Rotation of rigid bodies problem
Hello everyone,
I came across a problem recently which, in hindsight I believe I do not fully understand, and as a consequence of this I cannot seem to get the correct solution no matter how I tackle it. As I have gone through several different attempted solutions to the problem I will just list below the problem and my most recent attempted solution, and would appreciate it if someone could point me in the right direction as to how to solve the thing. Problem
You hang a thin hoop with radius R over a nail at the rim of the hoop. You displace it to the side (within the plane of the hoop) through an angle of B from it's equilibrium position and let it go. What is the angular speed when it returns to it's equilibrium position ? [Hint : Use equation  I = I (centre of mass) + m d (squared)]
Now I started off with the above equation then I assumed that the thin hoop corresponds to the same shape as a thinwalled hollow cylinder. So this would mean that the moment of Inertia of the hoop's around it's centre of mass is I (center of mass) = m R (squared)
Now since the pivot point of the new axis (the nail) is at the rim of the hoop then the value of 'd' (distance of new axis to centre of mass axis) in the original equation is equal to the radius of the hoop (R) .
If we then substitute these values into the equation above we get I = m R (squared) + m R (squared) = 2 m R (squared)
I we then apply the energy equations to this system we have .. K (i) + U (i) = K (f) + U (f) ....
where K (i), K (f) are initial and final kinetic energy values and
U (i), U (f) are initial and final potential energy values
Now K (i) and U(f) are both zero
and K (f) = 1/2 I w (squared) (w = angular speed)
and U (i) = m g y (g = acceleration (gravity), y= height displacement)
So we now have .. 0 + m g y = 1/2 I w (squared) + 0 m g y = 1/2 I w (squared)
Giving ... w (squared) = 2 m g y / 2 m R (squared)
so w = sqrt (g y / R (squared))
However the answer the book gives form this problem is .. w = sqrt (g/r (1  cos B))
I don't understand this answer and don't know where i have gone wrong, can anyone help ?
Regards,
Jackthehat
