The spring has a force constant k. The pulley is light and smooth while the spring and string are light figure. If the block of mass m slightly displaced vertically and released, find the period of verticall oscillations.

The spring has a force constant k. The pulley is light and smooth whil

1.	The spring has a force constant k. The pulley is light and smooth while the spring and string are light figure. If the block of mass m slightly displaced vertically and released, find the period of verticall oscillations.
Answer» Solution :Let `x_(0)` be the extension in the spring in EQUILIBRIUM position and `F_(0)` be the tension in the spring. Then `F_(0)=kx_(0)` If `T_(0)` is the tension in the string, then `T_(0)=mf or 2T_(0)=F_(0)=kx_(0)`…(i) Let mass m be pulled downward through distnace x, then tension in spring and string be F and T respectively. Now `F=2T=k(x_(0)=x//2)=kx_(0)KX//2` or `2T=2T_(0)+kx//2`[from (i)] or `T-T_(0)=kx//4` Restoring force on the mass m is `f=T-mg=T-T_(0)=kx//4`...(ii) From (ii) , `fpropx and f` is a restoring force, therefore, if mass m is left FREE, it will execute SHM. Here, spring factore`=k//4,` inertia factore `=`m. TIME period, `T=2pisqrt((m)/(k//4))=4pisqrt((m)/(k))`