A U-tube filled with liquid is being rotated about an axis...

A U-tube filled with liquid is being rotated about an axis with angular speed $\omega$ as shown in the figure. The dilfercnce in heights of liquid in two arms is (a) $\frac{\omega^{2}}{2 g}\left(\begin{array}{ll}r_{1}^{2} & r_{2}^{2}\end{array}\right)$ (b) $\frac{\omega^{2}}{g}\left(\begin{array}{ll}r_{1}^{2} & r_{2}^{2}\end{array}\right)$ (c) $\frac{2 \omega^{2}}{g}\left(r_{1}^{2} \mid r_{2}^{2}\right)$ (d) $\frac{\omega^{2}}{2 g}\left(r_{1}^{2} \mid r_{2}^{2}\right)$

A U-tube filled with liquid is being rotated about an axis with angular speed $\omega$ as shown in the figure. The dilfercnce in heights of liquid in two arms is
(a) $\frac{\omega^{2}}{2 g}\left(\begin{array}{ll}r_{1}^{2} & r_{2}^{2}\end{array}\right)$
(b) $\frac{\omega^{2}}{g}\left(\begin{array}{ll}r_{1}^{2} & r_{2}^{2}\end{array}\right)$
(c) $\frac{2 \omega^{2}}{g}\left(r_{1}^{2} \mid r_{2}^{2}\right)$
(d) $\frac{\omega^{2}}{2 g}\left(r_{1}^{2} \mid r_{2}^{2}\right)$

Key Concepts

Rotating Reference Frames

When analyzing fluids in motion due to rotation, it is essential to work in a rotating reference frame. In such a frame, inertial or pseudo forces, including the centrifugal force, must be introduced to correctly describe the dynamics. This approach is key to understanding how fluids behave when the container rotates, as it allows us to compute the net forces acting on fluid elements.

Centrifugal Force

The centrifugal force is an apparent force experienced by masses in a rotating frame, acting outward from the axis of rotation. In fluid problems, it causes the free surface to assume a curved shape since the effective acceleration at each point is the vector sum of gravity and the centrifugal acceleration. The magnitude of this force is proportional to the square of the angular speed and the radial distance from the axis, making it pivotal in calculating height differences in fluids under rotation.

Hydrostatic Equilibrium in a Rotating System

Hydrostatic equilibrium in a rotating fluid entails the balance between the pressure gradient and the combined effect of gravitational and centrifugal forces. Under equilibrium, the pressure within the fluid is constant along a surface of constant effective potential. This analysis allows for the derivation of an expression relating the difference in liquid heights to the angular speed and the radial position.

Free Surface Profile of Rotating Fluids

In a rotating system, the free surface of a fluid conforms to a shape determined by the balance of gravitational and centrifugal potentials. The expression for the height of the fluid at a given distance from the axis often involves terms proportional to the square of the radial distance. This concept is critical for determining the height difference in configurations like a U-tube as well as other rotating fluid scenarios.

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