# Work & Power in Rotational Motion

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• 0:05 Rotational Motion:…
• 2:00 Rotational Motion:…
• 2:36 Example Calculation
• 3:51 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you will be able to explain what work and power are in the context of rotational motion and use equations to solve problems involving rotational work and power. A short quiz will follow.

## Rotational Motion: What Is Work?

Let's imagine that I give you a huge pile of heavy books and leave you standing in place for, say... 20 minutes. Then I come back and ask you a simple question: Are you doing work, right now? Considering that your arms would be aching and your back sore, you'll probably give a categorical 'Yes!' But in physics terms, you did no work at all. Sorry to disappoint you.

Work is a force applied over a distance and involves the transfer of energy from one type to another. Whenever energy is transferred between types, work is being done somewhere. But for you to do work on an object - for example, for you to do work on the books - you have to apply a force to them and cause them to move in the direction of that force. So if you had lifted the books, you would most certainly have been doing work. But since you were just holding them in place, you did no work at all. You applied a force, but your distance was zero.

Technically, that example, though often used by physics teachers, is a bit misleading. The fact that your muscles are tired tells you the work was being done somewhere. The food in your body (chemical energy) was being changed into heat energy in your muscles. But your body as a whole wasn't doing work; all the work was being done deep inside your body. You weren't doing work on the books.

The translational equation for work is force multiplied by distance. A greater force means greater work, but we're talking about rotation. Instead of force, in rotation, we have torque. And instead of moving a distance, in rotation, we rotate around an angle. So the rotational equation for work says that work, measured in joules, is equal to torque (tau), measured in newton-meters, multiplied by the change in angle (theta), measured in radians, not degrees.

## Rotational Motion: What Is Power?

Power is often a more useful measure in the real world. Telling us that a light bulb did 1000 Joules of work isn't very helpful, because if it did those 1000 Joules of work over say, a million years, the light bulb would have been very dim. And if it did that work over two seconds, it would be super bright.

Power is the work done per second, or in other words, the energy used per second, measured in watts. So if work is torque multiplied by the change in angle, the power will be the same thing divided by the time it took to do the work.

Distance over time is velocity. So the change in angle over time is also equal to the angular velocity. So, we can replace theta over t with the symbol for angular velocity, omega. And then here we have another possible equation for rotational work: torque multiplied by angular velocity. You can use whichever equation is most useful in a particular situation.

## Example Calculation

Now, let's do an example.

Let's say that the wind applies a 300 Newton to the outside edge of a wind turbine over a period of 10 seconds, causing the turbine to turn by pi radians; that's 180 degrees. If the wind turbine has a radius of 5 meters, how much work did the wind do on the wind turbine? And how much power was generated?

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