In figure find the acceleration of `m` assuming that there is friction between `m and M` and all other surfaces are smooth and pulleys light and `mu = `coefficient of friction between `m and M`

Answer

`(2sqrt(5)mg)/(m + 5m + 2mu m)`

Explanation

Let X be the leftward displacement of `M ` and `x and y` be the leftward and downward displacement of `m` as shown in the fig Then by constant relationwe have

` x = X rArr overset(..)x = overset(..)X rArr a_(x) = A_(x)`
and `l_(1) - x + l_(2) + _(3) - x + l_(4) + y = l_(1) + l_(2) + l_(3) + l_(4)`
where`l_(1), l_(2), l_(3), l_(4)`are the instantaneous length of the segments of the string
`rArr 2x = y rArr 2 overset(..)x = overset(..)y rArr 2a_(x) = a_(y)`
`N = ma_(x)` and `mg - mu N - T = ma_(y)` and `2T - N = MA_(x) = Ma_(x)`
Eliminating `T,A` and `N`
`a_(x) = (2mg)/(M + 5m + 2 mu m)` and `(4mg)/(M + 5m + 2 mu m)`
`:. a = sqrt(a_(x)^(2) + a_(y)^(2)) = (2sqrt(5)mg)/(m + 5m + 2mu m)`