Does the coefficient of static friction during uniform circular motion change depending on the radius?

Question

I am working on a problem where a button is placed on a turntable spinning in uniform circular motion. The turntable can be brought up to 40 revolutions per minute without the button slipping, provided it is no more than 0.150m from the center.


I am supposed to find the coefficient of static friction based on this information. Does this mean that the coefficient of static friction changes based on how far away the object is from the center? Am I supposed to be finding a formula for the coefficient in terms of radius R, or is there a fixed coefficient between the button and turntable no matter where the button is placed?

Assuming I've done my calculations right, I came up with coefficient of static friction = 1.79R. Is this my answer, or should I use the information in the question (Rmax=0.150m) to come up with 0.269?

Constructive feedback would be appreciated!

Answer 1

I forget the angular dynamics equations, but I can tell you with 100% certainty that the coefficient of friction is constant no matter what. This value is determined solely by the two materials in question. The force of friction is determined by the coefficient of friction and the force normal to the surface. (i.e. the weight of the button).

Given the radius and the angular frequency (rpms) you can determine the centripetal acceleration at the point where friction is overcome. This acceleration will be equivalent to the force created by the friction between the turntable and the button. Then you use the formula for force of friction, inserting the known values: centripetal acceleration and weight of the button to determine the coefficient.

I looked up the formulas on Wikipedia:
F = -mw^2 * r (w = omega = angular frequency - probably in radians per second)
F = uN (F = Force of Friction, u = mu = coefficient, N = force normal to the surface = mass of button * 9.8m/s^2)

Answer 2

I got .269

The force on the button varies with the rpm, assumed constant in this case at 40 rpm, and the radius of revolution. Think about it, as the radius gets larger the button must have a higher speed. The static friction is a phenomenon where the button will not move until the force gets high enough to overcome the friction. So the coefficient is constant, but the force varies to counteract the force due to spinning motion until it is overcome at Rmax.

Look at the force due to spinning
m*w^2*r
versus the frictional force
m*g*u
u=w^2*r/g
use .1047198 * 40 to get radians per second

j

Answer 3

You have to use the given info. µ is a constant, not a function of radius. Depends solely on the materials involved.
The C.F., µ is the ratio of the friction force to the normal force. Friction force = mrω². Normal force = mg. Divide the later into the former to find µ. ( = rω²/g)

I got .269