Question

The string the spring and the pulley shown in figure are light. Find the time period of the mass m.

Answer 1

Let l be the extension in the spring when mass m is hung.

Let T₁ be the tension in the string; its value is given by,
T_1 = kl = mg 
Let x be the extension in the string on applying a force F.
Then, the new value of tension T₂ is given by,
 T₂ = k(x + l)
Driving force is the difference between tensions T₁ and T_2.
∴ Driving force = T₂ − T₁ = k(x + l) − kl
                           = kx

\[\text{Acceleration},   a = \frac{kx}{m}\] 

\[\text { Time  period } \left( T \right)\]  is  given  by, 

\[T = 2\pi\sqrt{\frac{\text { displacement }}{\text { Acceleration }}}\] 

\[     = 2\pi\sqrt{\frac{x}{kx/m}} = 2\pi\sqrt{\frac{m}{k}}\]