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

In figure find the acceleration of m assuming that there is friction b

1.	In figure find the acceleration of m assuming that there is friction between m and M and all other surface are smooth and pulleys light and mu = coefficient of friction between m and M
Answer» Solution :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 rArroverset(..)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)`