Question

A mass $M$ attached to a spring oscillates with a period of $2 \,s$. If the mass is increased by $2 \,kg$, the period increases by one second. The initial mass $M$ (in $kg$ ) is ___

Answer 1

We know that for a spring block system $T=2 \pi \sqrt{\left(\frac{M}{k}\right)}$
Here $k =$ spring constant.
In first case, $2=2 \pi \sqrt{\left(\frac{M}{k}\right)}\,\,\,...(i)$
In second case, $3=2 \pi \sqrt{\left(\frac{M+2}{k}\right)}\,\,\,...(ii)$
Squaring of Eqs. (i) and (ii) and then dividing (ii) by (i), we have
$\frac{9}{4}=\frac{M+2}{M}=1+\frac{2}{M}$
Solving we get $M=1.6\, kg$.