# Decrease of the rotation speed of a wheel with inertia, when we stop applying rotational force to it [closed]

Let's say we have a wheel rotating at a constant speed $$s$$.

Suddenly at time $$t=1$$ we stop applying force to it.

What is the law for the decrease of the speed, until it stops?

I guess it will look like this: Question: Will the speed decrease be proportional to:

• $$1/t$$,
• $$1/t^2$$,
• $$\exp(-t)$$
• or I guess, more or less, a sigmoid function?
• Something else?

Note:

• we neglect the impact of the removal of the device which made it rotating.

• friction occurs between the plastic axle of the wheel and the frame, and also air resistance / friction.

• To the downvoter: can you please leave a note about how to improve the question? (wording? graph?)
– Basj
Sep 17, 2021 at 13:17

If there is no friction your wheel would continue to rotate indefinitely. So your question is really just about which kind of friction forces are applied. The correct model for the friction may very widely depending on what the specific situation is that you are imagining. For example, if we imagine the wheel turning slowly in a fluid of high viscosity, a good model would probably be some kind of stokes drag i.e. a friction force of the shape

$$F = - c \cdot v$$

for some constant $$c$$. In that case you would indeed obtain the exponential decay of velocity that you have guessed.

• Friction occurs between the plastic axle of the wheel and the frame, and also air resistance / friction.
– Basj
Sep 17, 2021 at 13:11
• To be more precise, what kind of exponential decay?
– Basj
Sep 17, 2021 at 13:18
• @Basj there are lots of different possible friction effects between axle and the axle mount -- you might have bearings there; you might have lubricants, etc. It's not a trivial problem. Sep 17, 2021 at 13:20
• @CarlWitthoft Let's take the simplest model with no lubricant and no bearings.
– Basj
Sep 17, 2021 at 13:21
• You still have to provide the form of the friction in order to actually be able to solve the problem. And then it would be best if you provided some thoughts/attempt on the solution yourself. Sep 17, 2021 at 16:14