Question

A mass $M$ is suspended form a spring of negligible mass. The spring is pulled a little and then released so that the mass executes simple harmonic oscillations with a time period $T$. If the mass is increased by $m$ then the time period becomes $\left(\frac{5}{4}T\right)$. Find the ratio $\frac{m}{M}$.

Answer 1

We know time period of spring mass system $T\propto \sqrt{M}$, where $M$= mass attached to string.

So, $\frac{T_2}{T_1}=\sqrt{\frac{M_2}{M_1}}$

Here, $T_1=T$ and $T_2=\frac{5}{4}T$

and, $M_1=M$ and $M_2=(M+m)$

So, $\frac{5}{4}=\sqrt{\frac{M+m}{M}}$

$\Rightarrow \frac{25}{16}=1+\frac{m}{M}$

$\Rightarrow \frac{m}{M}=\frac{9}{16}$