Rotational motion

11-1 Translation and Rotation

The graceful movement of figure skaters can be used to illustrate two kinds of pure, or unmixed, motion. If a skater is gliding across the ice in a straight line with constant speed, her motion is one of pure translation. If she spinning at a constant rate about a vertical axis, in a motion of pure rotation. Translation is motion along a straight line, which has been our focus up to now. Rotation is the motion of wheels, gears, motors, planets, the hands of clocks, the rotors of jet engines, and the blades of helicopters. It is our focus in this chapter.

11-2 The Rotational Variables

We wish to examine the rotation of a rigid body about a fixed axis. A rigid body is a body that can rotate with all its parts locked together and without any change in its shape. A fixed axis means 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 bowling alley, because the ball rotates about an axis that moves (the ball's motion is a mixture of rotation and translation).

Figure 11-2 shows a rigid body of arbitrary shape in rotation about a fixed axis, called the axis of rotation or the rotation axis. 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, every point of the body moves in a straight line, and every point moves through the same linear distance during a particular time interval. (Comparisons between angular and linear motion will appear throughout this chapter.)

We deal now - one at a time - with the angular equivalents of the linear quantities position, displacement, velocity, and acceleration.

Angular Position

Figure 11-2

Figure 11-2 shows a reference line, fixed in the body, perpendicular to the rotation axis, and rotating with the body. The angular position of this line is the angle of the line relative to a fixed direction, which we take as the zero angular position. In Fig. 11-3, the angular position is measured relative to the positive direction of the axis. From geometry, we know that is

Here is the length of arc (or the arc distance) along a circle and between the axis (the zero angular position) and the reference line; is the radius of that circle.

An angle defined in this way is measured in radians (rad) rather than in revolutions (rev) or degrees. The radian, being the ratio of two lengths, is a pure number and thus has no dimension. Because the circumference of a circle of radius is , there are radians in a complete circle:

Fig. 11-3

and thus

We do not reset to zero with each complete rotation of the reference line about the rotation axis. If the reference line completes two revolutions from the zero angular position, then the angular position of the line is

rad.

For pure translational motion along the direction, we can know all there is to know about a moving body if we are given , its position as a function of time. Similarly, for pure rotation, we can know all there is to know about a rotating body if we are given , the angular position of the body's reference line as a function of time.