Question

An early method of measuring the speed of light makes use of a rotating slotted wheel. A beam of light passes through one of the slots at the outside edge of the wheel, travels to a distant mirror, and returns to the wheel just in time to pass through the next slot in the wheel. One such slotted wheel has a radius of 6.0 cm and 400 slots at its edge. Measurements taken when the mirror was L = 500 m from the wheel indicated a speed of light of 3.0 ✕ 105 km/s. (a) What was the (constant) angular speed of the wheel?

Answer 1

To measure the speed of light using a rotating slotted wheel, the wheel's angular speed was calculated to be approximately 4714 rad/s based on the provided parameters.

Step-by-step explanation:

The task involves calculating the constant angular speed of a rotating slotted wheel used to measure the speed of light. Given that the light needs to travel 500 m to the mirror and back (1000 m total) and pass through the next slot of the rotating wheel, we can derive the wheel's angular speed.

The speed of light provided is 3.0 × 10⁵ km/s, which is 3.0 × 10⁸ m/s. The time (t) it takes for light to complete the journey can be calculated by dividing the total distance traveled by light (2L) by the speed of light (c):

t = Δd / c = 2L / c = 2 × 500 m / (3.0 × 10⁸ m/s) = 3.33 × 10⁻⁶ s

The wheel has 400 slots, so each slot corresponds to an angular displacement of θ = (360° / 400) = 0.9° per slot. To calculate the angular displacement in radians, we use:

θ = (0.9°) × ( π rad / 180°) = 0.0157 rad

Since the wheel rotates through one slot in the time the light takes to make the round trip, we find the angular speed (ω) as:

ω = θ / t = 0.0157 rad / 3.33 × 10-6 s = 4.714 × 10³ rad/s

Therefore, the angular speed of the wheel is approximately 4714 rad/s.

Answer 2

4.7 x 10³ rad / s

Step-by-step explanation:

During the time light goes and comes back , one slot is replaced by next slot while rotating before the light source

Time taken by light to travel a distance of 2 x 500 m is

= (2 x 500) / 3 x 10⁸

= 3.333 x 10⁻⁶ s .

In this time period, two consecutive slots come before the source of light one after another by rotation. There are 400 slots so time taken to make one rotation

= 3.333 x 10⁻⁶ x 400

= 13.33 x 10⁻⁴ s

This is the time period so

T = 13.33 X 10⁻⁴

Angular speed

= 2π / T

=
`(2*3.14)/(13.33*10^(-4))`

4.7 x 10³ rad / s