Question

A block of mass $M$ with a semi-circular track of radius $R$, rests on a horizontal frictionless surface. A uniform cylinder of radius $r$ and mass $m$ is released from rest at the top point $A$ (See fig.). 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 1

Using the concept of center of mass.

Let block m move x left and cylinder of mass m move $(R-r-x)$ right.

Now let center of mass and origin therefore

$X_{com}=\frac{m(R-r-x)-xM}{m+M}=0=m(R-r)-mx-xM=0x=\frac{m(R-r)}{(M+m)}$