How can I predict the stopping angle of a spinning wheel?

Question

I'm trying to predict the angle in which a spinning wheel will stop if it is slowed down to a rotational speed of 0 over a certain time.

For example;
If my wheel is spinning at +20 degrees per tick, and it reaches a set angle of say, 720 degrees, where I then begin to slow it down to a stop over 5 seconds, what angle will it rest at.

I have been looking at a lot of existing questions and material for this and am aware of the sort of maths I need to be working with; getting the deceleration speed and rotational velocity, but I'm struggling to use this information to predict an end angle.

This was a particularly useful resource but I am still unsure on how to translate this math into a predicted resting angle. https://physics.stackexchange.com/questions/142761/how-to-model-a-very-simple-spinning-wheel

Thanks in advance.

Answer 1

The answer depends on the mechanism that stops it, but if we assume that it's simple friction, then deceleration is constant and the answer is easy:

If it starts off at 20 degrees per tick, then the average speed during the slowdown period is 10 degrees per tick, so 720 + 10 * 5 * ticks_per_second degrees.

Answer 2

Assuming constant deceleration (dry friction), the angular speed decreases linearly. Hence, the average speed is the half* of the initial speed, or

A / T = Ω / 2

where A is the traversed angle, T the time to stop and Ω the initial angular speed.

Hence

A = T Ω / 2

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*For the same reason that the area of a right triangle is the half of the area of the bounding rectangle.