Question

A block of mass m welded with a light spring of stiffness k is in equilibrium on an smooth inclined plane with angle of inclination `theta`. If a variable external force is applied slowly till the spring comes to its relaxed position, find the work done by spring force.

Answer 1

Referring to the FBD as shown in figure, we have four forces (N, mg, `F_(spring)` and `F_(ext)` acting on the particle.

Initially, the block is at equilibrium.
`kx_i=mg sin thetaimpliesx_i=(mgsintheta)/(k)`
When the block is pulled by an external force to bring the spring to relaxed length `x_f=0`
Hence, work done by spring force,
`W_(sp)=-1/2k(x_f^2-x_i^2)`

`=-1/2[0-((mgsintheta)/(k))^2]`
`=1/2((mgsintheta)^2)/(2k)`