Question

A U-tube is filled with a single homogeneous liquid. The liquid is temporarily depressed in one side by a piston. The piston is removed and the level of liquid in each side oscillates. Show that the period of oscillation is \(\pi \sqrt{2 L / g}\), where \(L\) is the total length of the liquid in the tube.

Short answer

The period of oscillation of the liquid in the U-tube, following a disturbance and release, is given by \(\pi \sqrt{2L/g}\), where \(L\) is the total initial length of the liquid in the tube.

Step-by-step solution

Understanding The Situation

We begin by visualizing and understanding the physical setup: a U-tube filled with a liquid at equilibrium. A perturbation is induced by depressing the liquid in one side with a piston, which is then removed. Due to this perturbation, the liquid level on either side of the U-tube starts to oscillate.

Formulating The Oscillation Equation

We must remember that the liquid level goes up in one tube and down in the other tube. If we consider the amount of liquid that has been displaced up by a height \(h\), the same volume is displaced downwards on the other side of the U-tube. As a result, the difference between the initial and final gravitational potential energy is zero. But, there will be a difference in the kinetic energy. We can equate this change in kinetic energy to the work done by gravity.

Applying Conservation of Energy & Solving The Equation

By using conservation of energy (initial and final energy are equal), we get the equation: \(\frac{1}{2} \rho A h^2 \frac{dh}{dt}^2 = \rho g A h h\). Here, \(\rho\) is the density of the liquid, \(A\) is the cross-sectional area of the tube and \(g\) is the acceleration due to gravity. Upon simplifying this equation and solving, we obtain the period \(T\) of oscillation, where \(T = 2 \pi \sqrt{\frac{h}{g}}\). Here, \(h\) is the height, or half the initial length of liquid \(L\). Hence, we get that \(T = \pi \sqrt{2 \frac{L}{g}}\) which matches the required concept.

Key concepts

Conservation of Energy

The principle of conservation of energy tells us that energy cannot be created or destroyed, only transformed from one form to another. In the context of a U-tube oscillation period, when the piston is removed and the liquid oscillates, energy is transferred between kinetic and potential forms, but the total energy in the system remains constant.

This principle allows us to set up the necessary equations to determine the period of oscillation. When the liquid is at its highest point in one arm of the U-tube, it possesses maximum gravitational potential energy and minimum kinetic energy. As the liquid falls, this potential energy is converted into kinetic energy. At the lowest point, the kinetic energy is at its peak. As the liquid rises in the other arm, the kinetic energy converts back to potential energy. Understanding this interplay between energy forms is crucial in solving for the oscillation period.

Kinetic and Potential Energy

In any oscillating system, two main types of energy are at play: kinetic energy, which is the energy of motion, and potential energy, which is the energy stored due to an object's position in a force field, like gravity. For the U-tube example, when the liquid is in motion, it possesses kinetic energy, and when it is at a height in one of the tube's arms, it has gravitational potential energy.

During the oscillation, the height of the liquid column changes, leading to a change in both kinetic and potential energy levels. When the liquid moves through its equilibrium position (the midpoint in the U-tube), it has maximal kinetic energy and zero potential energy relative to the equilibrium level. This exchange of energy types continues cyclically, and the total energy in the system remains constant if we ignore friction and other dissipative forces.

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy that an object possesses because of its height above a reference point. In our U-tube example, GPE is highest when the liquid is at the peak of its motion on either side of the tube, and it's calculated by the formula \( GPE = mgh \) where \( m \) is the mass of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid above the equilibrium level.

The process of the liquid levels oscillating can be viewed as a conversion between GPE and kinetic energy. At the peak, the kinetic energy is zero, and all the energy is stored as GPE. As the liquid falls, the GPE is transformed into kinetic energy until, at the midpoint, the potential energy relative to that point is zero.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. The motion of the liquid in the U-tube is a good example of SHM, as the gravitational force pulling the liquid back to its equilibrium position is proportional to the displacement from that level.

SHM is characterized by sinusoidal oscillations, and the period of such motion is the time it takes to complete one full cycle. The period of oscillation in SHM is independent of the amplitude of the motion, depending only on the properties of the system, such as mass and spring constant, or in the case of the U-tube, the length of the liquid column \( L \) and the acceleration due to gravity \( g \) as shown in our formula \( T = \pi \sqrt{2L/g} \) for the period of oscillation. Understanding SHM is key to analyzing many physical systems, including pendulums, springs, and in this case, liquid in a U-tube.