A body of mass is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass is slightly pulled down and released, it oscillates with a time period of 3 s. When the mass is increased by 1 kg, the time

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Solution

Q.
A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. The spring has negligible mass. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of 3 s. When the mass $m$ is increased by 1 kg, the time period of oscillations becomes 5 s. The value of $m$ in kg is [2016]

1 $\frac{3}{4}$

2 $\frac{4}{3}$

3 $\frac{16}{9}$

4 $\frac{9}{16}$

Ans.
(4)

Time period of spring-block system, $T = 2 π \sqrt{\frac{m}{k}}$

For a given spring, $T \propto \sqrt{m}$

   $\frac{T_{1}}{T_{2}} = \sqrt{\frac{m_{1}}{m_{2}}}$

Here,   $T_{1} = 3 s, m_{1} = m, T_{2} = 5 s, m_{2} = m + 1, m = ?$

   $\frac{3}{5} = \sqrt{\frac{m}{m + 1}} or \frac{9}{25} = \frac{m}{m + 1}$

   $25 m = 9 m + 9 \Rightarrow 16 m = 9;$ $∴$ $m = \frac{9}{16}$ $kg$

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