Rotation of Rigid Bodies Chapter 9 Rotation of Rigid Bodies

A speedometer as our starting model A car’s analog speedometer gives us a very good example to begin defining rotational motion. Consider the clockwise (or counterclockwise) motion of a rigid, fixed-length speedometer needle about a fixed pivot point.

Angular motions in revolutions, degrees, and radians One complete cycle of 360° is one revolution. One complete revolution is 2π radians. Relating the two, 360° = 2 π radians or 1 radian = 57.3°.

Angular displacement is the angle being swept out Like a second hand sweeping around a clock, a radius vector will travel through a displacement of degrees, radians, or revolutions. We denote angular displacement as Θ (theta). It is the angular equivalent of x or y in earlier chapters.

Angular velocity The angular velocity is the angle swept out divided by the time it took to sweep out the angular displacement. Angular velocity is denoted by the symbol ω (omega). Angular velocity is measured in radians per second (SI standard) as well as other measures such as r.p.m. (revolutions per second). Refer to Example 9.1.

Angular velocity is a vector You can visualize the position of the vector by sweeping out the angle with the fingers of your right hand. The position of your thumb will be the position of the angular velocity vector. This is called the “right-hand rule.”

Angular acceleration The angular acceleration is the change of angular velocity divided by the time interval during which the change occurred. Use the symbol α (alpha) to denote radians per second2. Refer to Example 9.2.

Angular acceleration is a vector The angular acceleration vector will be parallel or antiparallel to the angular velocity vector (as determined by the RHR).

We have four fundamental equations for angular kinematics

The circular motion of an audio CD—Example 9.3 Users may not even think about it, but the CD in a personal player is rotating rapidly. Consider Example 9.3.

Linear and angular quantities related Once upon a time, we played music with flat vinyl disks rotating at 33 1/3 r.p.m. A bug walking from the spindle out toward the edge will experience the same angular displacement, velocity, and acceleration, but as the radius changes, the tangential variable changes (as the bug will discover when it flies off the record).

An athlete throwing the discus Refer to Example 9.4 and Figure 9.12.

An airplane propeller Refer to Example 9.5 and Figure 9.13.

Rotational energy Just like linear kinetic energy is ½ mv2, the angular energy will be determined by ½ Iω2.

Rotational energy changes if parts shift and I changes Even if the masses are equal, rearranging the components of a rotating system can change the moment of inertia and the rotational energy. Refer to Example 9.7.

Finding the moment of inertia for common shapes

Calculating rotational energy Consider Problem-Solving Strategy 9.1. Follow Example 9.8.

Variations on Example 9.8 Refer to Example 9.9.

Parallel Axis Theorem Example 9.10.