Consider the situation shown in the figure. A mass is hanging from a inextensible string which is passing over a pulley. The pulley itself is attached to a massless spring of stiffness k as shown in the figure. Find the time period of vertical oscillations of mass m if pulley is slightly displaced from its mean position. Assume that string does not slip over pulley.

Explanation

At equilibrium position let `x_(0)` be the deformation in the spring

`:. x_(0) = (3mg)/(k)`
Let the pulley is displaced by x then total energy of system
`E = (-mg x + (1)/(2) mv^(2) + (1)/(2) l omega^(2)) + ((1)/(2) m (2v)^(2) - mg(h + 2x)) + (1)/(2) k (x_(0) + x)^(2)`
As string does not slip over pulley so `omega = v//r`.
So `E = (-mg x + (1)/(2) mv^(2) + (1)/(4) mv^(2)) + (2mv^(2) - mg (h + 2x) + (1)/(2) k (x_(0) + x)^(2))`
`(dE)/(dt) = 0 , :. 0 = - mg + m (dv)/(dt) + (m)/(2) (dv)/(dt) - 2mg + 4m (dv)/(dt) + k (x_(0) + x)`
`rArr kx + (m + (m)/(2) + 4m) (dv)/(dt) = 0`
`rArr (dv)/(dt) = (2k)/(11m) x`
`rArr omega = sqrt((2k)/(11m)) or T = 2pi sqrt((11m)/(2k))`