Question

Figure shows spring + block + pulley system which are light. The time period of mass would be

• A. $2\pi \sqrt{\frac{k}{m}}$
• B. $\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
• C. $2\pi \sqrt{\frac{m}{k}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{m}{k}}$

Answer 1

The correct option is C $2\pi \sqrt{\frac{m}{k}}$

In equilibrium position the elongation of the spring is:

$k{x}_0{=mg\Rightarrow x}_0=\frac{mg}{k}$

Now for a small displacement of the block, the force equation becomes:

$k(x_0+x)-mg=ma\Rightarrow kx=ma\Rightarrow a=\frac{kx}{m}\Rightarrow {\omega }^{2}x=\frac{kx}{m}\Rightarrow \omega =\sqrt{\frac{k}{m}}\Rightarrow T=2\pi \sqrt{\frac{m}{k}}$