How Is Torque Generated by Friction on a Rotating Disc Calculated?

[johnysmithers]

If I have a spinning circular disc of uniform density, how would I find the torque generated by friction, if the disc is lying flat against a table with coefficient of friction μ? τ=Fxr, but what is F, and what is r in this case?

 

[Simon Bridge]

Welcome to PF;

You use calculus ... divide the disk into narrow rings.

 

[johnysmithers]

Thank you Mr. Bridge, for your reply. I am still confused however, as towhat the force on a differential element would be. The friction force would be μ N=μ m g(or would it be μ g dm?) The friction force would always act perpendicular to position, so torque would be upward with magnitude equal to the product of the distance from the center and the force. However, I am not sure what I would be integrating with respect to, and I am also unsure as to what the bounds on the integration would be. Please clarify? Thank you in advance.

 

[Simon Bridge]

That would be:
$d\tau = \mu g r \text{d}m$

The limits of the integration depend on how you define dm ...
hint: relate dm to dV (volume) and exploit the symmetry.

Friction force always acts opposite to the velocity vector.
Friction torque always twists the opposite way to the angular velocity.

 

[PJC01]

The torque generated by friction on a rotating disc can be calculated using the equation τ=Fxr, where F is the force of friction and r is the radius of the disc. In this case, the force of friction can be determined by multiplying the coefficient of friction (μ) by the normal force (N) exerted on the disc by the table. The normal force is equal to the weight of the disc, which can be calculated by multiplying the mass of the disc by the acceleration due to gravity (9.8 m/s^2).

Once the force of friction is determined, the radius of the disc can be used to calculate the distance (r) from the center of the disc to the point where the force of friction is acting. This distance is typically equal to half the diameter of the disc.

Once you have the force of friction and the distance from the center, you can calculate the torque generated by friction on the rotating disc. It is important to note that this equation assumes that the disc is rotating at a constant angular velocity and that the coefficient of friction remains constant throughout the rotation. Any changes in these parameters may affect the calculation of torque.