frequencies comparison

Question

I have a rather quiz question (sorry if this a wrong stack to ask such questions). A propeller with 3 blades makes exactly 24 spins in 1 second. Camera, that is filming it, takes 54 frames in 1 second. How many photos that differ from one another were taken in 1 second?

I'm not sure how to approach this question. Comparing two frequencies and finding the point where they meet?

If I get the time for one propeller spin 1/24 and divide with time needed to take one picture it will result in ~ 2,23 pictures per propeller spin. But how do I find the point where camera starts taking same pictures?

Answer 1

You need to consider:

• the time it takes the propeller to move from its starting position to an identical-looking position; call this $t_p$ seconds.
• the interval between successive camera frames; call this $t_c$ seconds.
• the least common multiple of these two times: this is when the images start repeating. Call this $t_r$ seconds.
• the number of frames taken in either $t_r$ seconds or in 1 second, depending on which is shorter.

Assume the propeller blades are identical, so the propeller looks the same after $\frac{1}{3}$ of a spin. 1 spin takes $\frac{1}{24}$ of a second, so its appearance repeats in a third of that time. That is, $t_p=\frac{1}{72}$.

From the problem description, we already know that the frame interval in seconds is $t_c=\frac{1}{54}$.

Now we need the least common multiple of $t_r$ and $t_c$.

Noting that $54=3×18$ and $72=4×18$, we see that $3t_c = 4t_p = \frac{1}{18}$. That is, in $\frac{1}{18}$ of a second the camera takes 3 frames and the propeller position moves on by 4 identical-looking positions.

The answer is therefore 3 different images before they repeat.