Rotation of adiabatic tube

Classical Mechanics Level 5

Consider an adiabatic tube which is divided into compartments by movable adiabatic pistons of negligible mass. The compartments have same length. Initially each of the compartments contains gas of same pressure and temperature.

Now, the tube starts to rotate at uniform angular velocity \(\omega\) about axis through center perpendicular to length. The length of the compartments are small enough to consider the pressure in a particular compartment constant.

After some period of time, the system comes to equilibrium.

Let, \(x\) represents the initial position (from the center of the tube) of a piston, and \(y\) represents its final position.

In the graph \(y~ vs~ x\), the slope at some point \((x,y)\), \(\frac{dy}{dx}\) can be expressed as \((Ay^2 + B)^n\), where \(A, B \) and \(n\) are constants.

Calculate the absolute value of \(nA\) in \(10^{-5}~m^{-2}\) unit.

Details and assumptions:

Initial temperature \( T_0 = 298~ K\)

Angular velocity \( \omega = 3~ rad \cdot s^{-1}\)

Molar gas constant \(R = 8.314~ J\cdot K^{-1} \cdot mol ^{-1}\)

Molecular mass of the gas \(M = 32 \times 10^{-3} kg\cdot mol^{-1}\) and the gas is diatomic.


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