Question

A U-shaped glass tube with a uniform crosssectional area, \(A\), is partly filled with fluid of density \(\rho\). Increased pressure is applied to one of the arms, resulting in a difference in elevation, \(h\), between the two arms of the tube, as shown in the figure. The increased pressure is then removed, and the fluid oscillates in the tube. Determine the period of the oscillations of the fluid column. (You have to determine what the known quantities are.)

Short answer

Answer: The period of oscillation of a fluid column in a U-shaped tube is given by the formula \(T = 2\pi\sqrt{\dfrac{L}{g}}\), where \(T\) is the period, \(L\) is the length of the oscillating fluid column, and \(g\) is the acceleration due to gravity.

Step-by-step solution

Identifying the forces on the fluid column

Let's assume the fluid oscillates in the horizontal direction along the tube and consider a small section of the fluid column at a depth \(y\) below the surface in one arm. This section would experience two forces: (1) weight force due to the mass of the fluid column above it and (2) the difference in fluid pressures from the two arms of the tube since they have different heights.

Writing the equation of motion

We assume that the height of the fluid in one arm is \(y\) and the height of the fluid in the other arm is \(y + h\). Then the pressure difference between two arms of the tube is given by \(\Delta P = \rho g (y + h - y) = \rho g h\). The pressure difference acts on the area \(A\) and results in a net force of \(F = \Delta P \cdot A = \rho g h A\). The acceleration of the section of the fluid column due to the net force can be obtained by applying Newton's second law of motion: \(F = m a\), where \(m\) is the mass of the oscillating fluid column. Since \(m = \rho V\) (where \(V\) is the volume of the oscillating fluid column) and \(V = A L\) (where \(L\) is the length of the fluid column that oscillates), we can write the equation of motion as \(\rho g h A = (\rho A L) a\).

Relating the acceleration and displacement

Since the fluid oscillates, we can relate its acceleration, \(a\), to the displacement, \(h\), using simple harmonic motion equations. In simple harmonic motion, the acceleration is proportional to the displacement from the equilibrium position and acts in the opposite direction. Mathematically, we can write \(a = -\omega^2 h\), where \(\omega\) is the angular frequency of oscillation.

Finding the angular frequency

Now, substitute the expression for \(a\) in terms of \(h\) from step 3 into the equation of motion derived in step 2: \(\rho g h A = - (\rho A L) \omega^2 h\). Simplify the equation to find the angular frequency: \(\omega^2 = \dfrac{g}{L}\). Then, the angular frequency is given by \(\omega = \sqrt{\dfrac{g}{L}}\).

Calculating the period of oscillation

Lastly, we need to find the period of oscillation, \(T\). The angular frequency and the period of oscillation are related as \(\omega = \dfrac{2\pi}{T}\). Solving for \(T\), we get \(T = \dfrac{2\pi}{\omega}\). Substituting the expression for \(\omega\) from step 4, the period of the fluid's oscillation is given by \(T = 2\pi\sqrt{\dfrac{L}{g}}\).

Key concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. It's important to visualize this concept: picture a spring with a mass attached to one end. If you displace the mass from its equilibrium position and let go, the spring will pull it back towards the center, but then it overshoots and oscillates back and forth.

In the context of fluid oscillations in a U-tube, the fluid acts similarly to the mass on a spring. When displaced, gravity provides the restoring force that tries to bring the fluid back to its original level. The characteristics of SHM, such as the oscillation's frequency and period, can be determined by the system's physical properties, just like the mass and spring constant in the spring-mass system.

Understanding SHM is crucial for predicting the behavior of many physical systems, from the oscillation of fluids to the motion of pendulums and the vibration of atoms in a crystal.

Equation of Motion

The equation of motion is the mathematical relationship that describes the motion of an object. It connects forces acting on the object, its mass, and its acceleration. In the U-tube scenario, the equation of motion for the fluid column derived from Newton's second law states that the net force acting on the fluid is equal to the mass of the fluid times its acceleration. To establish this equation, we account for the gravitational force acting on the displaced fluid and the restoring force that attempts to return the fluid to its equilibrium position, effectively modeling the U-tube as an oscillator.

In applying the SHM principles, we use the equation of motion to express the acceleration of the fluid in terms of its displacement from the equilibrium position and the angular frequency of oscillation. Through this relationship, we can solve for quantities like the angular frequency and subsequently the period of the oscillation.

Angular Frequency of Oscillation

The angular frequency of oscillation, often denoted by the Greek letter \(\omega\), is a measure of how fast something oscillates. It relates to how many radians an object will go through in its oscillatory path per unit of time. For SHM, it is found by taking the square root of the ratio of the restoring force's constant to the mass of the oscillating object.

In our U-tube example, after determining the equation of motion, we rearrange it to solve for \(\omega^2\) and obtain that \(\omega^2 = \dfrac{g}{L}\). Given that \(\omega = \sqrt{\dfrac{g}{L}}\), we now have a direct and simple quantitative measure of the fluid oscillation's rapidity within the tube. Understanding angular frequency is not just academic; it has real-world applications in designing systems that must manage or exploit oscillatory behaviors, such as shock absorbers in vehicles or the design of musical instruments.