins of the forces in this
example?
F r
F t
F
particle must also have a tangential and a radial component. Because the total acceleration
is a � a r � a t, the total force exerted on the particle is F � F r � F t, as
shown in Figure 6.10. The vector F r is directed toward the center of the circle and is
responsible for the centripetal acceleration. The vector F t tangent to the circle is responsible
for the tangential acceleration, which represents a change in the speed of
the particle with time. The following example demonstrates this type of motion.
EXAMPLE 6.8
6.2 Nonuniform Circular Motion 159
Figure 6.10 When the force acting on a particle moving
in a circular path has a tangential component F t, the
particle’s speed changes. The total force exerted on the
particle in this case is the vector sum of the radial force
and the tangential force. That is, F � F r � F t.
Keep Your Eye on the Ball
A small sphere of mass m is attached to the end of a cord of
length R and whirls in a vertical circle about a fixed point O,
as illustrated in Figure 6.11a. Determine the tension in the
cord at any instant when the speed of the sphere is v and the
cord makes an angle � with the vertical.
Solution Unlike the situation in Example 6.7, the speed is
not uniform in this example because, at most points along the
path, a tangential component of acceleration arises from the
gravitational force exerted on the sphere. From the free-body
diagram in Figure 6.11b, we see that the only forces acting on