Question

The string, the spring and the pulley shown in figure (12−E9) are light. Find the time period of the mass m.

Figure

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
$$\begin{gathered} \mathrm{Acceleration},a=\frac{kx}{m} \\ \mathrm{Time}\mathrm{period}\left(T\right)\mathrm{is}\mathrm{given}\mathrm{by}, \\ T=2\mathrm{\pi }\sqrt{\frac{\mathrm{displacement}}{\mathrm{Acceleration}}} \\ =2\mathrm{\pi }\sqrt{\frac{x}{kx/m}}=2\mathrm{\pi }\sqrt{\frac{m}{k}} \end{gathered}$$