# Solving KEtot Ratios: ω=v/r?

In summary, the conversation revolved around two homework problems involving finding the ratio of translational kinetic energy to total kinetic energy and rotational kinetic energy to total kinetic energy. The first problem involved a sphere rolling down a hill and the second involved a string around a hockey puck. The participant was able to solve the first problem by substituting v/r for ω, but was incorrect in doing so for the second problem. It was then explained that this substitution is only valid when there is no slipping involved. The conversation ended with the participant thanking for the clarification.

I have two homework problems that I've solved, but I can't reconcile the answers.

The first problem involved a sphere rolling down the hill, I must find the ratio of the translational kinetic energy to the total kinetic energy.

KEtot = KEtrans + KErot
= 1/2mv^2 + 1/2Iω^2
= 1/2mv^2 + 1/2(2/5mr^2)(v/r)^2

Working through the algebra gives me a ratio of 5/7, which was accepted as correct by the software.

The second problem is similar, this time with a string wound around a hockey puck sitting on a frictionless surface and pulling with constant force. I must determine the ratio of rotational kinetic energy to the total kinetic energy. I proceeded to do this problem the same way as the first:

KEtot = KEtrans + KErot
= 1/2mv^2 + 1/2Iω^2
= 1/2mv^2 + 1/2(1/2mr^2)(v/r)^2

This time I got a ratio of 1/3, but this is incorrect. The tutorial in the software indicated that ω does not equal (v/r), but rather is dependant on the moment of inertia of the object being rotated. Since the puck is a cylinder, ω = (2v/r). Plugging this value into the formula above gave me the accepted ratio of 2/3.

To me these look like identical problems, however in the first problem I was able to simply substitute v/r for ω, but wasn't able to do so for this problem. How do I know when to make that substitution vs. working out the relationship based on the inertia of the object?

Thanks!

The first problem involves rolling without slipping, which implies v = ωr. The second problem has no such condition.

Great, thank you!

## 1. What is the formula for calculating KEtot ratios?

The formula for calculating KEtot ratios is ω = v/r, where ω represents the angular velocity, v represents the linear velocity, and r represents the radius.

## 2. Why is it important to solve KEtot ratios?

Solving KEtot ratios allows us to determine the relationship between the angular and linear velocities of a rotating object. This information is essential in understanding the motion and energy of rotating systems.

## 3. How do I calculate the angular velocity (ω)?

The angular velocity (ω) can be calculated by dividing the linear velocity (v) by the radius (r).

## 4. Can KEtot ratios be used to determine the total kinetic energy of a rotating object?

Yes, KEtot ratios can be used to determine the total kinetic energy of a rotating object. This can be done by using the formula KEtot = 1/2 * I * ω^2, where I is the moment of inertia of the object.

## 5. Are there any real-world applications of solving KEtot ratios?

Yes, solving KEtot ratios has many real-world applications, such as in the design and analysis of mechanical systems, understanding the motion of celestial bodies, and calculating the energy output of wind turbines.