# Rotational Motion and Angular Momentum

## Dynamics of Rotational Motion: Rotational Inertia


### Learning Objectives

By the end of this section, you will be able to:
1. Understand the relationship between force, mass and acceleration.
2. Study the turning effect of force.
3. Study the analogy between force and torque, mass and moment of inertia, and linear acceleration and angular acceleration.
If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity as seen in . In fact, your intuition is reliable in predicting many of the factors that are involved. For example, we know that a door opens slowly if we push too close to its hinges. Furthermore, we know that the more massive the door, the more slowly it opens. The first example implies that the farther the force is applied from the pivot, the greater the angular acceleration; another implication is that angular acceleration is inversely proportional to mass. These relationships should seem very similar to the familiar relationships among force, mass, and acceleration embodied in Newton’s second law of motion. There are, in fact, precise rotational analogs to both force and mass.
To develop the precise relationship among force, mass, radius, and angular acceleration, consider what happens if we exert a force  on a point mass  that is at a distance  from a pivot point, as shown in . Because the force is perpendicular to , an acceleration  is obtained in the direction of . We can rearrange this equation such that  and then look for ways to relate this expression to expressions for rotational quantities. We note that , and we substitute this expression into , yielding 
Recall that torque is the turning effectiveness of a force. In this case, because  is perpendicular to , torque is simply . So, if we multiply both sides of the equation above by , we get torque on the left-hand side. That is,



or 
This last equation is the rotational analog of Newton’s second law (), where torque is analogous to force, angular acceleration is analogous to translational acceleration, and  is analogous to mass (or inertia). The quantity  is called the rotational inertia or moment of inertia of a point mass  a distance  from the center of rotation.

### Rotational Inertia and Moment of Inertia

Before we can consider the rotation of anything other than a point mass like the one in , we must extend the idea of rotational inertia to all types of objects. To expand our concept of rotational inertia, we define the moment of inertia

 of an object to be the sum of

 for all the point masses of which it is composed. That is,

. Here

 is analogous to

 in translational motion. Because of the distance

, the moment of inertia for any object depends on the chosen axis. Actually, calculating

 is beyond the scope of this text except for one simple case—that of a hoop, which has all its mass at the same distance from its axis. A hoop’s moment of inertia around its axis is therefore

, where

 is its total mass and  its radius. (We use  and  for an entire object to distinguish them from  and  for point masses.) In all other cases, we must consult  (note that the table is piece of artwork that has shapes as well as formulae) for formulas for  that have been derived from integration over the continuous body. Note that  has units of mass multiplied by distance squared (), as we might expect from its definition.
The general relationship among torque, moment of inertia, and angular acceleration is 
or
where net  is the total torque from all forces relative to a chosen axis. For simplicity, we will only consider torques exerted by forces in the plane of the rotation. Such torques are either positive or negative and add like ordinary numbers. The relationship in  is the rotational analog to Newton’s second law and is very generally applicable. This equation is actually valid for any torque, applied to any object, relative to any axis.
As we might expect, the larger the torque is, the larger the angular acceleration is. For example, the harder a child pushes on a merry-go-round, the faster it accelerates. Furthermore, the more massive a merry-go-round, the slower it accelerates for the same torque. The basic relationship between moment of inertia and angular acceleration is that the larger the moment of inertia, the smaller is the angular acceleration. But there is an additional twist. The moment of inertia depends not only on the mass of an object, but also on its distribution of mass relative to the axis around which it rotates. For example, it will be much easier to accelerate a merry-go-round full of children if they stand close to its axis than if they all stand at the outer edge. The mass is the same in both cases, but the moment of inertia is much larger when the children are at the edge.

### Test Prep for AP Courses


### Section Summary

1. The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
2. If we exert a force  on a point mass  that is at a distance  from a pivot point and because the force is perpendicular to , an acceleration  is obtained in the direction of . We can rearrange this equation such that             
                and then look for ways to relate this expression to expressions for rotational quantities. We note that
 

          
3. Torque is the turning effectiveness of a force. In this case, because  is perpendicular to , torque is simply . If we multiply both sides of the equation above by , we get torque on the left-hand side. That is,
    
    or
    
4. The moment of inertia  of an object is the sum of  for all the point masses  of which it is composed. That is,
    
5. The general relationship among torque, moment of inertia, and angular acceleration is
    or 

### Conceptual Questions


### Problems & Exercises
