Question

... CP Oscillations of a Piston. A vertical cylinder of radius rcontains an ideal gas and is fitted with a piston of mass m that is free to move (Fig. P18.79). 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. In equilibrium, the piston sits at a height habove 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 above the bottom of the cylinder, where (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?

Short answer

(A)The absolute pressure of the gas trapped below the piston when in equilibrium is

(B)The net force acting on the piston when its base is a distance above the bottom of the cylinder is

(c)The frequency of the small oscillations is

(d)If the displacement are not very small then motion of the piston is not simple harmonic motion

Step-by-step solution

Step 1:Determine the absolute pressure of the gas trapped below the piston when in equilibrium  

The pressure in the cylinder will be equal to pressure outside in addition with the pressure exerted by the piston weight on cylinder gas therefore

where m is the mass of the piston and r is the radius

Therefore The absolute pressure of the gas trapped below the piston when in equilibrium is

:Determine the net force acting on the piston when its base is a distance above the bottom of the cylinder

The piston is pulled by small distance y so pressure is

we know

Solve for Net force taking positive direction upward

As y increases net force increase in opposite direction

therefore gas is at lower pressure than equilibrium pressure and net force tends to restore the piston toequilibrium

therefore the net force acting on the piston when its base is a distance above the bottom of the cylinder is

Step 3:Determine the frequency of the small oscillations.

solve for angular frequency first

Solve for frequency

Therefore the frequency of the small oscillations is

Determine If the displacement is not small, are the oscillations simple harmonic

If thedisplacementare not very small then motion of thepistonis not simple harmonic motion

if thedisplacementthen the gas is compressed to very small volume in this casetheforce due to pressure of gas willnotremainproportionaltodisplacementTherefore motion will not remain simple harmonic

If the displacement is force due to pressure of gas is very small In this caserestoringforce andweightbecome same this indicate that motion is not simple harmonic

Therefore If the displacement are not very small then motion of the piston is not simple harmonic motion