Two blocks of masses `m_(1)` and `m_(2)` are connected by a spring of stiffness k. The coefficient of friction between the blocks and the surface is `mu`. Find the minimum constant force F to be applied to `m_(1)` in order to just slide the mass `m_(2)`.

Explanation

At any intermediate position the net external force is equal to
`F.=F-(mu m_(1)g+kx)` (numerically)
`rArr ` The work done by the net external force F. in displacing the block `m_(1)` through a length
`x_(0)=W_("ext")= int F..dx`
`rArr W_("ext")=int_(0)^(x_(0)) {F-(mu mg+kx)}dx`
where `x_(0)=` compression of the spring required to just move the block `m_(2)`.
`rArr kx_(0)=mu m_(2)g`
`rArr x_(0)=(mu m_(2)g)/(k)`.
Since the KE of the system just before moving the block `m_(1)` & just before moving the block `m_(2)` is zero, the change in `KE=0`
`rArr DeltaKE=0`
`:.W_("ext")=DeltaKE`
`rArr int_(0)^(x_(0))F-(mu m_(1)g+kx) dx=0`
`rArr Fx_(0)-(mu m_(1)gx_(0)+(kx_(0)^(2))/(2))=0`
`rArr F= mu m_(1)g+(kx_(0))/(2)=mu m_(1)g+(mu m_(2)g)/(2)=[m_(1)+(m_(2))/(2)]mu g`.