Question

You give tours on the Great Lakes and talk about points of interest. A lighthouse typically has 2 beacons that rotate together—but not necessarily facing opposite directions. In that way, it can be a shorter time between the first and second flashes than between the second flash and the third flash (when the first beacon comes around for the second time). One way of identifying which lighthouse you are looking at is to find the ratio of the short time between flashes to the long time between flashes. If the beacons are set 120° apart along the rotation, as shown below, what is this ratio?

Answer 1

Since the beacons rotate together, the angular speed is constant.

From the time you see the first beacon, you have to wait for the lighthouse to make 120°, i.e. 1/3 of a whole turn.

So, if T is the time it takes for the lighthouse to make a whole turn, the interval between the first and the second beacon is T/3.

The interval between the second and third beacon is the amount of time it takes to the lighthouse to make the remaining 2/3 of a turn, so that you'll see the first beacon again. So, this time is 2T/3

So, the ratio is

$\dfrac{2T}{3}\div\dfrac{T}{3}=\dfrac{2T}{3}\cdot\dfrac{3}{T}=2$

So, the time between the second and third beacon is twice as much the time between the first two.