Question

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

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

Answer 1

The correct option is D $T=2\pi \sqrt{\frac{m}{adg}}$

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