A spring has a certain mass suspended from it and its period for vertical oscillations is ' $T_1$ '. The…
A spring has a certain mass suspended from it and its period for vertical oscillations is ' $T_1$ '. The spring is now cut in to two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations is now ' $T_2$ '. The ratio $T_1 / T_2$ is
$2$
$\sqrt{2}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{2}$
Solution
$\mathrm{T}_1=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}} .$
Also, spring constant $(\mathrm{k}) \propto \frac{1}{\text { Length }(l)}$
When the spring is half in length, then $\mathrm{k}$ becomes twice.
$\begin{aligned}
& \therefore \quad \mathrm{T}_2=2 \pi \sqrt{\frac{\mathrm{m}}{2 \mathrm{k}}} \\
& \therefore \quad \frac{\mathrm{T}_1}{\mathrm{~T}_2}=\frac{1}{\frac{1}{\sqrt{2}}}=\sqrt{2}
\end{aligned}$