Problem 69 - University Physics with Modern Physics

Question

CP Oscillations of a Piston. A vertical cylinder of radius \(r\) contains a quantity of ideal gas and is fitted with a piston with mass \(m\) that is free to move (Fig. P19.69). The piston and the walls of the cylinder are frictionless, and the entire cylinder is placed in a constant-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 these small oscillations. If the displacement is not small, are the oscillations simple harmonic? How can you tell?

Step-by-Step Solution

Answer

(a) \( p = p_0 + \frac{mg}{\pi r^2} \); (b) \( F = -\pi r^2 \frac{p}{h}y \); (c) \( f = \frac{1}{2\pi}\sqrt{\frac{\pi r^2 p}{mh}} \), non-linear for large displacements.

Step 1: Find Equilibrium Absolute Pressure

In equilibrium, the pressure of the gas inside the piston, \( p \), must balance the weight of the piston and the outside atmospheric pressure \( p_0 \). Hence, the force exerted by the gas is equal to the force due to gravity on the piston plus the force due to outside pressure. The force equation can be written as:\[ p \cdot A = mg + p_0 \cdot A \]where \( A = \pi r^2 \) is the cross-sectional area of the cylinder. Solving for \( p \), we get:\[ p = p_0 + \frac{mg}{\pi r^2} \]

Step 2: Determine Net Force on Displacement

When the piston is displaced by a distance \( y \), the new position is \( h + y \) and the gas pressure changes slightly. By considering the change in volume, the pressure in the gas after displacement, \( p + \Delta p \), can be found using the ideal gas law, assuming temperature is constant:\[ pV = (p + \Delta p)(V + \Delta V) \]where \( V = \pi r^2 h \) and \( \Delta V = \pi r^2 y \). Solving for \( \Delta p \), we use the approximation \( P \Delta V = V \Delta P \) (for small \( \Delta P \)), resulting in:\[ \Delta p = -\frac{p}{h}y \]The net force \( F \) acting on the piston is given by:\[ F = A \cdot (-\Delta p) = -\pi r^2 \frac{p}{h}y \]

Step 3: Calculate Frequency of Small Oscillations

The net force equation resembles Hooke's Law, \( F = -ky \), where:\[ k = \frac{\pi r^2 p}{h} \]The equation of motion is \( m\ddot{y} = -ky \), representing simple harmonic motion with an angular frequency \( \omega \) given by:\[ \omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{\pi r^2 p}{mh}} \]Thus, the frequency \( f \) is:\[ f = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\pi r^2 p}{mh}} \]If the displacement is not small, the restoring force becomes nonlinear, thus the oscillations are not simple harmonic.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. In the context of an ideal gas inside a cylinder with a movable piston, thermodynamics helps us to understand how the gas behaves under certain conditions.
In our exercise, we're dealing with a situation where the gas is kept at constant temperature, which is an example of an isothermal process. During an isothermal process, the product of the pressure and volume remains constant for an ideal gas, as described by the ideal gas law:

• Ideal Gas Law: \(PV = nRT\) where \(P\) is the pressure, \(V\) is the volume, \(n\) is the amount of gas in moles, \(R\) is the ideal gas constant, and \(T\) is the temperature.

Since the piston can move, the volume \(V\) of the gas will change, impacting the pressure \(P\) exerted by the gas. Thermodynamics gives us the tools to calculate changes in pressure when the volume changes, especially under the constraint of constant temperature.

Harmonic Oscillations

Harmonic oscillations describe the type of periodic motion seen in systems like springs and pendulums. These systems exhibit oscillations that are characterized by consistent oscillatory motion about an equilibrium point.
In this exercise, after the piston is slightly displaced from its equilibrium position and released, it starts to oscillate up and down. These are harmonic oscillations where the piston moves back and forth in a periodic manner. This type of motion can be represented by Hooke's Law, which states that the force acting on a displaced object in simple harmonic motion is proportional and opposite to its displacement:

• Hooke's Law: \( F = -ky \) where \(F\) is the force, \(k\) is the force constant, and \(y\) is the displacement.

In our scenario, the gas provides a restoring force that acts on the displaced piston, causing it to oscillate. The properties of these oscillations rely on the mass of the piston and the stiffness of the gas force comparable to a spring constant. When displacements are small, the motion closely resembles simple harmonic oscillation, characterized by the sinusoidal nature of the piston’s position with time.

Equilibrium Pressure

Equilibrium pressure is the pressure exerted by a gas when the system is in a state of balance. In the context of the exercise, it means that the pressure exerted by the gas inside the piston is balanced by the gravitational force on the piston and any external atmospheric pressure.
To find the absolute pressure in equilibrium, we consider the forces exerted on the piston. There are two main components:

• The weight of the piston, which depends on its mass \(m\) and gravity \(g\).
• The atmospheric pressure \(p_0\) acting from above.

By balancing these forces, the equilibrium pressure \(p\) of the gas can be calculated as:\[ p = p_0 + \frac{mg}{\pi r^2} \] Using this expression, we see that the equilibrium pressure is higher than the surrounding atmospheric pressure by an amount determined by the weight of the piston over the area. This ensures the piston maintains a steady height when not disturbed, setting the foundation for analyzing its oscillations once displaced from this equilibrium.