A horizontal tube of length l closed at both ends contains...

A horizontal tube of length $l$ closed at both ends contains IIc gas. The tube is rotaled at a constant angular velocity $\omega$ aboul a vertical axis passing through an cnd. $\Lambda$ ssuming temperature is uniform and consunt, which of the following is correct? (a) $P_{1}>P_{2}$ (b) $P_{2}>P_{1}$ (c) $P_{1}=P_{2}$ (d) Cannot say

A horizontal tube of length $l$ closed at both ends contains IIc gas. The tube is rotaled at a constant angular velocity $\omega$ aboul a vertical axis passing through an cnd. $\Lambda$ ssuming temperature is uniform and consunt, which of the following is correct?
(a) $P_{1}>P_{2}$
(b) $P_{2}>P_{1}$
(c) $P_{1}=P_{2}$
(d) Cannot say

Key Concepts

Isothermal Ideal Gas Behavior

When the gas is maintained at a uniform temperature, it behaves as an isothermal ideal gas. In this case, the pressure and density are directly proportional through the ideal gas law, which means that any spatial variation in density due to the centrifugal force will correspond to a similar variation in pressure. This relation is crucial in determining the equilibrium pressure distribution along the tube.

Rotational Dynamics and Centrifugal Force

In a rotating system, each fluid element experiences a centrifugal force that acts radially outward. This apparent force, arising from the rotation, effectively modifies the body forces acting on the gas. It plays a crucial role in determining how pressure distributes within the container, as regions farther from the rotation axis experience a higher outward acceleration.

Hydrostatic Equilibrium in a Rotating Frame

For a gas in a steady state within a rotating frame, the pressure gradient must balance the effective forces, including the centrifugal force. This condition, known as hydrostatic equilibrium, implies that the spatial change in pressure is directly related to the local mass density and the centrifugal acceleration. The resulting pressure variation is similar to that seen in gravitational fields but with the centrifugal acceleration replacing gravitational acceleration.

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