Pinnaka, I dared to solve d 1st part of ur problem but I am not too sure about the answer. I think it should be (5/7)h.
Explanation:
Let the marble be released from a vertical height h (height of the bowl).
During the downward motion, it is rolling purely. so
v=rw
Let the mass be m.
Pot. E at h = K.E. at the bottom
So, mgh = 1/2 mv2 + 1/2 Iw2 (w is the angular speed at the bottom)
mgh = 1/2 mw2r2 + 1/2 X 2/5 mr2w2
gh = 7/10 r2w2
i.e., r2w2 = 10gh/7 .............................(i)
Since no external torque acts on the upward journey so w remains same.
After reaching max height H
v becomes 0.
So, K.E. possessed by the body = 1/2.2/5mr2w2
So, K.E. lost = 7/10 mr2w2 - 1/5 mr2w2 = 1/2 mr2w2
This lost energy is gained in d form of P.E.
So,
mgH= 1/2 mr2w2
So, H= 1/2 . 10/7gh (from (i))
H = 5/7 h
Hey everyone out there, please check and tell me if I am correct or wrong.