To solve the problem of measuring the speed of light using a rotating toothed wheel, we need to break it down into two parts: first, calculating the angular speed of the wheel, and second, determining the linear speed at its edge. Let's tackle both parts step by step.

Calculating the Angular Speed of the Wheel

The angular speed of the wheel can be derived using the relationship between the distance traveled by light and the rotation of the wheel. When the light beam passes through the first slot, it travels to the mirror and back, passing through the next slot just in time. This means the time taken for the light to travel to the mirror and back is equal to the time it takes for the wheel to rotate through one full slot.

Given parameters:

• Distance to the mirror, L = 500 m
• Speed of light, c = 3.0 × 10⁵ km/s = 3.0 × 10⁸ m/s
• Number of teeth on the wheel, N = 500
• Radius of the wheel, r = 5.0 cm = 0.05 m

The total distance light travels to the mirror and back is:

D = 2L = 2 × 500 m = 1000 m

Now, we can find the time taken for this round trip:

Time (t) = Distance / Speed = D / c = 1000 m / (3.0 × 10⁸ m/s) = 3.33 × 10^-6 s

Since the wheel has 500 teeth, the time for one full rotation (which corresponds to passing through one slot) can be calculated as:

Time per rotation = t / N = (3.33 × 10^-6 s) / 500 = 6.67 × 10^-9 s

Angular speed (ω) is given by the formula:

ω = 2π / T, where T is the time for one complete rotation.

Plugging in our value:

ω = 2π / (6.67 × 10^-9 s) = 9.45 × 10⁸ rad/s (approximately)

Finding the Linear Speed of a Point on the Edge

The linear speed (v) of a point on the edge of the wheel can be calculated using the formula:

v = r × ω

Substituting in our values:

r = 0.05 m and ω = 9.45 × 10⁸ rad/s

v = 0.05 m × 9.45 × 10⁸ rad/s = 4.72 × 10⁷ m/s

Summary of Findings

To summarize:

• The angular speed of the wheel is approximately 9.45 × 10⁸ rad/s.
• The linear speed of a point on the edge of the wheel is approximately 4.72 × 10⁷ m/s.

This method of using a rotating wheel provides an innovative way to measure the speed of light by correlating rotational speed with the distance light travels. It highlights the fascinating interplay between mechanics and electromagnetism in experimental physics.