Question

An ideal gas is enclosed in a vessel of volume `V_0` fitted with a piston of cross-sectional area `A` and mass `M`. The piston is slightly depressed and returned. Find the frequency of vibration of the piston?

Final Volume = `V`
Final pressure = `P`
Atmospheric pressure = `P_0`

Answer 1

`f = 1/(2 pi) sqrt((p_o gamma A^2)/(m V_o ))`

Explanation:

Before applying a simple harmonic motion (SHM) model to this system, note that in the SHM, say of a spring mass system, no allowance is made for the spring heating up, or for that energy leaving the system - both of which will happen in the real world.

So an idealization here is that this is an isentropic process, ie:

• `pV^gamma = p_o V_o^gamma qquad square , qquad "where "gamma = c_p/c_V`

Let `V_o = x_o A` and so for small depression `-x`:

• `V = (x_o - x) A = V_o - x A`

From `square` the resultant change in pressure is:

`implies Delta p = p - p_o = p_o ( (V_o/V)^gamma - 1)`

`= p_o ( (1 - x A/V_o)^(-gamma) - 1)`

A binomial expansion for small `x` also linearizes the system, essential for SHM:

`Delta p = p_o ( 1 + gamma x A/V_o + mathbb O (x^2) - 1)`

`implies Delta p approx (p_o gamma A)/V_o x`

The restorative force on the piston is linear so this is now idealized SHM:

• `F = - Delta p A`

And the equation of motion is:

• `F = M ddot x= - Delta p A`

`implies ddot x + (p_o gamma A^2)/(M V_o ) x = 0 qquad triangle`

Writing `triangle` as `ddot x + omega^2 x = 0`, with `omega = sqrt((p_o gamma A^2)/(M V_o ))`, the standard SHM solution is:

• `x = A cos (omega t + phi), qquad "with " omega = 2 pi f`

So, "frequency of vibration of the piston" is:

• `f = 1/(2 pi) sqrt((p_o gamma A^2)/(M V_o ))`

A dimensional Analysis of `sqrt((p_o gamma A^2)/(M V_o ) )` reveals:

• `sqrt( ((M L T^(-2))/(L^2) (1) (L^2)^2)/(M L^3 ) )= sqrt( T^(-2) ) = T^(-1)`

....Wich is what is to be expected.

But which does not guarantee that the answer is correct :(