Question

A solid cube of side $a$ and density ${\rho }_0$ floats on the surface of a liquid of density $\rho$. If the cube is slightly pushed downward, then it oscillates simple harmonically with a period of:

• A. $2\pi \sqrt{\frac{{\rho }_0}{\rho }\frac{a}{g}}$
• B. $2\pi \sqrt{\frac{\rho }{{\rho }_0}\frac{a}{g}}$
• C. $2\pi \sqrt{\frac{a}{(1-\frac{\rho }{{\rho }_0})g}}$
• D. $2\pi \sqrt{\frac{a}{(1+\frac{\rho }{{\rho }_0})g}}$

Answer 1

The correct option is A $2\pi \sqrt{\frac{{\rho }_0}{\rho }\frac{a}{g}}$

$T=2\pi \sqrt{\frac{m}{k}}$
Here Mass of the cube is $m=\left({\rho }_0\right)\left(a^3\right)$
Since the cube is pushed very slightly so the part dipped into the water is very negligible

Spring constant for a negligible distance $k={\rho }_1\times$ bottom surface area of cube $\times g=\rho a^{2}g$

$\therefore$ $T=2\pi \sqrt{\frac{{\rho }_{0}a}{\rho g}}$