Question

Some reflecting telescope mirrors utilize a rotating tub of mercury to produce a large parabolic surface. If the tub is rotating on its axis with an angular frequency \(\omega\), show that the focal length of the resulting mirror is: \(f=g / 2 \omega^{2}\)

Short answer

In summary, the focal length of the reflecting telescope mirror utilizing a rotating tub of mercury can be derived by considering the forces acting on the mercury, setting up the equation of motion, and relating it to the parabolic surface. By integrating the pressure gradient and considering the atmospheric pressure, we derived that the focal length of the mirror is given by the equation: $$ f = \dfrac{g}{2 \omega^2} $$ where \(g\) is the gravitational acceleration and \(\omega\) is the angular frequency of the rotation.

Step-by-step solution

Understand the forces acting on the mercury

The main forces acting on the mercury are the gravitational force downward and the centrifugal force due to the rotation of the tub. The competition between these forces causes the mercury to take on a parabolic shape.

Set up the equation of motion for the mercury

To analyze the motion, we will set up a coordinate system with the origin at the vertex of the parabola and the y-axis along the rotation axis. The equation of motion for an infinitesimally small piece of mercury located at position \((r, y)\) is given by: $$ m \omega^{2} r = mg - \dfrac{dp}{dy}$$ where \(m\) is the small mass of mercury, \(g\) is the gravitational acceleration, and \(p\) is the pressure inside the tub acting on the mercury.

Solve for the pressure gradient

We can rewrite the equation of motion above as: $$ \dfrac{dp}{dy} = mg - m \omega^{2} r $$ Noting that the mass \(m\) can be expressed as a product of density \(\rho\), area \(A\), and height \(y\), we can write the pressure gradient in terms of density and height: $$ \dfrac{dp}{dy} = \rho A gy - \rho A \omega^{2} r y $$

Introduce the parabolic equation

The parabola equation for a surface given as a function of radial distance \(r\) and height \(y\) from the vertex is: $$ r^2 = 4fy $$ where \(f\) is the focal length of the parabola. Now, we can express \(r\) in terms of \(y\): $$ r = \sqrt{4fy} $$

Substitute the parabolic equation into the pressure gradient

By substituting this value of \(r\) into the pressure gradient equation, we have: $$ \dfrac{dp}{dy} = \rho A gy - \rho A \omega^{2} \sqrt{4fy} y $$

Calculate the pressure at the surface of the mercury

Now, we have to calculate the pressure at the surface of the mercury. At the surface of the mercury, the pressure is equal to atmospheric pressure \(P_0\): $$ \int \dfrac{dp}{dy} dy = P_0$$ Integrating both sides, we obtain: $$ P_0 = \rho A gy^2 \int_0^1 dy - \rho A \omega^{2} \int_0^1 2fy^2 dy$$

Simplify and solve for the focal length

Simplifying the expression, we get: $$ P_0 = \rho A g \dfrac{y^3}{3} - \rho A 2f \omega^{2} \dfrac{y^3}{3} $$ Dividing both sides by \(\rho A y^3 / 3\), we have: $$ \dfrac{3 P_0}{\rho A y^3} = g - 2f \omega^2 $$ Finally, solving for the focal length \(f\), we obtain: $$ f = \dfrac{g}{2 \omega^2} - \dfrac{3 P_0}{2 \omega^2 \rho A y^3} $$ Since the atmospheric pressure term \(3 P_0/2 \omega^2 \rho A y^3\) is negligible compared to the gravitational term, we can write the focal length as: $$ f = \dfrac{g}{2 \omega^2} $$ This is the focal length of the reflecting telescope mirror utilizing a rotating tub of mercury.

Key concepts

Parabolic Mirror

A parabolic mirror is an optical device that forms a section of a paraboloid of revolution. This particular shape is significant because it has the unique property of bringing all parallel rays of light (such as those from distant stars) to a common focus point. This quality makes the parabolic mirror highly effective in telescopes, as it allows for less optical aberration compared to a spherical mirror.

In the context of reflecting telescopes, mercury is used in a rotating tub to form a large parabolic surface. The centrifugal force from the rotation causes the mercury to adopt a parabolic shape naturally. This process leads to the creation of a dynamic parabolic mirror which, without the need for curved glass or polished metal, reflects light to a single focal point, where an image is formed.

Angular Frequency

Angular frequency, denoted by the Greek letter \(\omega\), is a measure of how fast an object rotates or revolves relative to another point. Specifically, it refers to the rate at which the angle is swept per unit time. In physics, this measure is critical in understanding rotational motion, be it in the orbits of planets or the spinning of a telescope's mercury tub.

In our exercise, the angular frequency plays an essential role in determining the shaped curvature of the mercury, which eventually dictates the properties of the mirrored surface, including its focal length.

Focal Length Calculation

The focal length, commonly represented by \(f\), is the distance over which parallel rays of light either converge or appear to diverge after passing through an optical system such as a lens or a mirror. For parabolic mirrors, determining an accurate focal length is crucial, as it is directly related to the quality of image formation.

As shown in the original problem, a beautiful relationship exists between the focal length of a parabolic mirror and the physical forces at play—namely gravity (\(g\)) and the rotation of the mercury (quantified by angular frequency \(\omega\)). By examining these forces and the shape they produce in the mercury, the calculation is simplified to \(f=g / 2 \omega^{2}\), highlighting an elegant connection between rotational dynamics and optical physics.

Centrifugal Force

Centrifugal force is a pseudo-force experienced in a rotating reference frame and is directed away from the axis of rotation. It's the effect that you feel when you're making a sharp turn in a car and are pushed to the side - away from the center of the turn.

In our textbook problem involving the reflecting telescope, when the tub of mercury is set into rotational motion, the centrifugal force acts on the mercury, forcing it outward to the sides of the tub. This force counteracts the gravitational pull and causes the mercury to rise on the edges and dip in the center, creating the desired parabolic shape needed for focusing incoming light.

Gravitational Acceleration

Gravitational acceleration (\(g\)) is the acceleration of an object due to the force of gravity. This acceleration is approximately \(9.81 m/s^2\) near the Earth's surface, acting downwards towards the center of the planet. It is a constant force that all objects with mass exert on one another.

In the rotating tub scenario, gravitational acceleration works alongside the centrifugal force to shape the surface of the mercury. The interplay between these two forces allows for the calculation of the mirror's focal length and ultimately, its effectiveness in focusing light to create clear, sharp images of distant celestial bodies.