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Find V1 and V2 when the ball reaches the point A.
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Asked by patra04011965 | 11 Dec, 2019, 01:57: PM
answered-by-expert Expert Answer
Let origin of X-Y coordinate system coincide with center of mass of big block that has mass M.
 
A small ball of mass M/2 is rolling on the surface of semi-spherical ( or it can be semi-hollow cylindrical surface )
made in the big block as shown in figure.
 
When the ball is at a distance -x from y axis, x-coordinate xcm of center of mass of combined system of ( big block + ball )
 
is given by,     xcm =  (M/2) (-x ) / [ M + (M/2) ]  = -x/3  ......................(1)
 
X-component velocity of centre of mass of combined system ,   ( dxcm / dt )  = [ (M/2) V1 + M V2 ] / [ M + (M/2) ]  .............(2)
 
where V1 is the speed of ball and V2 is the speed of block
 
But , if there is no external force acting on the system, centre of mass will not move,
hence x-component velocity of centre of mass of combined system is zero.
 
Hence from eqn.(2), we have,  (V1/2) + V2 = 0   or  V2 = - ( V1 /2 )
 
when the ball reaches the point A,  velocity V1 = (2gR)1/2
 
( Potential energy difference is converted to kinetic energy )
 
velocity V2 = - [ (gR) / 2 ]1/2
 
Answered by Thiyagarajan K | 11 Dec, 2019, 11:10: PM

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