A block of mass `M` with a semicircular track of radius `R` rests on a horizontal frictionless surface shown in figure. A uniform cylinder of radius `r` and mass `m` is released from rest at the point `A`. The cylinder slips on the semicircular frictionless track. How far has the block moved when the cylinder reaches the bottom 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)))`