Figure shown a small block `A` of mass `m` kept at the left end of `a` plank `B` of mass `M = 2m` and length `l` . The system can slide on a horizontal road. The system is started towards right with the initial velocity `v`. The friction coefficient between the road and the plank is `1//2` and that between the plank and the block is `1//4` . Find

(a) the time elapsed before the block separates from the plank
(b) displacement of block and plank relative to ground till that moment.

Answer

`t=4sqrt((l)/(3g)) (b)S_(A)=4Vsqrt((l)/(3g))-(2)/(3)l,S_(B)=4Vsqrt((l)/(3g))-(5)/(3)l`,

Explanation

System start moving towards right with velocity `v`.

`=((1)/(2)(3m)g-(1)/(4)mg)
/(2m)=(5g)/(8)`
`U_(rel)=0`
`a_(rel)=(5g)/(8)-(g)/(4)=(3g)/(8)`
`:. l=(1)/(2)(3)/(8)g t^(2)`
`t=4sqrt((l)/(3g))`
`S_(1)=vt-(1)/(2)(g)/(4)t^(2)=4vsqrt((l)/(3g))-(2l)/(3)s`
`S_(2)=vt-(1)/(2)(5)/(8)g t^(2)=4vsqrt((l)/(3g))-(5)/(3)l`.