Edited:This may be a simple standard qn but im not gettin the final ans.An ideal gas is contained in a piston cylinder arrangement. Area of piston is A and its mass is M.The surrounding atm pressure is P0. Under equilbrium condition, volume of the gas is V0 . Find the angular frequency of small vertical oscillations of the piston assuming expansion and compression of the gas to be adiabatic. assume adiabatic constant =.The ans is

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Edited:
This may be a simple standard qn but im not gettin the final ans.
An ideal gas is contained in a piston cylinder arrangement. Area of piston is A and its mass is M.
The surrounding atm pressure is P₀. Under equilbrium condition, volume of the gas is V₀ . Find the angular frequency of small vertical oscillations of the piston assuming expansion and compression of the gas to be adiabatic. assume adiabatic constant =.

The ans is
$\sqrt{\frac{\gamma.A}{V_{0}}\times(g+\frac{P_{0}A}{m})}$

Shane Macguire , 11 Years ago

Grade upto college level

1 Answers

Askiitians Tutor Team

Last Activity: 5 Months ago

To determine the angular frequency of small vertical oscillations of the piston in an ideal gas system, we need to analyze the forces acting on the piston and how they relate to the properties of the gas. Let's break this down step by step.

Understanding the System

We have a piston with area A and mass M, which is in equilibrium under the influence of the atmospheric pressure P₀ and the pressure of the gas inside the cylinder. The volume of the gas at equilibrium is V₀.

Forces Acting on the Piston

At equilibrium, the force exerted by the gas pressure must balance the weight of the piston and the atmospheric pressure. The pressure of the gas can be expressed as:

Pressure of gas, P = P₀ + \frac{Mg}{A}

Here, g is the acceleration due to gravity. When the piston moves slightly, the volume of the gas changes, which in turn affects the pressure according to the adiabatic condition.

Adiabatic Process and Pressure-Volume Relationship

For an adiabatic process, the relationship between pressure and volume is given by:

P V^γ = constant

where γ (gamma) is the adiabatic constant. If the piston moves a small distance x, the new volume becomes V = V₀ + Ax. The corresponding pressure can be expressed as:

P = P₀ + \frac{Mg}{A} - \frac{dP}{dx} x

Small Oscillations and Angular Frequency

For small oscillations, we can apply Hooke's law analogy. The restoring force acting on the piston due to the change in pressure can be approximated as:

F = -k x

where k is the effective spring constant. The angular frequency ω of the oscillation can be derived from the formula:

ω = √(k/M)

Finding the Spring Constant

To find k, we differentiate the pressure with respect to volume:

k = -A \frac{dP}{dV}

Using the adiabatic relation, we can express k in terms of the initial conditions:

k = \frac{γ(P₀ + \frac{Mg}{A})}{V₀}

Final Expression for Angular Frequency

Substituting k into the angular frequency formula gives:

ω = √( \frac{γ(P₀ + \frac{Mg}{A})}{MV₀} )

Thus, the angular frequency of small vertical oscillations of the piston can be expressed as:

ω = √( \frac{γ(P₀ + \frac{Mg}{A})}{MV₀} )

This formula encapsulates the relationship between the mass of the piston, the area of the piston, the pressure of the gas, and the volume of the gas under adiabatic conditions. By plugging in the values for γ, P₀, M, A, and V₀, you can calculate the angular frequency of the oscillations.