Question

A meter stick swinging in vertical plane about a fixed horizontal axis passing through its one end undergoes small oscillation of frequency $f_0$. If the bottom half of the stick were cut off, if its new frequency of small oscillation is found to be $\sqrt{x}{f}_0$. The value of $x$ is ____

Answer 1

Answer: 2

We use $f_0=\frac{1}{2\pi }\sqrt{\frac{mgl}{I}}$
where $l$ is distance between point of suspension and centre of mass of the body.
Thus, for the stick of length $L$ and mass $m$ frequency is
$f_0=\frac{1}{2\pi }\sqrt{\frac{m\cdot g\cdot \frac{L}{2}}{\left(m{L}^2/12\right)}}=\frac{1}{2\pi }\sqrt{\frac{6g}{L}}$
when bottom half of the stick is cut off we use
$f_1=\frac{1}{2\pi }\sqrt{\frac{\frac{m}{2}\cdot g\cdot \frac{L}{4}}{\frac{m}{2}\frac{(L/2{)}^2}{12}}}$
$=\frac{1}{2\pi }\times \sqrt{\frac{12g}{L}}=\sqrt{2}{f}_0$