With suitable schematic diagram, show that the height of the free surface of any liquid in a rotating container does not depend on the density of that fluid.

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To understand this concept, let's consider a cylindrical container filled with a liquid and rotating with a uniform angular velocity ω about its vertical axis.

Step 1: Draw the cylindrical container Draw a cylindrical container with its axis vertical. The container is filled with a liquid and is rotating with a uniform angular velocity ω.

Step 2: Identify the forces acting on the liquid There are two forces acting on the liquid in the container. The first is the gravitational force acting downwards, and the second is the centrifugal force acting radially outwards due to the rotation of the container.

Step 3: Draw the free surface When the container is rotating, the liquid will move away from the axis of rotation due to the centrifugal force. This causes the free surface of the liquid to take the shape of a paraboloid. The height of the liquid at any point is given by h = (ω²r²)/2g, where r is the distance from the axis of rotation, ω is the angular velocity, and g is the acceleration due to gravity.

Step 4: Show that the height does not depend on the density of the liquid The height of the free surface of the liquid depends on the square of the distance from the axis of rotation (r²), the square of the angular velocity (ω²), and inversely on the acceleration due to gravity (g). It does not depend on the density of the liquid. This is because both the gravitational force and the centrifugal force depend on the mass (and therefore the density) of the liquid, but the ratio of these two forces (which determines the shape of the free surface) does not depend on the mass or density.

Therefore, the height of the free surface of any liquid in a rotating container does not depend on the density of that fluid.