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 $3s$. when the mass m is increased by $1$ kg the time period of oscillations becomes $5s$. Then value of $m$ in kg is

• A. $\frac{9}{16}$
• B. $\frac{3}{4}$
• C. $\frac{4}{3}$
• D. $\frac{16}{9}$

Answer 1

The correct option is A $\frac{9}{16}$

For SHM of a hanging mass by a spring,

$F=-k_{s}x$

where $k_s=m{\omega }^2$

$⟹\omega =\sqrt{\frac{k_s}{m}}$

Hence time period of SHM=$T=\frac{2\pi }{\omega }$
$=2\pi \sqrt{\frac{m}{k_s}}\propto \sqrt{m}$

Hence $\frac{5}{3}=\sqrt{\frac{m+1}{m}}$

$⟹m=\frac{9}{16}kg$