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10-3 RELATING THE LINEAR AND ANGULAR VARIABLES 269

The Position

If a reference line on a rigid body rotates through an angle u, a point within the
body at a position rfrom the rotation axis moves a distance salong a circular arc,
wheresis given by Eq. 10-1:

sur (radian measure). (10-17)

This is the first of our linear – angular relations.Caution:The angle uhere must be
measured in radians because Eq. 10-17 is itself the definition of angular measure
in radians.

The Speed

Differentiating Eq. 10-17 with respect to time — with rheld constant — leads to

However,ds/dtis the linear speed (the magnitude of the linear velocity) of the
point in question, and du/dtis the angular speed vof the rotating body. So

vvr (radian measure). (10-18)

Caution:The angular speed vmust be expressed in radian measure.
Equation 10-18 tells us that since all points within the rigid body have the
same angular speed v, points with greater radius rhave greater linear speed v.
Figure 10-9areminds us that the linear velocity is always tangent to the circular
path of the point in question.
If the angular speed vof the rigid body is constant, then Eq. 10-18 tells
us that the linear speed vof any point within it is also constant. Thus, each point
within the body undergoes uniform circular motion. The period of revolution T
for the motion of each point and for the rigid body itself is given by Eq. 4-35:

. (10-19)

This equation tells us that the time for one revolution is the distance 2prtraveled
in one revolution divided by the speed at which that distance is traveled.
Substituting for vfrom Eq. 10-18 and canceling r, we find also that

(radian measure). (10-20)

This equivalent equation says that the time for one revolution is the angular dis-
tance 2prad traveled in one revolution divided by the angular speed (or rate) at
which that angle is traveled.

The Acceleration

Differentiating Eq. 10-18 with respect to time — again with rheld constant —
leads to
(10-21)

Here we run up against a complication. In Eq. 10-21,dv/dtrepresents only the
part of the linear acceleration that is responsible for changes in the magnitude v
of the linear velocity. Like , that part of the linear acceleration is tangent to
the path of the point in question. We call it the tangential component atof the lin-
ear acceleration of the point, and we write

atar (radian measure), (10-22)

v: v:

dv

dt



dv

dt

r.

T

2 p

v

T

2 pr

v

ds

dt



du

dt

r.

Figure 10-9The rotating rigid body of Fig. 10-2,

shown in cross section viewed from above.

Every point of the body (such as P) moves

in a circle around the rotation axis. (a)The

linear velocity of every point is tangent to

the circle in which the point moves. (b) The

linear acceleration of the point has (in

general) two components: tangential atand

radialar.

:a

:v

x

y

r

Rotation

axis

P

Circle

traveled by P

(a)

v

The velocity vector is

always tangent to this

circle around the

rotation axis.

x

y

ar

P

(b)

at

Rotation

axis

The acceleration always

has a radial (centripetal)

component and may have

a tangential component.