Question

One proposal for a space-based telescope is to put a large rotating liquid mirror on the Moon. Suppose you want to use a liquid mirror that is \(100.0 \mathrm{~m}\) in diameter and has a focal length of \(347.5 \mathrm{~m} .\) The gravitational acceleration on the Moon is \(1.62 \mathrm{~m} / \mathrm{s}^{2}\). a) What angular velocity does your mirror have? b) What is the linear speed of a point on the perimeter of the mirror? c) How high above the center is the perimeter of the mirror?

Short answer

Question: Calculate the angular velocity, linear speed, and height of a rotating liquid mirror on the Moon with a diameter of 100 meters and a focal length of 347.5 meters. Answer: The angular velocity of the rotating liquid mirror is 0.180 rad/s, the linear speed of a point on the perimeter is 9.0 m/s, and the height above the center of the mirror is approximately 1.80 meters.

Step-by-step solution

a) Finding the angular velocity

To find the angular velocity of the mirror, we can use the following equation derived from the relationship between centripetal acceleration and gravitational acceleration: \(g_{moon} = a_c = R\omega^2.\) Here, \(g_{moon}\) is the gravitational acceleration on the Moon, \(R\) is the radius of the mirror, and \(\omega\) is the angular velocity we are trying to find. First, let's find the radius of the mirror. Given the diameter \(D\) is \(100.0 \mathrm{~m}\), we can use the formula: \(R = \frac{D}{2}\) which gives \(R= 50.0 \mathrm{~m}\) Now, we can rearrange the equation to solve for \(\omega\): \(\omega = \sqrt{\frac{g_{moon}}{R}}\) Plugging in the given values, we get: \(\omega = \sqrt{\frac{1.62 \mathrm{~m}/\mathrm{s}^2}{50.0 \mathrm{~m}}} = 0.180 \mathrm{~rad/s}\)

b) Finding the linear speed

To find the linear speed of a point on the perimeter of the mirror, we use the following equation: \(v = R\omega\) where \(v\) is the linear speed and \(\omega\) is the angular velocity we found in step (a). \(v = (50.0 \mathrm{~m})(0.180 \mathrm{~rad/s}) = 9.0 \mathrm{~m/s}\) So, the linear speed of a point on the perimeter of the mirror is \(9.0 \mathrm{~m/s}\).

c) Finding the height above the center

To find the height above the center of the mirror, we can use the properties of a parabolic liquid surface. For a parabolic mirror, the equation of the parabola is given by: \(y = \frac{x^2}{4f}\) where \(y\) is the height above the center, \(x\) is the distance from the center (which is equal to the radius of the mirror), and \(f\) is the focal length of the mirror. In our case, \(x = R = 50.0 \mathrm{~m}\) and \(f = 347.5 \mathrm{~m}\). Plugging in these values, we get: \(y = \frac{(50.0 \mathrm{~m})^2}{4(347.5 \mathrm{~m})} = \frac{2500}{1390} \mathrm{~m} \approx 1.80 \mathrm{~m}\) So, the height above the center of the mirror is approximately \(1.80 \mathrm{~m}\).

Key concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves around a central point. In the context of a liquid mirror telescope, angular velocity describes how fast the mirror spins around its axis. More formally, angular velocity, denoted as \(\omega\), is defined as the rate at which the angular position of a point on the mirror changes with time.

In the solution provided, we calculated the angular velocity of a liquid mirror on the Moon using the relation between the centripetal acceleration necessary to maintain the mirror's shape and the gravitational acceleration present on the Moon. It's essential for students to grasp that angular velocity is not the same as linear speed, though the two are related: a point on the mirror's perimeter travels the circumference of the mirror in the same time that it takes for one full rotation, defined by the angular velocity.

Linear Speed

Linear speed, often referred to simply as speed, is the distance traveled per unit of time. It represents how fast an object is moving along a path. In circular motion, such as that of the rotating liquid mirror telescope, linear speed is related to the angular velocity and the radius of the circular path.

Calculating the linear speed involves multiplying the angular velocity by the radius of the mirror, as seen in the solution provided. This value gives us the speed at which a point on the perimeter of the mirror moves in a linear direction as the mirror rotates. The concept of linear speed is crucial for understanding the dynamics of objects in rotation since it directly affects the necessary centripetal force to maintain circular motion.

Parabolic Liquid Surface

The Parabolic Shape Explained

When a liquid rotates, the surface naturally forms a parabola. This characteristic feature is exploited in the design of a rotating liquid mirror telescope. The parabolic liquid surface is critical for focusing incoming light to a single focal point. The exact shape required for the parabola is governed by the speed of rotation and the properties of the liquid.

In the context of the exercise, to determine the height above the center, we used the parabolic equation, factoring in the mirror's radius and the designated focal length. It's pivotal for students to understand that the parabola's properties dictate the effectiveness of the telescope, especially its ability to focus light precisely.

Centripetal Acceleration

Centripetal acceleration is the acceleration that is needed to keep an object moving in a circular path and is directed towards the center of the circle. It's this kind of acceleration that maintains the circular motion of the liquid mirror. For the mirror to retain its shape and function correctly, centripetal acceleration must balance with the gravitational pull on the Moon.

In our example, the gravitational acceleration of the Moon factors into the equation, illustrating that the centripetal force needed comes from the Moon's gravity. Students must comprehend that without sufficient centripetal acceleration, the mirror would not maintain its parabolic shape, which is essential for its operation as a telescope. Understanding how this force works are fundamental to understanding the physics of not just telescopes but also planets, satellites, and other objects in circular motion.