9781118230725.pdf

Uniform Circular Motion

From Module 4-5, recall that when a body moves in a circle (or a circular arc) at
constant speed v, it is said to be in uniform circular motion. Also recall that the
body has a centripetal acceleration (directed toward the center of the circle) of
constant magnitude given by

(centripetal acceleration), (6-17)

whereRis the radius of the circle. Here are two examples:

1. Rounding a curve in a car.You are sitting in the center of the rear seat of a car
   moving at a constant high speed along a flat road. When the driver suddenly
   turns left, rounding a corner in a circular arc, you slide across the seat toward the
   right and then jam against the car wall for the rest of the turn. What is going on?
   While the car moves in the circular arc, it is in uniform circular motion;
   that is, it has an acceleration that is directed toward the center of the circle.
   By Newton’s second law, a force must cause this acceleration. Moreover, the
   force must also be directed toward the center of the circle. Thus, it is a cen-
   tripetal force,where the adjective indicates the direction. In this example, the
   centripetal force is a frictional force on the tires from the road; it makes the
   turn possible.
   If you are to move in uniform circular motion along with the car, there
   must also be a centripetal force on you. However, apparently the frictional
   force on you from the seat was not great enough to make you go in a circle
   with the car. Thus, the seat slid beneath you, until the right wall of the car
   jammed into you. Then its push on you provided the needed centripetal force
   on you, and you joined the car’s uniform circular motion.


2. Orbiting Earth.This time you are a passenger in the space shuttle Atlantis.As
   it and you orbit Earth, you float through your cabin. What is going on?
   Both you and the shuttle are in uniform circular motion and have acceler-
   ations directed toward the center of the circle. Again by Newton’s second law,
   centripetal forces must cause these accelerations. This time the centripetal
   forces are gravitational pulls (the pull on you and the pull on the shuttle) ex-
   erted by Earth and directed radially inward, toward the center of Earth.

a

v^2

R

6-3 UNIFORM CIRCULAR MOTION 133

6-3UNIFORM CIRCULAR MOTION

After reading this module, you should be able to...
6.06Sketch the path taken in uniform circular motion and
explain the velocity, acceleration, and force vectors
(magnitudes and directions) during the motion.
6.07ldentify that unless there is a radially inward net force
(a centripetal force), an object cannot move in circular motion.

6.08For a particle in uniform circular motion, apply the rela-

tionship between the radius of the path, the particle’s

speed and mass, and the net force acting on the particle.

●If a particle moves in a circle or a circular arc of radius Rat
constant speed v,the particle is said to be in uniform circular
motion. It then has a centripetal acceleration with magni-
tude given by

a

v^2

R

.

a:

●This acceleration is due to a net centripetal force on the

particle, with magnitude given by

,

wheremis the particle’s mass. The vector quantities and

are directed toward the center of curvature of the particle’s path.

F

:

a:

F

mv^2

R

Learning Objectives

Key Ideas