A small ball `(` uniform solid sphere `)` of mass `m` is released from the top of a wedge of the same mass `m`. The wedge is free to move on a smooth horizontal surface. The ball rolld without sliding on the wedge. The required height of the wedge are mentioned in the figure.

The total kinetic energy of the ball just before it falls on the ground

Options

A. `2mgh`
B. `mgh`
C. `(13)/(18)mgh`
D. None of these

Answer

D

Explanation

`mgh=(1)/(2)mv^(2)+(1)/(2)mv^(2)+(1)/(2).(2)/(5)mr^(2)((2v)/(r))^(2)`
`=(1)/(2)mv^(2)[1+1+(8)/(5)]=(1)/(2)mv^(2)(18)/(5)=(9mv^(2))/(5)`
`rArr v=sqrt((5)/(9)gh)`
`KE` of the ball `=mg 2h`
`-(1)/(2)m(sqrt((5gh)/(9)))^(2)=(31)/(18)mgh`
`=mg2h-(1)/(2)m(sqrt((5gh)/(9)))^(2)=(31)/(18)mgh`