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)`

Explanation

Considering cylinder and track as a system, the system is not having any external force in horizontal direction. Hence the centre of mass of (M + m) will not move in horizontal direction. Let M moves towards left by a distance x them m will move towards right by a distance (R – r – x) (with respect to ground). For centre of mass to not move along horizontal we should have
`Mx = m(R - r - x) , " " x = (m(R - r))/(M + m)` .