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## Homework Statement

A cylinder with a mass of M and radius R is attached to a spring with a constant of k. The attachment always stays the same as described in the picture. The cylinder is moved to the right and the released, and it rolls without slipping (pure rolling). What will be the frequency of its movement? I

_{CM}=0.5MR^2 and the center of mass is at the middle.

## Homework Equations

ma=F

Iα=Sum of torques.

## The Attempt at a Solution

I tried to the following.

first I think that the forces relevant are the forces applied by the spring and the friction from the floor. Then I wrote these equations:

Ma

_{CM}=-kx+f where f is the friction.

I

_{CM}α=r

_{1}×F

_{1}+r

_{2}×F

_{2}=-0.5Rkx-Rf.

Then when I tried to say that a=αR (pure rolling) I get that f=0, which is not possible and gives a wrong answer.

Where is my mistake?

EDIT: From what I understand by reading the solution, the problem is that for some reason αR=a is incorrect but 1.5αR=a is correct. I have absolutely no idea why.

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