## 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.

**KinematicMeasurements**

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* = *rω* 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* = *r***ω**, we can derive the relationship that connects angular acceleration and linear acceleration.

*a* = *r*α

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 *a*_{t} to distinguish it from centripetal acceleration (*a*_{c}).

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

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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?

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By definition

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

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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 *a*_{c} = *v*^{2}/*r*. Using the fundamental equation *v* = *r*ω, we find that