First slide

Simple hormonic motion

Question

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 $1 k g$ , the time period of oscillations becomes $5 s$ . The value of m in kg is

Moderate

A

$\frac{3}{4}$
B

$\frac{4}{3}$
C

$\frac{16}{9}$
D

$\frac{9}{16}$

Solution

Time period of spring - block system,
$T = 2 π \sqrt{\frac{m}{k}}$
$For given spring, T \propto \sqrt{m}$
$\frac{T_{1}}{T_{2}} = \sqrt{\frac{m_{1}}{m_{2}}}$
$Here, T_{1} = 3 s, m_{1} = m, T_{2} = 5 s, m_{2} = m + 1, m = ?$
$\begin{array}{l} \frac{3}{5} = \sqrt{\frac{m}{m + 1}} or, \frac{9}{25} = \frac{m}{m + 1} \\ 25 m = 9 m + 9 \Rightarrow 16 m = 9 \Rightarrow m = \frac{9}{16} kg \end{array}$

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