Question

A test tube of cross-sectional area $a$ has some lead shots in it. The total mass is $m$. It floats upright in a liquid of density $d$. When pushed down a little and released, it oscillates up and down with a period $T$. Choose the correct relationship from the following.

• A. $T=2\pi \sqrt{\frac{ma}{gd}}$
• B. $T=2\pi \sqrt{\frac{mg}{ad}}$
• C. $T=2\pi \sqrt{\frac{md}{ag}}$
• D. $T=2\pi \sqrt{\frac{m}{agd}}$

Answer 1

The correct option is D

$T=2\pi \sqrt{\frac{m}{agd}}$

The dimension of $\frac{m}{adg}=\frac{M}{L^2\times M{L}^{-3}\times L{T}^{-2}}=T^2$
$\therefore$ Dimension of $2\pi \sqrt{\frac{m}{adg}}=T$
Hence, the correct choice is (d).