# USAPhO 2014 F=ma exam #14, (rotational dynamics)

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1. The problem statement, all variables and given/known data

A disk of moment of inertia I, mass M, and radius R has a cord wrapped around it tightly as
shown in the diagram. The disk is free to slide on its side as shown in the top down view. A
constant force of T is applied to the end of the cord and accelerates the disk along a frictionless
surface.

After the disk has accelerated some distance, determine the ratio of the translational KE to total KE of the disk,

KEtranslational / KEtotal =

2. Relevant equations

Torque = Iα = F x R

KE = 1/2 mv2 or 1/2 Iω2

3. The attempt at a solution

My conceptual understanding is the problem here, I think. If there is absolutely no friction, then would there be any rotation? I think there would be, but I’m not sure.

Here’s an attempt:

KEtranslational / KEtotal = Mv2 / (Mv2 + Iω2)

Substituting v = Rω, cancelling the ω2 terms

= MR2 / (MR2 + I)

So I get an answer close but not exactly the correct answer. The answer I got is choice E on the actual exam, meaning my attempt probably has a common mistake.

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