Problem 17 - Physics

Question

Michelson's setup for measuring the speed of light with the mirrors placed on Mt. San Antonio and Mt. Wilson in California, which are \(35 \mathrm{~km}\) apart. Using a value of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\) for the speed of light, find the minimum angular speed (in rev \(/ \mathrm{s}\) ) for the rotating mirror.

Step-by-Step Solution

Answer

The minimum angular speed is approximately 2145 rev/s.

Step 1: Understand the setup

In Michelson's setup, a beam of light is sent from one mountain to another, hits a mirror, and returns to the original point. The total distance traveled by the light is twice the distance between the mountains, or 70 km.

Step 2: Convert distance

Convert the distance from kilometers to meters because the speed of light is given in meters per second. Thus, the distance is \(70 \, \text{km} = 70,000 \, \text{meters}\).

Step 3: Determine time for light to travel

To find the time it takes for the light to travel to the second mirror and back, use the formula: \( t = \frac{d}{v} \), where \(d\) is the total distance (70,000 m) and \(v\) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)). Substitute the values to find \(t\).

Step 4: Calculate time for one revolution

Since we want the mirror to make half a revolution in the time \(t\) (so it can reflect the light back), the time for one full revolution is \(2t\).

Step 5: Find angular speed

The angular speed in revolutions per second is \( \omega = \frac{1}{2t} \), since one revolution takes \(2t\) seconds. Substitute the expression for \(t\) from Step 3 to find \(\omega\).

Step 6: Perform the Calculations

1. Calculate \(t = \frac{70,000}{3 \times 10^8} = 2.33 \times 10^{-4} \, \text{seconds}\). 2. The time for one full revolution is \(2 \times 2.33 \times 10^{-4} = 4.66 \times 10^{-4} \, \text{seconds}\). 3. Thus, \(\omega = \frac{1}{4.66 \times 10^{-4}} = 2145 \, \text{rev/s}\).

Key Concepts

Michelson Experiment

The Michelson Experiment is a groundbreaking method devised by American physicist Albert A. Michelson to measure the speed of light. Conducted in the late 19th century, this experiment utilizes a setup involving mirrors placed at significant distances from each other. The specific setup mentioned in this exercise places mirrors on Mt. San Antonio and Mt. Wilson, separated by 35 km. The fundamental idea is sending a beam of light from one location to another, where it bounces off the mirror and returns to the original point. This entire setup provides a total round-trip distance of 70 km for the light to travel. The aim is to precisely measure this travel using the speed of light and clever optical instrumentation, paving the way for accurate velocity calculations of light.

Rotating Mirror

The core instrument in Michelson's experiment is the rotating mirror. This mirror is not static but instead rotates at a specific angular speed. Its primary function within the setup is to reflect the light beam so that it travels back to its source after hitting the distant mirror. The rotation is crucial because it needs to align perfectly with the returning light beam to ensure accurate measurement. This means that during the time it takes for the light to travel the round-trip distance, the mirror should have rotated such that it can redirect the light correctly back. Hence, calculating the precise rotating speed is essential for the experiment's success in measuring the speed of light accurately. The mirror's rotation compensates for the time lapse during the light's journey.

Angular Speed Calculation

Calculating the angular speed of the rotating mirror is vital. To ensure that the mirror redirects the beam of light accurately, its speed needs to be calculated based on the time it takes for the light to complete its round trip. In this exercise, the angular speed \( \omega \) refers to revolutions per second. This is determined by the relationship: \[ \omega = \frac{1}{2t} \] where \( t \) is the time taken for light to travel to the mirror and back. 1. First, calculate the travel time \( t \) using the formula \( t = \frac{d}{v} \), where \( d \) represents the total distance (70,000 meters) and \( v \) is the speed of light. 2. Determining \( t \) gives the time for half of the revolution.3. Finally, one full revolution is twice \( t \), leading tothe calculation of \( \omega \) using the formula above. This ensures the mirror adjusts swiftly enough to reflect the incoming beam efficiently.

Light Travel Time

Light travel time is a core concept in this experiment and must be calculated accurately to determine the necessary mirror rotation speed. Light travel time refers to the duration it takes for the beam to travel from its source, hit the distant mirror, and return.- This time \( t \) is derived using the formula: \[ t = \frac{d}{v} \] * \( d \) represents the round-trip distance between the mirrors, i.e., 70,000 meters. * \( v \) is the speed of light presumed to be \(3 \times 10^{8}\) meters per second.After calculating \( t \), we understand how much time elapses for the light to complete the journey. This interval dictates how fast the mirror should rotate to catch and correctly redirect the returning light beam. Precise measurement of this time is crucial for accurate speed of light calculations.