Problem 68 - University Physics with Modern Physics

Question

Oscillations of a Piston. A vertical cylinder of radins \(r\) contains a quantity of ideal gas and is fitted with a piston with mass \(m\) that is free to move (Fig. 19.34\()\) . The piston and the walls of the cylinder are frictionless and the entire cylinder is placed in aconstant-temperature bath. The outside air pressure is \(p_{0}\) . In equilibrium, the piston sits at a height \(h\) above the bottom of the cylinder. (a) Find the absolute pressure of the gas trapped below the piston when in equilibrium. (b) The piston is pulled up by a small distance and released. Find the net force acting on the piston when its base is a distance \(h+y\) above the bottom of the cylinder, where \(y\) is much less than \(h\) (c) After the piston is displaced from equilibrium and released, it oscillates up and down. Find the frequency of nthese small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?

Step-by-Step Solution

Answer

(a) The absolute pressure is \(p_0 + \frac{mg}{\pi r^2}\). (b) The net force is \(-\frac{mg}{h}y\). (c) Frequency is \(\frac{1}{2\pi}\sqrt{\frac{g}{h}}\). For large displacements, oscillations are not simple harmonic.

Step 1: Determine Pressure in Equilibrium

In equilibrium, the forces acting on the piston must balance. These forces include the force due to the gas pressure inside the cylinder, the gravitational force due to the mass of the piston, and the external atmospheric pressure. The pressure inside the cylinder is denoted by \(P\). Hence, the force equation is:\[ P \cdot A = mg + p_0 \cdot A \]Where \(A = \pi r^2\) is the cross-sectional area of the piston. Rearranging to find the absolute pressure \(P\), we have:\[ P = p_0 + \frac{mg}{A} = p_0 + \frac{mg}{\pi r^2} \]

Step 2: Find Net Force on Displaced Piston

When the piston is displaced by a small distance \(y\), the height above the cylinder base changes to \(h+y\). The pressure in the gas changes due to this displacement. Using the ideal gas law and the constant temperature assumption, we approximate it as a small change:\[ PV = \text{const} \implies (p_0 + \frac{mg}{\pi r^2})A(h+y) = \text{const} \]The change in pressure \(\Delta P\) is tied to the change in volume, leading to a net force \(F\) as:\[ F = -A \cdot \Delta P \approx -\frac{\gamma mg}{h}y \]where \(\gamma\) is a specific ratio for small changes.

Step 3: Oscillation Frequency Calculation

For small oscillations, the motion of the piston can be considered as simple harmonic. From Hooke's law, the acceleration of the piston \(a\) follows the equation:\[ m\frac{d^2y}{dt^2} = -\frac{mg}{h}y \]Comparing with \(-ky = m\frac{d^2y}{dt^2}\) where \(k = \frac{mg}{h}\), the angular frequency \(\omega\) becomes:\[ \omega = \sqrt{\frac{g}{h}} \]Thus, the frequency \(f\) is:\[ f = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{g}{h}} \]

Step 4: Evaluate Nature of Oscillations for Large Displacements

For small oscillations, the piston follows simple harmonic motion because the restoring force is directly proportional to displacement. However, for larger displacements, the assumption of linearity in restoring force breaks down due to nonlinear terms coming from larger gas compression or expansion, leading to non-simple harmonic behavior.

Key Concepts

Ideal Gas

An ideal gas is a theoretical gas that follows the ideal gas law, which is a good approximation for many real gases at high temperatures and low pressures. The ideal gas law is written as: \[PV = nRT\]where:

• \( P \) is the pressure exerted by the gas.
• \( V \) is the volume the gas occupies.
• \( n \) is the number of moles of the gas.
• \( R \) is the ideal gas constant.
• \( T \) is the temperature in Kelvin.

For the problem of the oscillating piston, the ideal gas behaves predictably, allowing us to calculate changes in pressure and volume as the piston moves. Since the system is in a constant-temperature bath, the temperature remains constant, simplifying our calculations using the ideal gas law. This is crucial when considering small oscillations of the piston because the gas responds almost perfectly adiabatically to any changes in volume.

Equilibrium Pressure

In equilibrium, the piston in the cylinder is balanced by a combination of forces. These forces include the weight of the piston, the atmospheric pressure outside, and the internal pressure exerted by the trapped gas. To find the equilibrium pressure inside the cylinder, we use the force balance equation:\[P \cdot A = mg + p_0 \cdot A\]where:

• \( P \) is the equilibrium pressure of the gas.
• \( A = \pi r^2 \) is the cross-sectional area of the piston.
• \( mg \) represents the gravitational force due to the piston's mass.
• \( p_0 \) is the atmospheric pressure.

Solving for \( P \), we get:\[P = p_0 + \frac{mg}{\pi r^2}\]This shows that the gas pressure inside has to support both the piston's weight and counteract the atmospheric pressure, thus it is slightly higher than \( p_0 \). When the piston is pulled up or down, this equilibrium pressure is disrupted, leading to oscillations.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of oscillatory motion where the restoring force is directly proportional to the displacement from an equilibrium position and is always directed towards that position. In this exercise, the piston's motion can be considered as SHM when it is displaced by a small distance from its equilibrium position. The force equation becomes:\[m\frac{d^2y}{dt^2} = -\frac{mg}{h}y\]where:

• \( y \) is the small displacement from the equilibrium position.
• \( h \) is the equilibrium height of the piston above the bottom of the cylinder.

The negative sign indicates that the force is a restoring force, aiming to bring the piston back to its original position. SHM in this scenario is characterized by a consistent period and frequency, dependent on the properties of the system like mass \( m \) and equilibrium height \( h \). It's important to note that SHM assumes linear behavior, which holds true only for small displacements. Larger displacements may result in more complex, non-linear behavior.

Oscillation Frequency

The frequency of oscillation tells us how often the piston completes one cycle of movement per unit time. For small oscillations of the piston, the motion can be modeled as simple harmonic, with a frequency given by:\[f = \frac{1}{2\pi}\sqrt{\frac{g}{h}}\]where:

• \( f \) is the frequency of oscillation.
• \( g \) is the acceleration due to gravity.
• \( h \) is the equilibrium height.

This formula results from comparing the motion of the piston to that of a harmonic oscillator, where \( \omega = \sqrt{\frac{g}{h}} \) is the angular frequency. The derived frequency formula reflects how the gravitational force and equilibrium height establish the oscillation characteristics. If the displacements were large, the system would no longer exhibit simple harmonic behavior, as non-linear forces would dominate, causing unpredictable oscillation patterns. This in-depth understanding of oscillation frequency helps in predicting the piston's motion over time.