A U-tube is of non-uniform cross-section. The area of cross...

A U-tube is of non-uniform cross-section. The area of cross sections of two sides of tube are $A$ and $2 A$ (see fig.). It contains non-viscous liquid of mass $m$. The liquid is displaced slightly and free to oscillate. Its time period of oscillations is : (a) $T=2 \pi \sqrt{\frac{m}{3 \rho g A}}$ (b) $T=2 \pi \sqrt{\frac{m}{2 \rho g A}}$ (c) $T=2 \pi \sqrt{\frac{m}{\rho g A}}$ (d) None of these

A U-tube is of non-uniform cross-section. The area of cross sections of two sides of tube are $A$ and $2 A$ (see fig.). It contains non-viscous liquid of mass $m$. The liquid is displaced slightly and free to oscillate. Its time period of oscillations is :
(a) $T=2 \pi \sqrt{\frac{m}{3 \rho g A}}$
(b) $T=2 \pi \sqrt{\frac{m}{2 \rho g A}}$
(c) $T=2 \pi \sqrt{\frac{m}{\rho g A}}$
(d) None of these

Key Concepts

U-tube Oscillations

This concept deals with the behavior of a fluid in a U-shaped tube when it is displaced from its equilibrium position. The fluid oscillates back and forth under the influence of gravity, and the period of this oscillation depends on factors such as the gravitational acceleration, the density of the liquid, and the geometry of the tube. The analysis typically involves understanding how the displacement in one arm is coupled with the displacement in the other due to the continuity of the fluid.

Simple Harmonic Motion

In many cases, especially for small displacements, the oscillatory motion of the liquid in a U-tube can be approximated as simple harmonic motion (SHM), where the restoring force is proportional to the displacement from equilibrium. This proportionality leads to oscillations with a well-defined period that can be derived from the system’s physical parameters. The assumption of SHM greatly simplifies the analysis by allowing the use of standard formulas.

Conservation of Mass

This principle is crucial when analyzing fluid oscillations in a U-tube because the liquid is incompressible. A displacement in one arm of the tube must be matched by an opposite displacement in the other arm, ensuring that the total volume of the liquid remains constant. In tubes with different cross-sectional areas, this relationship is used to link the displacements in each arm, directly influencing the dynamics of the oscillation.

Effective Restoring Force and Energy Considerations

The oscillation of the liquid arises from the gravitational potential energy change when the liquid is displaced. Calculating the change in potential energy leads to determining the effective restoring force, which in turn is used to derive the equations of motion for the oscillating liquid. In cases with non-uniform cross-sections, the geometry alters the relationship between the displacement and the change in potential energy, affecting the effective mass involved in the motion and the resulting time period of the oscillation.

Transcript

Please give Ace some feedback