Solution to Question 8
Principle:
A liquid rotating with constant angular velocity $\omega$ takes the shape of a paraboloid of revolution: $ y = \frac{\omega^2 x^2}{2g} $.
This surface acts as a concave parabolic mirror. For a distant star, the incoming light rays are parallel to the principal axis. A parabolic mirror focuses all such parallel rays to a single point, the Focus (F), located on the axis of symmetry.
Fig 1: Parallel rays focus at point F located above the vertex of the paraboloid.
Calculating the Focal Length:
The standard equation of a parabola opening upwards is $ y = \frac{x^2}{4f} $, where $f$ is the focal length (distance from the vertex to the focus).
Comparing this with the liquid surface equation $ y = \frac{\omega^2 x^2}{2g} $: $$ \frac{\omega^2}{2g} = \frac{1}{4f} $$
Solving for $f$: $$ 4f = \frac{2g}{\omega^2} \implies f = \frac{g}{2\omega^2} $$
To get a clear picture of the distant star, the photo film must be placed exactly at the focus.