A uniform rod of length 6a and mass 8m lies on a smooth horizontal table. Two particle of masses m and 2m , moving in the same horizontal plane but in opposite directions with speeds 2v and v respectively strike and rod normally as shown in figure and stick to the rod. Denoting angular velocity ( about the centre of mass), total energy and transnational velocity of centre of mass by `omega`, E and `v_(c)` respectively after the collision.

Options

A. `(v)/(5a)`
B. `(v)/(15a)`
C. `(v)/(3a)`
D. `(v)/(10a)`

Answer

A

Explanation

`(i)` Cons. Linear momentum

`-2m.v+2v.m=0=MV_(cm)`
`V_(cm)=0`
`(ii)` As ball sticks to Red
Conserving angular momentum about `C`
`2v.m 2a+2ma=Iomega`

`=((8m.36a^(2))/(12)+2m.a^(2)+m4a^(2))`
`6mv.a=30ma^(2).omega`
`rArr omega=(v)/(5a)`
`(iii) KE=(1)/(2)Iomega^(2)=(1)/(2).30ma^(2)xx(v^(2))/(25a^(2))`
`=(3mv^(2))/(5)`