In the shown arrangement , both pulleys and the string are massless and all the surface are frictionless. Find the acceleration of the wedge.

Explanation

`FBD` of `m_(1)` (showing only the horizontal force) Equation of motion for `m_(1)` is

`T - N = m_(1)a_(1)` …(i)
Equation of motion for `m_(2)` are
`N = m_(2) a_(1)` ...(i)
and `m_(2) g - T = m_(2) a_(2)` ...(ii)
Equation of motion for `m_(3)` are
`m_(2) g - T = m_(3) a_(3)` ...(iv)
Further from constrain equation we can find the reation,
`a_(1) = a_(2) + a_(3)` ...(v)
We have five unknown `a_(1), a_(2),a_(3) T` and `N` solving we get
`a_(1) = (2 m_(1)m_(3) g)/((m_(2) + m_(3)) (m_(1) + m_(2)) + m_(2)m_(3))`