# Rotational Motion and Angular Momentum

## Rotational Kinetic Energy: Work and Energy Revisited


### Learning Objectives

By the end of this section, you will be able to:
1. Derive the equation for rotational work.
2. Calculate rotational kinetic energy.
3. Demonstrate the Law of Conservation of Energy.
In this module, we will learn about work and energy associated with rotational motion.  shows a worker using an electric grindstone propelled by a motor. Sparks are flying, and noise and vibration are created as layers of steel are pared from the pole. The stone continues to turn even after the motor is turned off, but it is eventually brought to a stop by friction. Clearly, the motor had to work to get the stone spinning. This work went into heat, light, sound, vibration, and considerable rotational kinetic energy.
Work must be done to rotate objects such as grindstones or merry-go-rounds. Work was defined in Uniform Circular Motion and Gravitation for translational motion, and we can build on that knowledge when considering work done in rotational motion. The simplest rotational situation is one in which the net force is exerted perpendicular to the radius of a disk (as shown in ) and remains perpendicular as the disk starts to rotate. The force is parallel to the displacement, and so the net work done is the product of the force times the arc length traveled:
To get torque and other rotational quantities into the equation, we multiply and divide the right-hand side of the equation by , and gather terms:
We recognize that  and , so that
This equation is the expression for rotational work. It is very similar to the familiar definition of translational work as force multiplied by distance. Here, torque is analogous to force, and angle is analogous to distance. The equation  is valid in general, even though it was derived for a special case.
To get an expression for rotational kinetic energy, we must again perform some algebraic manipulations. The first step is to note that , so that
Now, we solve one of the rotational kinematics equations for . We start with the equation
Next, we solve for :
Substituting this into the equation for net  and gathering terms yields
This equation is the work-energy theorem for rotational motion only. As you may recall, net work changes the kinetic energy of a system. Through an analogy with translational motion, we define the term  to be rotational kinetic energy  for an object with a moment of inertia  and an angular velocity :
The expression for rotational kinetic energy is exactly analogous to translational kinetic energy, with  being analogous to  and  to . Rotational kinetic energy has important effects. Flywheels, for example, can be used to store large amounts of rotational kinetic energy in a vehicle, as seen in .
Helicopter pilots are quite familiar with rotational kinetic energy. They know, for example, that a point of no return will be reached if they allow their blades to slow below a critical angular velocity during flight. The blades lose lift, and it is impossible to immediately get the blades spinning fast enough to regain it. Rotational kinetic energy must be supplied to the blades to get them to rotate faster, and enough energy cannot be supplied in time to avoid a crash. Because of weight limitations, helicopter engines are too small to supply both the energy needed for lift and to replenish the rotational kinetic energy of the blades once they have slowed down. The rotational kinetic energy is put into them before takeoff and must not be allowed to drop below this crucial level. One possible way to avoid a crash is to use the gravitational potential energy of the helicopter to replenish the rotational kinetic energy of the blades by losing altitude and aligning the blades so that the helicopter is spun up in the descent. Of course, if the helicopter’s altitude is too low, then there is insufficient time for the blade to regain lift before reaching the ground.

### How Thick Is the Soup? Or Why Don’t All Objects Roll Downhill at the Same Rate?

One of the quality controls in a tomato soup factory consists of rolling filled cans down a ramp. If they roll too fast, the soup is too thin. Why should cans of identical size and mass roll down an incline at different rates? And why should the thickest soup roll the slowest?
The easiest way to answer these questions is to consider energy. Suppose each can starts down the ramp from rest. Each can starting from rest means each starts with the same gravitational potential energy , which is converted entirely to , provided each rolls without slipping. , however, can take the form of  or , and total  is the sum of the two. If a can rolls down a ramp, it puts part of its energy into rotation, leaving less for translation. Thus, the can goes slower than it would if it slid down. Furthermore, the thin soup does not rotate, whereas the thick soup does, because it sticks to the can. The thick soup thus puts more of the can’s original gravitational potential energy into rotation than the thin soup, and the can rolls more slowly, as seen in .
Assuming no losses due to friction, there is only one force doing work—gravity. Therefore the total work done is the change in kinetic energy. As the cans start moving, the potential energy is changing into kinetic energy. Conservation of energy gives
More specifically,
or
So, the initial  is divided between translational kinetic energy and rotational kinetic energy; and the greater  is, the less energy goes into translation. If the can slides down without friction, then  and all the energy goes into translation; thus, the can goes faster.

### Test Prep for AP Courses


### Section Summary

1. The rotational kinetic energy  for an object with a moment of inertia  and an angular velocity  is given by

2. Helicopters store large amounts of rotational kinetic energy in their blades. This energy must be put into the blades before takeoff and maintained until the end of the flight. The engines do not have enough power to simultaneously provide lift and put significant rotational energy into the blades.
3. Work and energy in rotational motion are completely analogous to work and energy in translational motion.
4. The equation for the work-energy theorem for rotational motion is,

### Conceptual Questions


### Problems & Exercises
