A bullet of mass m is fired horizontally into a large sphere of mass M and radius R resting on a smooth horizontal table.

The bullet hits the sphere at a height h from the table and sticks to its surface. If the sphere starts rolling without slippng immediately on impact, then

Options

A. `(h)/(R)= (4m+ 3M)/(2(m+M))`
B. `(h)/(R)= (m+3M)/(m+2M)`
C. `(h)/(R)= (10m+ 7m)/(5(m+M))`
D. `(h)/(R)= (4m+ 3M)/(m+ M)`

Answer

C

Explanation

Apply conservation of angular momentum `mv sin theta R= ((2)/(5) MR^(2) + mR^(2)) omega_(0)`
`mv= ((h-R)/(R)) R= ((2M+ 5m))/(5) omega_(0) R^(2) rArr (m+ M) (h- R) omega_(0)`
`R= ((2M + 5m))/(5) omega_(0)R^(2) rArr (h)/(R) = ((10m+ 7m))/(5(m+ M))`