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

**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,

KE_{translational} / KE_{total} =

Answer) I / (MR^{2} +I)

**2. Relevant equations**

Torque = Iα = F x R

KE = 1/2 mv^{2} or 1/2 Iω^{2}

**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:

KE_{translational} / KE_{total} = Mv^{2} / (Mv^{2} + Iω^{2})

Substituting v = Rω, cancelling the ω^{2} terms

= MR^{2} / (MR^{2} + 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.

http://ift.tt/1p4ZRu5

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