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268 CHAPTER 10 ROTATION

Relating the Linear and Angular Variables

In Module 4-5, we discussed uniform circular motion, in which a particle travels at con-

stant linear speed valong a circle and around an axis of rotation. When a rigid body,

such as a merry-go-round, rotates around an axis, each particle in the body moves in its

own circle around that axis. Since the body is rigid, all the particles make one revolu-

tion in the same amount of time; that is, they all have the same angular speed v.

However, the farther a particle is from the axis, the greater the circumference

of its circle is, and so the faster its linear speed vmust be. You can notice this on a

merry-go-round. You turn with the same angular speed vregardless of your dis-

tance from the center, but your linear speed vincreases noticeably if you move to

the outside edge of the merry-go-round.

We often need to relate the linear variables s,v, and afor a particular point in

a rotating body to the angular variables u,v, and afor that body. The two sets of

variables are related by r, the perpendicular distanceof the point from the

rotation axis. This perpendicular distance is the distance between the point and

the rotation axis, measured along a perpendicular to the axis. It is also the radius r

of the circle traveled by the point around the axis of rotation.

(b) How much time did the speed decrease take?

Calculation:Now that we know a, we can use Eq. 10-12 to
solve for t: 46.5 s. (Answer)

t

vv 0

a



2.00 rad/s3.40 rad/s

0.0301 rad/s^2

10-3RELATING THE LINEAR AND ANGULAR VARIABLES

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

10.15For a rigid body rotating about a fixed axis, relate the angular
variables of the body (angular position, angular velocity, and an-
gular acceleration) and the linear variables of a particle on the
body (position, velocity, and acceleration) at any given radius.

10.16Distinguish between tangential acceleration and radial

acceleration, and draw a vector for each in a sketch of a

particle on a body rotating about an axis, for both an in-

crease in angular speed and a decrease.

●A point in a rigid rotating body, at a perpendicular distance
rfrom the rotation axis, moves in a circle with radius r. If the
body rotates through an angle u, the point moves along an
arc with length sgiven by

sur (radian measure),

whereuis in radians.

●The linear velocity of the point is tangent to the circle; the
point’s linear speed vis given by

vvr (radian measure),

wherevis the angular speed (in radians per second) of the body,
and thus also the point.

v:

●The linear acceleration of the point has both tangential

and radial components. The tangential component is

atar (radian measure),

whereais the magnitude of the angular acceleration (in radi-

ans per second-squared) of the body. The radial component

of is

(radian measure).

●If the point moves in uniform circular motion, the period Tof

the motion for the point and the body is

T (radian measure).

2 pr

v



2 p

v

ar

v^2

r

v^2 r

:a

:a

Learning Objectives

Key Ideas

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