SAT Physics Subject Test

Chapter 6 Curved and Rotational Motion

ROTATIONAL KINEMATICS

If we mark several dots along a radius on a disk and call this radius the reference line, and the disk rotates around its center, we can use the movement of these dots to talk about angular displacement, angular velocity, and angular acceleration.

Kinematic
Measurements

The angular kinematic
qualities of θ, ω, α,
and τ are analogous to the
linear kinematics s, v, a,
and F for distance, velocity,
acceleration, and force,
respectively.

If the disk rotates as a rigid body, then all three dots shown have the same angular displacement, ∆θ. In fact, this is the definition of a rigid body: In a rigid body, all points along a radial line always have the same angular displacement.

Just as the time rate-of-change of displacement gives velocity, the time rate-of-change of angular displacement gives angular velocity, symbolized by ω (omega).


The definition of the average angular velocity is


Finally, just as the time rate-of-change of velocity gives acceleration, the time rate-of-change of angular velocity gives angular acceleration, or α (alpha).


The definition of the average angular acceleration is


On the rotating disk illustrated on the previous page, we said that all points undergo the same angular displacement at any given time interval; this means that all points on the disk have the same angular velocity, ω, but not all points have the same linear velocity, v. This follows from the definition of radian measure. Expressed in radians, the angular displacement, ∆θ, is related to the arc length, ∆s, by the equation

Rearranging this equation and dividing by ∆t, we find that

Therefore, the greater the value of r, or v =  the greater the value of v. Points on the rotating body farther from the rotation axis move faster than those closer to the rotation axis.

From the equation v = rω, we can derive the relationship that connects angular acceleration and linear acceleration.

a = rα

It’s important to realize that the acceleration a in this equation is not centripetal acceleration, but rather tangential acceleration, which arises from a change in speed caused by an angular acceleration. By contrast, centripetal acceleration does not produce a change in speed. Often, tangential acceleration is written as at to distinguish it from centripetal acceleration (ac).

13. A rotating, rigid body makes one complete revolution in 2 s. What is its average angular velocity?

Here’s How to Crack It

One complete revolution is equal to an angular displacement of 2π radians, so the body’s average angular velocity is

14. The angular velocity of a rotating disk increases from 2 rad/s to 5 rad/s in 0.5 s. What’s the disk’s average angular acceleration?

Here’s How to Crack It

By definition

15. Derive an expression for centripetal acceleration in terms of angular speed.

Here’s How to Crack It

For an object revolving with linear speed v at a distance r from the center of rotation, the centripetal acceleration is given by the equation ac = v2/r. Using the fundamental equation v = rω, we find that