# Angular velocity of a yo-yo!

## Homework Statement

A certain yo-yo can be modeled as a uniform cylindrical
disk with mass M and radius R and a lightweight hub of
radius ½R. A light string is wrapped around the hub.

(a) First, the yo-yo is allowed to fall. Find the angular
velocity of the yo-yo when the string has unwrapped a
distance L.

(b) Now, imagine that that you pull upward on the string such
that the yo-yo remains in the same place. Find the angular
velocity of the yo-yo when you have pulled the string
upward a distance of L.

(c) Explain in words why it makes sense that the answers to
parts (a) and (b) are different.

## Homework Equations

K (total) = .5 * I (center of mass) *w^2 + .5MR^2

I cm for a uniform cylindrical hub = .5M(R^2 + (.5R)^2)

## The Attempt at a Solution

(A)
K (total) = .5 * I (center of mass) *w^2 + .5MR^2 = MgL

W^2 = MgL/ (.5 *I (cm) + .5MR^2)

I cm for a uniform cylindrical hub = .5M(R^2 + (.5R)^2)
So..

W^2 = MgL/(.5 * (.5M(R^2 + (.5R)^2) + .5MR^2)

W^2 = MgL/(1/4Mr^2 + 1/16MR^2 + 1/2MR^2)

W^2 = gL/(13/16R^2)

Does that seem about right? (obviously need to make it the square root but just leaving it squared for now)

(b)
I assume I cannot use conservation of energy, so maybe I could solve this with the Work that is done? I dunno, I'm confused I guess I don't know where to begin

Thanks for any help fellow physics buds!