share/mechanics/hw/fizeau-cogwheel.tex

   1 In 1849, Fizeau carried out the first terrestrial measurement of the speed of light;
   2 previous measurements by Roemer and Bradley had involved astronomical observation.
   3 The figure shows a simplified conceptual representation of Fizeau's experiment.
   4 A ray of light from a bright source was directed through the teeth at the edge of
   5 a spinning cogwheel. After traveling a distance $L$, it was reflected from a mirror
   6 and returned along the same path. The figure shows the case in which the ray passes
   7 between two teeth, but when it returns, the wheel has rotated by half the spacing
   8 of the teeth, so that the ray is blocked. When this condition is achieved, the
   9 observer looking through the teeth toward the far-off mirror sees it go completely
  10 dark. Fizeau adjusted the speed of the wheel to achieve this condition and recorded
  11 the rate of rotation to be $f$ rotations per second. Let the number of teeth on the
  12 wheel be $n$.\\
  13 (a) Find the speed of light $c$ in terms of $L$, $n$, and $f$.\answercheck\hwendpart
  14 (b) Check the units of your equation using the method shown in example \ref{eg:checking-units}
  15 on p.~\pageref{eg:checking-units}. (Here $f$'s units of rotations per second should be
  16 taken as inverse seconds, $\sunit^{-1}$, since the number of rotations in a second is
  17 a unitless count.)\hwendpart
  18 (c) Imagine that you are Fizeau trying to design this experiment. The speed of light is
  19 a huge number in ordinary units. Use your equation from part a to determine whether
  20 increasing $c$ requires an increase in $L$, or a decrease. Do the same for $n$ and $f$.
  21 Based on this, decide for each of these variables whether you want a value that is as
  22 big as possible, or as small as possible.\hwendpart
  23 (d) Fizeau used $L=8633\ \munit$, $f=12.6\ \sunit^{-1}$, and $n=720$. Plug in to your
  24 equation from part a and extract the speed of light from his data.\answercheck\hwendpart