# Rotational Motion and Angular Momentum

## Angular Acceleration


### Learning Objectives

By the end of this section, you will be able to:
1. Describe uniform circular motion.
2. Explain non-uniform circular motion.
3. Calculate angular acceleration of an object.
4. Observe the link between linear and angular acceleration.
Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity  was defined as the time rate of change of angle :
where  is the angle of rotation as seen in . The relationship between angular velocity  and linear velocity  was also defined in Rotation Angle and Angular Velocity as
or
where  is the radius of curvature, also seen in . According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative 
Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer’s hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration, in which  changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration  is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:
where  is the change in angular velocity and  is the change in time. The units of angular acceleration are , or . If  increases, then  is positive. If  decreases, then  is negative.
If the bicycle in the preceding example had been on its wheels instead of upside-down, it would first have accelerated along the ground and then come to a stop. This connection between circular motion and linear motion needs to be explored. For example, it would be useful to know how linear and angular acceleration are related. In circular motion, linear acceleration is tangent to the circle at the point of interest, as seen in . Thus, linear acceleration is called tangential acceleration .
Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction. We know from Uniform Circular Motion and Gravitation that in circular motion centripetal acceleration,

, refers to changes in the direction of the velocity but not its magnitude. An object undergoing circular motion experiences centripetal acceleration, as seen in . Thus,

 and

 are perpendicular and independent of one another. Tangential acceleration

 is directly related to the angular acceleration

 and is linked to an increase or decrease in the velocity, but not its direction.
Now we can find the exact relationship between linear acceleration  and angular acceleration . Because linear acceleration is proportional to a change in the magnitude of the velocity, it is defined (as it was in One-Dimensional Kinematics) to be
For circular motion, note that , so that
The radius  is constant for circular motion, and so . Thus,
By definition, . Thus,
or
These equations mean that linear acceleration and angular acceleration are directly proportional. The greater the angular acceleration is, the larger the linear (tangential) acceleration is, and vice versa. For example, the greater the angular acceleration of a car’s drive wheels, the greater the acceleration of the car. The radius also matters. For example, the smaller a wheel, the smaller its linear acceleration for a given angular acceleration .
So far, we have defined three rotational quantities— , and . These quantities are analogous to the translational quantities , and .  displays rotational quantities, the analogous translational quantities, and the relationships between them.

### Section Summary

1. Uniform circular motion is the motion with a constant angular velocity .
2. In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is .
3. Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as .
4. For circular motion, note that , so that
    
5. The radius r is constant for circular motion, and so . Thus,
    
6. By definition, . Thus,
    or
    

### Conceptual Questions


### Problems & Exercises
