# Rotating wheel, proving that power is proportional to square of torque

• djfusion777
In summary, the conversation discusses a mechanics question involving a wheel starting from rest and rotating with constant angular acceleration about a fixed axis. The question asks for a proof that the power at any given time is proportional to the square of the net torque about the axis and that the power at any given angular displacement is proportional to the 3/2 power of the total torque about the axis at that angular displacement. The participants discuss the equations for torque and rotational kinetic energy and consider different approaches to solving the problem.
djfusion777
I have a mechanics question.

"A wheel starts from rest and rotates with constant angular acceleration about a fixed axis.

1) Prove that the power at any given time is proportional to the square of the net torque about the axis.

2) Prove that the power at any given angular displacement is proportional to the 3/2 power of the total torque about the axis at that angular displacement"

I know that Torque = r x F

Power in this case is rate of change of rotational Kinetic Energy, Ek = 1/2I (omega)^2 where I = moment of inertia.

No idea where to go from here, any help much appreciated!

That's not the only thing torque equals. This problem doesn't mention force, it does mention angular acceleration though, hmmmmz...

A decent place to start, since you have an expression for Ek and you know you're looking for something dealing with the rate of change of Ek, is to find that rate of change of Ek, eh?

ignore force, go for angular momentum

torque can be defined as the rate of change of angular momentum

Does that help?

I wouldn't take that approach, I was thinking torque=I*angular acceleration, then take the time of derivative of Ek, and you're about one substitution away from beautiful victory

## 1. What is a rotating wheel?

A rotating wheel is a circular object that spins on an axis. It can be powered by various means, such as an engine or motor, and is commonly used in machinery and vehicles.

## 2. How is power related to torque in a rotating wheel?

The power of a rotating wheel is directly proportional to the square of its torque. This means that as the torque increases, the power also increases exponentially.

## 3. What is torque?

Torque is a measure of the rotational force or twisting force that is applied to an object. In the context of a rotating wheel, it is the force that causes the wheel to spin.

## 4. How can the relationship between power and torque be proven?

This relationship can be proven through experiments that involve varying the torque applied to a wheel and measuring the resulting power output. By plotting the data on a graph, it can be seen that the power is indeed proportional to the square of the torque.

## 5. What are some practical applications of this concept?

The relationship between power and torque in a rotating wheel is important in many industries, such as automotive and manufacturing. It is used in the design of engines, motors, and other machinery to ensure optimal performance and efficiency.

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