On applying a force F the mass M is displaced vertically down by from equilibrium position, Find the force F in terms of the force constant k of the spring and displacement y for the cases (A) and (B) as shown in

Explanation

Case (A) for the equilibrium of mass M
`F=T` …(i)
and for the equilibrium of pulley
`2T=F'` …(ii)
But as due to shift of mass M by y, the spring strentches by `(y//2)` so
`F' = k(y//2)` ….(iii)
So, substituting F' from Eqn. (iii) in Eqn. (ii) and then T from Eqn (ii) in Eqn. (i) we get

Case (B) As tension in massless string and spring will be same,
`T=F'`
And for equilibrium of pulley `T+F'=F` ...(i)
So, from Eqns (i) and (ii) `F=2F'` ....(ii)
Now if the mass M shifts by y the spring will stretch by 2y (as string is inextensible)
`F' =k(2y)` ....(iv)
So, subsituting f' from Eqn (iv) in Eqn (iii)
`F=(4k)y` ...(b)