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10-1 ROTATIONAL VARIABLES 259

letters) to sort out. (2) Although you are very familiar with linear motion (you
can get across the room and down the road just fine), you are probably very
unfamiliar with rotation (and that is one reason why you are willing to pay so
much for amusement park rides). If a homework problem looks like a foreign
language to you, see if translating it into the one-dimensional linear motion of
Chapter 2 helps. For example, if you are to find, say, an angulardistance, tem-
porarily delete the word angularand see if you can work the problem with the
Chapter 2 notation and ideas.

Rotational Variables

We wish to examine the rotation of a rigid body about a fixed axis. A rigid bodyis
a body that can rotate with all its parts locked together and without any change in
its shape. A fixed axismeans that the rotation occurs about an axis that does not
move. Thus, we shall not examine an object like the Sun, because the parts of the
Sun (a ball of gas) are not locked together. We also shall not examine an object
like a bowling ball rolling along a lane, because the ball rotates about a moving
axis (the ball’s motion is a mixture of rotation and translation).
Figure 10-2 shows a rigid body of arbitrary shape in rotation about a fixed
axis, called the axis of rotationor the rotation axis.In pure rotation (angular
motion), every point of the body moves in a circle whose center lies on the axis of
rotation, and every point moves through the same angle during a particular time
interval. In pure translation (linear motion), every point of the body moves in a
straight line, and every point moves through the same linear distanceduring a
particular time interval.
We deal now — one at a time — with the angular equivalents of the linear
quantities position, displacement, velocity, and acceleration.

Angular Position

Figure 10-2 shows a reference line, fixed in the body, perpendicular to the rotation
axis and rotating with the body. The angular positionof this line is the angle of
the line relative to a fixed direction, which we take as the zero angular position.
In Fig. 10-3, the angular position uis measured relative to the positive direction of
thexaxis. From geometry, we know that uis given by

(radian measure). (10-1)

Heresis the length of a circular arc that extends from the xaxis (the zero angular
position) to the reference line, and ris the radius of the circle.

u

s

r

Figure 10-2A rigid body of arbitrary shape in pure rotation about the zaxis of a coordinate
system. The position of the reference linewith respect to the rigid body is arbitrary, but it is
perpendicular to the rotation axis. It is fixed in the body and rotates with the body.

z

O

Reference line

Rotation

axis

x

y

Body This reference line is part of the body

and perpendicular to the rotation axis.

We use it to measure the rotation of the

body relative to a fixed direction.

Figure 10-3The rotating rigid body of

Fig. 10-2 in cross section, viewed from

above. The plane of the cross section is

perpendicular to the rotation axis, which

now extends out of the page, toward you.

In this position of the body, the reference

line makes an angle uwith the xaxis.

x

y

Referenceline

θ

r

s

Rotation

axis

The body has rotated

counterclockwise

by angle. This is the

positive direction.

θ

This dot means that

the rotation axis is

out toward you.