Question

A nonviscous liquid of mass m and density $\rho$ is filled in a tube with $A$ as the area of cross-section, as shown in the figure. If the liquid is slightly depressed in one of the arms, the liquid column oscillates with a frequency:

• A. $\frac{1}{2\pi }\sqrt{\frac{\rho A\sin \left(\frac{{\theta }_1+{\theta }_2}{2}\right)}{m}}$
• B. $\frac{1}{2\pi }\sqrt{\frac{\rho gA(\sin {\theta }_1-\sin {\theta }_2)}{m}}$
• C. $\frac{1}{2\pi }\sqrt{\frac{\rho gA(\sin {\theta }_1+\sin {\theta }_2)}{m}}$
• D. $\frac{1}{2\pi }\sqrt{\frac{\rho A\sin \left(\frac{{\theta }_1-{\theta }_2}{2}\right)}{m}}$

Answer 1

Answer: (C)

$\frac{1}{2\pi }\sqrt{\frac{\rho gA(\sin {\theta }_1+\sin {\theta }_2)}{m}}$

Let the level of liquid rises by $x$ in one column and falls by $x$ in another column.
then, the difference in levels of liquid,
$A=x(\sin {\theta }_1+\sin {\theta }_2)$
$\therefore$ pressure difference $P=h\rho g=\rho g(\sin {\theta }_1+\sin {\theta }_2)x$
$F=-PA=-\rho gA(\sin {\theta }_1+\sin {\theta }_2)x$
$F=-kx,$ frequency $=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$
$\Rightarrow f=\frac{1}{2\pi }\sqrt{\frac{\rho gA(\sin {\theta }_1+\sin {\theta }_2)}{m}}$