Question

A spring has force constant k and a mass m is suspended from it. The spring is cut in two parts in the ratio 1:3, and the same mass is suspended from the smaller part. If the frequency of oscillation in the first case in n, then the frequency in the second case will be:

• A. $\sqrt{2}n$
• B. $3n$
• C. $\frac{n}{\sqrt{3}}$
• D. $\frac{\sqrt{3}}{2}n$

Answer 1

The correct option is A $\sqrt{2}n$

$\begin{array}{c}Timeperiodofoscillationforspringmasssystem-\\ T=2\pi \sqrt{\frac{m}{K}}\\ wherein:-\\ m=massofblock\\ K=springcons\tan t\\ T=2\pi \sqrt{\frac{m}{K}}\Rightarrow n^{\prime }=\frac{1}{2\pi }\sqrt{\frac{K}{M}}\\ n=\frac{1}{2}\sqrt{\frac{K}{m}}\\ whenspringcutintotwohalfthentheforcecons\tan twillbedoubled\\ n_1=\frac{1}{2}\sqrt{\frac{2K}{m}}\\ \therefore \frac{n_1}{n}=\frac{\sqrt{2}}{1}=n_1=\sqrt{2}n\end{array}$

Hence,

option $\left(A\right)$ is correct answer.