Ratio of translational velocity to rotational velocity.

[spursfan2110]

1. A ball of mass m and radius R is both sliding and spinning on a horizontal surface so
that its rotational kinetic energy equals its translational kinetic energy.What is the ratio of the ball’s center-of-mass speed to the speed due to rotation only of a point on the ball’s surface? The moment of inertia of the ball is 0.56mR² .
(For ease, I will refer to omega as w from here on out)

Homework Equations

KE = .5(I)(w)² = .5mv²

The Attempt at a Solution

So if I understand it correctly the problem basically wants the ratio of linear velocity to rotational velocity, v to w. So, I set .5Iw² = 1/2mv²

From here, I plugged in the given moment of inertia of the ball.

.5(.56mR²)(w)² = .5mv²

Then I canceled out the .5 and the m,

.56R²w² = v²

Square rooted both sides,

sqrt(.56)Rw = v

but from here I am unsure of how I could possibly eliminate the R, and this causes big problems when finding a ratio. Any suggestions? Did I go wrong somewhere prior to this point? If I could just get that R, it should be easy I would think, but its a variable...

 

[kuruman]

Read the problem carefully. You are looking for the ratio of the speed of the center of mass to the speed of a point on the surface of the sphere. The latter quantity is not ω; it is a linear speed (in m/s) so the final answer must be a dimensionless quantity.