A block of mass M with a semicircualr of radius R, rests on a horizontal frictionless surface. A uniform cylinder of radius r and mass m is released from rest the top point A The cylinder slips on the semicircular frictionless track. How far has the block moved when the cylinder reaches the bottom (point B) of the track ? How fast is the block moving when the cylinder reaches the bottom of the track?

Answer

`(m(R - r))/(M + m) , msqrt(2g(R - r))/(M (M + m))`

Explanation

`Deltax_(cm) = ((MDeltax_(1) + m Deltax_(2) + m Delta x_(2))/(M + m)) rArr 0 = (Mx + m(x + R - r)/(M + m))`
Distance moved by the cylinder `x = -(m(R - r))/(M + m)`
For motion along x-axis `0 = m(v_(1) + v_(2)) + mv_(2)` …..(i)
`mg (R -r) = (1)/(2) m (v_(1) + v_(2))^(2) + (1)/(2) Mv_(2)^(2)` ......(ii)
`:. v_(2) = msqrt((2g(R - r))/(M(M + m)))`