In the arrangement shown in the figure, the blocks have masses `3m,m` and `2m`, the friction is absent, the masses of the pulleys and the threads are negligible. Find the acceleration of mass `3m`.

Explanation

Let the pulley `Q` falling with acceleration `a_(0)` with respect to the pulley, its acceleration is `a_(0)`
Let the acceleration of block `A` (mass `3m`) w.r.t. the pulley be `a`
`a_(A//g)=a_(A//p)+a_(p//g)=(a+a_(0))` in downward direction
`a_(B//g)=a_(B//p)-a_(p//g)=(a-a_(0))` in upward direction

`
3mg-T=3m(a+a_(0))` (i)

`T-mg=m(a-a_(0))` (ii)

`2T=2ma_(0)` (iii)
From (iii), `T=ma_(0)` in (i) and (ii)
`3mg-ma_(0)=3m(a+a_(0))rArr3a+4a_(0)=3g` (iv)
`ma_(0)-mg=m(a-a_(0))rArra-2a_(0)=-g` (v)
Solving (iv) and (v), we get
`a=(8)/(5),a_(0)=(3g)/(5)`
Acceleration of mass `3m` with respect to the ground is
`a+a_(0)=(4g)/(5)`