Chapter 34, Problem 1P

In an experiment to measure the speed of light using the apparatus of Armand H. L. Fizeau (see Fig. 34.2), the distance between light source and mirror was 11.45 km and the wheel had 720 notches. The experimentally determined value of c was 2.998 × 10⁸ m/s when the outgoing light passed through one notch and then returned through the next notch. Calculate the minimum angular speed of the wheel for this experiment.

To determine

The minimum angular speed of the wheel for this experiment.

Answer to Problem 1P

The minimum angular speed of the wheel is $114\text{rad}/\text{s}$.

Explanation of Solution

Given info: The distance between the light source and the mirror was $11.45\text{km}$, no of notches in the wheel is $720$ notches, experimentally determined value of $c$ was $2.998\times {10}^8\text{m}/\text{s}$.

The formula to calculate the angular speed of the wheel is,

$\omega =\frac{d\theta }{dt}$ (1)

Here,

$\omega$ is the angular speed of the wheel.

$d\theta$ is the degree movement of the wheel in consecutive notches.

$dt$ is the time for the light to travel to the stationary mirror and back.

The formula to calculate $d\theta$ is,

$d\theta =\frac{2\pi }{\text{number of notches}}$

Here,

$d\theta$ is the degree movement of the wheel in consecutive notches.

Substitute $720$ for the number of notches.

$d\theta =\frac{2\pi }{720}$ (2)

The formula to calculate $dt$ is,

$dt=\frac{2d}{c}$ (3)

Here,

$d$ is the number of notches in the wheel.

$c$ is the experimentally determined value of speed of light.

Substitute $\frac{2\pi }{720}$ for $d\theta$ and $\frac{2d}{c}$ for $dt$ in equation (1).

$\omega =\frac{\left(\frac{2\pi }{720}\right)}{\left(\frac{2d}{c}\right)}$ (4)

Substitute $11.45\text{km}$ for $d$ and $2.998\times {10}^8\text{m}/\text{sec}$ for $c$ in equation (4).

$\begin{array}{c}=\frac{2\pi /720\text{rad}}{\frac{2\times \left(11.45\text{km}\times \frac{{10}^3\text{m}}{1\text{km}}\right)}{2.998\times {10}^8\text{m}/\text{sec}}}\\ =114\text{rad}/\text{sec}\end{array}$.

Conclusion:

Therefore, the minimum angular speed of the wheel is $114\text{rad}/\text{sec}$.