Find the time period of mass M when displaced from its equilibrium position and then reelased for the system shown in figure.

Find the time period of mass M when displaced from its equilibrium pos

1.	Find the time period of mass M when displaced from its equilibrium position and then reelased for the system shown in figure.
Answer» Solution :Let k be the spring constant of the spring. When mass Mis suspended from the given system as SHOWN in figure.let l be the length through which mass M MOVES down before itcomes to rest. In this SITUATION, both the spring and string will be stretched by length l. Since string is inextensible, so spring isstretched by length 2l. The tension along the string is the same. So, RESTORING force is `F=-2kxx(2l)=-4kl` ...(i) In equilibrium position,this restoring balances weigth of the mass. i.e., `Mg=4kl` If mass M is pulled doen through small distance y, then restoring force `F^(')` is given by `F^(')=-2k(2l+2y)` Effective restoring force on mass M is given by `f=F^(')-F=-2(2l+2y)-(-4kl)=-4ky`...(iii) As `fpropy` and f is directed towards equilibrium position. hence if massis left free, it will execute linear SHM. From (ii), spring factor`=4k`. Inertia factor `=M` As time period, `T=2pisqrt((i n ertia fact o R)/(spri ng fact o r))=2pisqrt((M)/(4k))`