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The wagon wheel effect is an annoying effect we see in Western movies, when stagecoach wheels rotate in a way that is against human intuition. We expect that the wheel should be turning forward however in a moment we see that they are rotating in the opposite direction. In another moment we see the wheel is standing still or rotating forward much slower than we anticipated.

This effect is caused by the number of spokes in the wheel and by the number of frames per second the movie is recorded in. Since movies are recorded in 24 frames per second and if the wheel has 6 spokes it will look like the wheel is standing still if it is rotating 60 degrees at 1 24th of a second. We get the number 60 by dividing the number of spokes into 360. For 6 spoked wheel for every fame refresh the given spoke will move the location of the next spoke position if angle increment is 60. We think that the wheel is standing still. However, if the angle is 59 it will seem like the wheel is rotating backwards and for 61 degrees it will seem like the wheel is rotating forward very slowly.

In this animation we have six wheels with different number of spokes, the number of spokes are 2, 3, 4, 5, 6, and 8. The black arrows are to show the starting location of the wheel rotation. The small red circles are to show the angle step we are using during the animation and we control its value by using numerical stepper input box. The black dot shows the location of the spoke at the start point. This value constantly increases by adding another step value shown in the input box. Therefore the angle value of the black dot is increasing constantly.

The number above the wheels indicates the increment of the angle when we see the given wheel perceived by us standing still. Of course for a zero value all the wheels will stand still, but if an angle coincides with any angle above the wheel other than zero, we perceive the wheel to be standing still when actually it is rotating. These cases are indicated by the large red circles.

Please note that the first angle we see the wheel standing still is obtained by dividing the number of spokes by 360, the rest of the angles are obtained by multiplying the first value by 2, 3, 4 and so on. For example, for an eight spoked wheel, 360 divided by 8 gives us 45, then multiply this by two and we get 90. If you multiply 45 by 3 we get 135. In all these angles we will see the 8 spoked wheel as standing still. This logic will be true for different numbered spoked wheels as well.

There is also another interesting phenomenon, for the eight spoked wheel we found that 45 is the first position where the wheel seems to be standing still. What would happen if I use 22.5 as the angle (half of 45?) We will see the wheel as standing still, but remember the number of spokes we see is doubled to 16. Also, below 22.5, such as 22, the wheel will be seen as rotating backwards slowly and above 23 the wheel will be seen as rotating forward slowly. These cases are indicated with green circles in this animation.

Please also notice that when we see a wheel standing still the black dot is still rotating. If we had a black dot for every spoke we would see those black dots as standing still as well.

Finally, when we see a wheel completely standing still the wheel is colored red and for every spoke numbered cycle the red and black dots will coincide. For example, for an 8 spoked wheel every other 8 cycle he red and black dots will coincide. For the wheels that are colored green, in this case, it will be every other cycle.

As you can see this annoying effect has some physic behind it. It is also a very useful phenomenon. It was used in old turn tables to fine tune the rotation speed of the turn table. It is also used by engineers and technicians to find the rotation speed of an object without any physical contact.

This is also a very dangerous phenomenon and may cause death. Imagine you a lathe machine and a florescent lamp above it. Imagine further that you are listening to music with your headphones during your work. In this worst case scenario imagine that you turned the speed of your lathe to a speed that is the same frequency as your florescent lamp, at that moment you somehow you decided to get some coffee while listening music. When you come back, you may think that the lathe has stopped because you do not hear the machine running and you try to reach into the work piece. You can imagine the possibilities. It is best to use multiple lamps with different frequencies or some lamp that is not flickering to protect yourself from such an unfortunate possibility.

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