# Velocity given centripetal force, accounting for friction

moonman239
Hello,

I would like to know how to calculate the velocity of an object in a circular path, accounting for friction. I know that in a frictionless environment, the equation for the velocity is v2=F*R/M.

## Answers and Replies

Stonebridge
If you introduce friction into this scenario, then the object will no longer move in a circle because its velocity will decrease. This would result in the value of R having to decrease if you keep F constant. The object would spiral towards the centre of the circle.
If you keep R constant then you would need to reduce F to maintain circular motion.
Are you considering either of these scenarios?
Perhaps you could just elaborate a little on what you are thinking of.

moonman239
Okay, then let's say that I'm keeping R constant.

moonman239
Then what's the formula?

Gold Member
Okay, then let's say that I'm keeping R constant.

You haven't specified the kind of friction.
As a crude approximation you can use a friction that is proportional to velocity.

The next step is to work out the velocity as a function of time in the case of deceleration with a linear-to-velocity friction. If the velocity is halved the frictional force is halved; the velocity will decay. That is, in the crude approximation with a linear-to-velocity friction the velocity as a function of time will be exponential decay.

The general formula for a function that describes exponential decay is as follows:

$$y = e^{-x}$$

If the specification is that R must remain the same throughout then the centripetal force must be adjusted all the time to the current velocity. Alternatively, you can opt to use as approximation that the frictional force is proportional to the square of the velocity.

But demanding that R remains the same gives a physics problem that is rather uninteresting. In essence the problem is the same as the case of linear motion.

What if the centripetal force is not adjusted, but a constant force? For example, take the case of an object sliding over a surface, with a chord tugging at it, with the chord running over a pulley, and at the other end a weight.

Then as the object slows down the centripetal force start reeling it in, and in the process of being pulled closer to the center the speed tends to increase again. What happens then can't be expressed with just a simple expression.