# Does the Capstan Equation Apply to Rotating Pulleys?

Consider the following set up:

Here there is friction between the axel and the pulley, and effort is applied as T1 with the T2 end fixed to the ground as shown. The spring represents some system the load would be applied to that would deform to allow the pulley to rotate and maintain the tension in the fixed line.

I am currently analyzing a static case for this system by summing the moments as follows:

$$\sum_M = 0 = T_1R_o - T_2R_o \pm \mu NR_i$$

My question is this: Should I be including a term in this equation for the belt friction as characterized by the Capstan Equation? I don't think I need to because if the pulley rotates the differential tension described by the Capstan eqn would not apply. If weight is added to the T1 side sufficient to overcome the static friction, the pulley will rotate, increasing T2 until the equation I wrote is balanced by the (now) kinetic friction.

In general, does the Capstan equation affect the tension in conventional pulley systems assuming the pulleys are free to rotate?

• The belt-pulley friction is assumed to be large enough so slipping doesn't occur before the pulley starts rotating? Am I understanding correctly that you are considering additional friction at $R_1$? Commented Mar 25, 2021 at 1:58
• @BioPhysicist Yes it is assumed that the belt friction is sufficient to prevent slipping. I am considering only axel friction at Ri Commented Mar 25, 2021 at 16:05

The Captstan Equation gives you a relation between $$T_1$$ and $$T_2$$. Both of these forces appear in your equation already so there is no need to include another term. If you did not know one of these forces then you could use the Capstan Equation to find the other force provided that the rope is on the point of slipping around the pulley and the pulley is not accelerating.