A block of mass `m_(1) ` connected with another of mass `m_(2)` by a light spring of natural length `l_(0)` and stiiffness k is kept stationary on a rough horizontal surface . The coefficient of friction between `m_(1)` and surface is `mu` and the block `m_(2)` is smooth .The block `m_(2)` is moved with certain the block `m_(1)` in horizontal plane Find the (a) maximum to `m_(1)` and (b) acceleration of `m_(2)` in part(a)

Answer

`sqrt((mu km_(2)g)/(m_(2)(l_(0) + mu m_(2)g)))` b. ` (mu m_(2)g)/(m_(2))`

Explanation

a. From FBD of `m_(2)`
`kx = m_(2) omega^(2) (l_(0)+ 1)`
From `F.B.D` of `m_(1)`
`kx = mu N = mu m_(1)g`
`x = (mu m_(1)g)/(k)`
Substiting the value of x is Eqs (i)
`k ((mu m_(1)g)/(k)) = m_(2) omega^(2)(l_(0)+ (mu m_(1)g)/(k))`
`rArr omega^(2) = (mu m_(1)g)/(m_(2) (l_(0) + (mu m_(1)g)/(k))) rArr omega = sqrt((mu km_(1)g)/(m_(2)(l_(0) + mu m_(2)g)))`
b. Acceleration of `m_(2) ,a_(2) = (kx)/(m_(2)) = (mu m_(1)g)/(m_(2))`