An ideal gas is enclosed in a vertical cylindrical container and supports a freely moving piston of

Q: Why is the process considered adiabatic?

The problem states that the system is completely isolated from its surroundings. This means no heat exchange can occur ($dQ = 0$), which is the definition of an adiabatic process.

Q: Why is there a negative sign in the relation $dp = -\gamma \frac{p}{V} dV$?

The negative sign indicates that pressure and volume change in opposite directions. When the volume decreases ($dV 0$), which provides the restoring force necessary for SHM.

Q: How does the atmospheric pressure $p_0$ affect the frequency of oscillation?

Atmospheric pressure increases the equilibrium pressure of the gas, which makes the gas 'stiffer' (higher effective spring constant). Thus, a higher atmospheric pressure $p_0$ leads to a higher frequency of oscillation.

Q: What would change if the cylinder was horizontal instead of vertical?

In a horizontal cylinder, gravity does not act along the line of motion of the piston. Therefore, the $Mg$ term would be zero, and the frequency would simplify to $f = \frac{1}{2\pi}\sqrt{\frac{\gamma p_0 A^2}{V_0 M}}$.

Q: Why do we assume the displacement $x$ is very small?

We assume $x$ is small so that the changes in pressure and volume are differential ($dp$ and $dV$). For large displacements, the relationship between force and displacement becomes non-linear, and the motion would no longer be simple harmonic.

Question

An ideal gas is enclosed in a vertical cylindrical container and supports a freely moving piston of mass $M$. The piston and the cylinder have equal cross-sectional area $A$. Atmospheric pressure is $p_0$ and when the piston is in equilibrium, the volume of the gas is $V_0$. The piston is now displaced slightly from its equilibrium position. Assuming that the system is completely isolated from its surroundings, show that the piston executes simple harmonic motion and find the frequency of oscillation.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a vertical cylindrical container containing an ideal gas.

This cylinder is closed at the top by a piston of mass $M$ and cross-sectional area $A$ that can move without friction.

Initially, the system is in a state of perfect harmony—**equilibrium**.

At this equilibrium position, the piston is completely stationary.

This means the net force acting on it must be zero. Let's list the forces acting on the piston:

1. **Atmospheric Pressure ($p_0$):** Pushes downward on the piston with a force of $p_0 A$.

2. **Gravity ($Mg$):** Pulls the piston downward with its weight $Mg$.

3. **Gas Pressure ($p_{eq}$):** Pushes upward on the piston with a force of $p_{eq} A$.

Equating the upward and downward forces gives us our first master equation:

$$p_{eq} A = p_0 A + Mg$$

Dividing by the area $A$, we find the equilibrium pressure of the gas:

$$p_{eq} = p_0 + \frac{Mg}{A}$$

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### The Adiabatic Constraint

Now, the problem states that the system is **completely isolated** from its surroundings.

This is a major clue!

Isolation means no heat can enter or leave the cylinder during the motion.

Therefore, any displacement of the piston will result in an **adiabatic process**.

The state equation for an adiabatic process is:

$$p V^\gamma = 	ext{constant}$$

where $\gamma$ is the adiabatic index of the gas.

To see how a small change in volume ($dV$) affects the pressure ($dp$), we differentiate both sides of this equation:

$$V^\gamma dp + \gamma p V^{\gamma-1} dV = 0$$

Solving for $dp$, we get:

$$dp = -\gamma \frac{p}{V} dV$$

This negative sign is physically beautiful—it tells us that if we compress the gas (decrease volume, $dV < 0$), the pressure will increase ($dp > 0$), pushing the piston back up.

This is the origin of our restoring force!

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### Displacing the Piston

Let's perturb the system.

Suppose we push the piston downward by a small distance $x$ from its equilibrium position.

This downward displacement decreases the volume of the gas.

Since the cross-sectional area is $A$, the change in volume is:

$$dV = -A x$$

Now, let's substitute this change in volume, along with the equilibrium values $p = p_{eq}$ and $V = V_0$, into our differential adiabatic relation:

$$dp = -\gamma \frac{p_{eq}}{V_0} (-A x) = \gamma \frac{p_{eq} A}{V_0} x$$

Notice how the two negative signs cancel out, resulting in a positive pressure increase $dp$ that is directly proportional to the displacement $x$.

---

### Finding the Restoring Force and Acceleration

This extra pressure $dp$ acts on the piston's area $A$, creating an upward force that opposes the downward displacement.

The net restoring force $F$ acting on the piston is:

$$F = - (dp) A = -\gamma \frac{p_{eq} A^2}{V_0} x$$

Now, let's substitute our expression for the equilibrium pressure $p_{eq} = p_0 + \frac{Mg}{A}$ into this force equation:

$$F = -\gamma \frac{\left(p_0 + \frac{Mg}{A}ight) A^2}{V_0} x = -\gamma \left(\frac{p_0 A^2 + MgA}{V_0}ight) x$$

Using Newton's second law, $F = Ma$, we can find the acceleration $a$ of the piston:

$$a = \frac{F}{M} = -\left[\frac{\gamma(p_0 A^2 + MgA)}{V_0 M}ight] x$$

Since the acceleration is directly proportional to the displacement $x$ and is directed opposite to it ($a \propto -x$), the piston will execute **Simple Harmonic Motion (SHM)**!

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### Calculating the Frequency of Oscillation

By comparing our acceleration equation with the standard SHM equation $a = -\omega^2 x$, we can identify the angular frequency $\omega$:

$$\omega^2 = \frac{\gamma(p_0 A^2 + MgA)}{V_0 M} \implies \omega = \sqrt{\frac{\gamma(p_0 A^2 + MgA)}{V_0 M}}$$

Finally, the frequency of oscillation $f$ is given by:

$$f = \frac{\omega}{2\pi} = \frac{1}{2\pi}\sqrt{\frac{\gamma(p_0 A^2 + MgA)}{V_0 M}}$$

This is our final, elegant result! It beautifully combines the principles of thermodynamics and mechanics.

Visualized Solution

Analyzing the Equilibrium State

Force Balance at Equilibrium

The Adiabatic Constraint

Displacing the Piston

Calculating the Restoring Force

Proving Simple Harmonic Motion

Finding the Frequency of Oscillation

Exploring Variations

The Sigma Insight: Thermodynamic Processes

Analyzing the Setup

The Adiabatic Constraint

Displacing the Piston

Finding the Restoring Force and Acceleration

Calculating the Frequency of Oscillation

Similar Questions

A given quantity of an ideal gas is at pressure $p$ and absolute temperature $T$ . The isothermal bulk modulus of the gas is

The $p$ - $T$ diagram for an ideal gas is shown in the figure, where $A C$ is an adiabatic process, find the corresponding $p$ - $V$ diagram

Visualized Solution

Analyzing the Equilibrium State

Force Balance at Equilibrium

The Adiabatic Constraint

Displacing the Piston

Calculating the Restoring Force

Proving Simple Harmonic Motion

Finding the Frequency of Oscillation

Exploring Variations

The Sigma Insight: Thermodynamic Processes

Analyzing the Setup

The Adiabatic Constraint

Displacing the Piston

Finding the Restoring Force and Acceleration

Calculating the Frequency of Oscillation

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