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.