Angular Velocity Of A Sphere Rotating Under Gravity

[Woolyabyss]

Homework Statement

A solid sphere of mass m and radius a can rotate freely about a point A on its surface. The sphere is held initially at rest with the line OA through A and the centre of the sphere O horizontal and is released under gravity. Find the angular velocity of the system when OA first becomes vertical.
(You may assume that the moment of inertia of the sphere about an axis through O is (2/5)ma^2.)

Homework Equations

energy change
.5*Iz*w^2 = mgh

where Iz is the moment of inertia about z
w is angular velocity
mgh potential energy

The Attempt at a Solution

parallel axis theorem
Iz = (2/5)ma^2 + ma^2 = (7/5)ma^2

energy change

.5*Iz*w^2 = mgh (where h is the change in height of centre of mass)

(7/10)*m*a^2*w^2 = mga

rearranging

w = (100/(7a))^.5

I'm not sure how to find a or if I even have enough information to solve for a.

Should I try angular momentum?

 

[TSny]

Your work looks good. Perhaps they just want the answer to be expressed in terms of the radius a. If so, it might be preferable to also leave g in symbolic form rather than substitute a value for g. That way, your answer will be correct for any system of units (as long as you use radians for angle).

 

[Woolyabyss]

thanks.