A bent tube of uniform cross-section area A has a...

A bent tube of uniform cross-section area $A$ has a non-viscous liquid of density $\rho$. The mass of liquid in the tube is $m$. The time period of oscillation of the liquid is : (a) $2 \pi \sqrt{\frac{m}{\rho g A}}$ (b) $2 \pi \sqrt{\frac{m}{2 \rho g A}}$ (c) $2 \pi \sqrt{\frac{2 m}{\rho g A}}$ (d) None of these

A bent tube of uniform cross-section area $A$ has a non-viscous liquid of density $\rho$. The mass of liquid in the tube is $m$. The time period of oscillation of the liquid is :
(a) $2 \pi \sqrt{\frac{m}{\rho g A}}$
(b) $2 \pi \sqrt{\frac{m}{2 \rho g A}}$
(c) $2 \pi \sqrt{\frac{2 m}{\rho g A}}$
(d) None of these

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Key Concepts

Energy Considerations in Oscillatory Motion

By considering the conversion between kinetic energy and gravitational potential energy in an oscillating liquid, one can derive the equation for the time period of the oscillation. The overall mass of the liquid and the effective displacement due to the area of the tube are key parameters in linking the energy methods to the oscillatory motion.

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion in which the restoring force is directly proportional to the displacement from the equilibrium position, leading to oscillations that follow a sine or cosine function with a definite period. This concept helps identify the dynamics of systems like the oscillating liquid as it follows a predictable time period based on its mass and the effective force constant.

Restoring Force in Fluid Systems

In fluid systems such as the bent or U-tube, the restoring force arises from the difference in hydrostatic pressures caused by a displacement of the liquid from its equilibrium level. When one side of the liquid rises, the gravitational potential energy increases, and this energy imbalance creates a force that acts to restore the liquid back to its original level, thereby generating oscillatory motion.

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