Question

The frequency of oscillation of a mass $m$ suspended by a spring is $\nu$. If the length of the spring is cut to one third then the same mass will oscillate with frequency

• A. $\sqrt{2}\nu$
• B. $\sqrt{3}\nu$
• C. $\sqrt{5}\nu$
• D. $\sqrt{6}\nu$

Answer 1

The correct option is B $\sqrt{3}\nu$

We know that the angular frequency of the oscillation is given by,
$\omega =\sqrt{\frac{k}{m}}$
For the same mass,
$\omega \propto \sqrt{k}$
$\Rightarrow f\propto \sqrt{k}$ ............$\left(1\right)$
$[\because \omega =2\pi f]$
Also, spring constant,
$k\propto \frac{1}{l}$ ............$\left(2\right)$
$[kl=$ constant for any spring$]$
From $\left(1\right)$ and $\left(2\right)$,
$f\propto \sqrt{\frac{1}{l}}$
So,
$\frac{f_1}{f_2}=\sqrt{\frac{l_2}{l_1}}$
$\Rightarrow \frac{\nu }{f_2}=\sqrt{\frac{l/3}{l}}$
$\Rightarrow f_2=\sqrt{3}\nu$

Why this question? It intends to test your understanding of the change in value of spring constant $\left(k\right)$, if the spring is cut and its affect on frequency of oscillation.