9781118230725.pdf

76 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS

Uniform Circular Motion

A particle is in uniform circular motionif it travels around a circle or a circular

arc at constant (uniform) speed. Although the speed does not vary,the particle is

acceleratingbecause the velocity changes in direction.

Figure 4-16 shows the relationship between the velocity and acceleration

vectors at various stages during uniform circular motion. Both vectors have

constant magnitude, but their directions change continuously. The velocity is

always directed tangent to the circle in the direction of motion. The accelera-

tion is always directed radially inward.Because of this, the acceleration associ-

ated with uniform circular motion is called a centripetal(meaning “center seek-

ing”)acceleration.As we prove next, the magnitude of this acceleration is

(centripetal acceleration), (4-34)

whereris the radius of the circle and vis the speed of the particle.

In addition, during this acceleration at constant speed, the particle travels the

circumference of the circle (a distance of 2pr) in time

(period). (4-35)

Tis called the period of revolution,or simply the period,of the motion. It is, in

general, the time for a particle to go around a closed path exactly once.

Proof of Eq. 4-34

To find the magnitude and direction of the acceleration for uniform circular

motion, we consider Fig. 4-17. In Fig. 4-17a, particle pmoves at constant speed

varound a circle of radius r. At the instant shown,phas coordinates xpandyp.

Recall from Module 4-2 that the velocity of a moving particle is always

tangent to the particle’s path at the particle’s position. In Fig. 4-17a, that means

is perpendicular to a radius rdrawn to the particle’s position. Then the angle

uthat makes with a vertical at pequals the angle uthat radius rmakes with

thexaxis.

v:

:v

:v

T

2

 r

v

a

v^2

r

a:

Figure 4-16Velocity and acceleration

vectors for uniform circular motion.

v

v

v

a

a a

The acceleration vector

always points toward the

center.

The velocity

vector is always

tangent to the path.

4-5UNIFORM CIRCULAR MOTION

4.17 Apply the relationships between the radius of the circu-

lar path, the period, the particle’s speed, and the particle’s

acceleration magnitude.

Learning Objectives

After reading this module, you should be able to...

4.16Sketch the path taken in uniform circular motion and ex-

plain the velocity and acceleration vectors (magnitude and

direction) during the motion.

arc, and is said to be centripetal. The time for the particle to

complete a circle is

Tis called the period of revolution, or simply the period, of the

motion.

T

2

 r

v

.

:a

Key Ideas

●If a particle travels along a circle or circular arc of radius rat

constant speed v, it is said to be in uniform circular motion

and has an acceleration of constant magnitude

The direction of is toward the center of the circle or circular:a

a

v^2

r

.

:a