# Circular and Rotational Motion

## Rotational Motion


### Section Key Terms


### Rotational Kinematics

In the section on uniform circular motion, we discussed motion in a circle at constant speed and, therefore, constant angular velocity. However, there are times when angular velocity is not constant—rotational motion can speed up, slow down, or reverse directions. Angular velocity is not constant when a spinning skater pulls in her arms, when a child pushes a merry-go-round to make it rotate, or when a CD slows to a halt when switched off. In all these cases, angular acceleration occurs because the angular velocity  
   changes. The faster the change occurs, the greater is the angular acceleration. Angular acceleration  
   is the rate of change of angular velocity. In equation form, average angular acceleration is 

where  
   is the change in angular velocity and  
   is the change in time. The units of angular acceleration are (rad/s)/s, or rad/s2. If  
   increases, then  
   is positive. If  
   decreases, then  
   is negative. Keep in mind that, by convention, counterclockwise is the positive direction and clockwise is the negative direction. For example, the skater in  is rotating counterclockwise as seen from above, so her angular velocity is positive. Acceleration would be negative, for example, when an object that is rotating counterclockwise slows down. It would be positive when an object that is rotating counterclockwise speeds up.

The relationship between the magnitudes of tangential acceleration, a, and angular acceleration,
These equations mean that the magnitudes of tangential acceleration and angular acceleration are directly proportional to each other. The greater the angular acceleration, the larger the change in tangential acceleration, and vice versa. For example, consider riders in their pods on a Ferris wheel at rest. A Ferris wheel with greater angular acceleration will give the riders greater tangential acceleration because, as the Ferris wheel increases its rate of spinning, it also increases its tangential velocity. Note that the radius of the spinning object also matters. For example, for a given angular acceleration  
    , a smaller Ferris wheel leads to a smaller tangential acceleration for the riders.
So far, we have defined three rotational variables:  
    ,  
    , and  
    .  These are the angular versions of the linear variables x, v, and a. The following equations in the table represent the magnitude of the rotational variables and only when the radius is constant and perpendicular to the rotational variable.  shows how they are related.
We can now begin to see how rotational quantities like  
    ,  
    , and  
       are related to each other. For example, if a motorcycle wheel that starts at rest has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. Putting this in terms of the variables, if the wheel’s angular acceleration  
      is large for a long period of time t, then the final angular velocity  
      and angle of rotation  
      are large. In the case of linear motion, if an object starts at rest and undergoes a large linear acceleration, then it has a large final velocity and will have traveled a large distance.
The kinematics of rotational motion describes the relationships between the angle of rotation, angular velocity, angular acceleration, and time. It only describes motion—it does not include any forces or masses that may affect rotation (these are part of dynamics). Recall the kinematics equation for linear motion:  
      (constant a).
As in linear kinematics, we assume a is constant, which means that angular acceleration  
      is also a constant, because  
     . The equation for the kinematics relationship between  
    ,  
    , and t is
 
     
  
where  
      is the initial angular velocity. Notice that the equation is identical to the linear version, except with angular analogs of the linear variables. In fact, all of the linear kinematics equations have rotational analogs, which are given in . These equations can be used to solve rotational or linear kinematics problem in which a and  
      are constant.
In these equations,  
      and  
      are initial values,  
      is zero, and the average angular velocity  
      and average velocity  
      are 

     

  

### Torque

If you have ever spun a bike wheel or pushed a merry-go-round, you know that force is needed to change angular velocity. The farther the force is applied from the pivot point (or fulcrum), the greater the angular acceleration. For example, a door opens slowly if you push too close to its hinge, but opens easily if you push far from the hinges. Furthermore, we know that the more massive the door is, the more slowly it opens; this is because angular acceleration is inversely proportional to mass. These relationships are very similar to the relationships between force, mass, and acceleration from Newton’s second law of motion. Since we have already covered the angular versions of distance, velocity and time, you may wonder what the angular version of force is, and how it relates to linear force.
The angular version of force is torque  
      , which is the turning effectiveness of a force. See . The equation for the magnitude of torque is 
where r is the magnitude of the lever arm, F is the magnitude of the linear force, and  
   is the angle between the lever arm and the force. The lever arm is the vector from the point of rotation (pivot point or fulcrum) to the location where force is applied. Since the magnitude of the lever arm is a distance, its units are in meters, and torque has units of N⋅m. Torque is a vector quantity and has the same direction as the angular acceleration that it produces.
Applying a stronger torque will produce a greater angular acceleration. For example, the harder the man pushes the merry-go-round in , the faster it accelerates. Furthermore, the more massive the merry-go-round is, the slower it accelerates for the same torque. If the man wants to maximize the effect of his force on the merry-go-round, he should push as far from the center as possible to get the largest lever arm and, therefore, the greatest torque and angular acceleration. Torque is also maximized when the force is applied perpendicular to the lever arm.

### Solving Rotational Kinematics and Torque Problems

Just as linear forces can balance to produce zero net force and no linear acceleration, the same is true of rotational motion. When two torques of equal magnitude act in opposing directions, there is no net torque and no angular acceleration, as you can see in the following video. If zero net torque acts on a system spinning at a constant angular velocity, the system will continue to spin at the same angular velocity.
Now let’s look at examples applying rotational kinematics to a fishing reel and the concept of torque to a merry-go-round.

### Practice Problems


### Check Your Understanding


### Section Summary

1. Kinematics is the description of motion.
2. The kinematics of rotational motion describes the relationships between rotation angle, angular velocity, angular acceleration, and time.
3. Torque is the effectiveness of a force to change the rotational speed of an object. Torque is the rotational analog of force.
4. The lever arm is the distance between the point of rotation (pivot point) and the location where force is applied.
5. Torque is maximized by applying force perpendicular to the lever arm and at a point as far as possible from the pivot point or fulcrum. If torque is zero, angular acceleration is zero.

### Key Equations


### Concept Items


### Critical Thinking


### Problems


### Performance Task


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
