[Physics] Contact pressure – rope on pulley


I've been trying to find a way to calculate the contact pressure between a rope under tension and a pulley (free to move, or fixed). I'm expecting to find an equation that considers the wrap angle – similar to the capstan equation – but I'm specifically looking for an expression for the contact angle. Not concerned about the mass of the rope, in my problems the force applied is far greater than the weight of the rope.

EDIT: I need this to consider friction between the rope and the pulley – just like the capstan equation.
Consider a fixed pulley with a very large radius, and a very high contact friction. In this case, you would expect the region close to the centre of the contact area to have a lower contact pressure.

Any suggestions?

Best Answer

In a pulley we can assume that the friction can be neglected, so that the tension in the rope can be assumed to be constant. Thus, consider the following diagram, showing the forces acting on the rope due to a turning angle $\mathrm{d}\theta$:

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From the symmetry of the problem (any part of the rope that is in contact with the pulley will feel the same pressure, since the curvature and the tension are constant), you conclude that the force per unit length in the normal direction is given by:

$p = \frac{\mathrm{d}F_N}{\mathrm{d}\theta R}$

since the length that supports the force $\mathrm{d}F_N$ is $\mathrm{d}\theta R$.

But since, from the diagram, $\mathrm{d}F_N = 2 T \sin{(\mathrm{d}\theta/2)} \sim T \mathrm{d}\theta$ when the angle tends to zero, the above equation becomes:

$p = \frac{T}{R}$

Therefore, the pressure does not depend on the total angle bent, but rather on the curvature at which you are bending it (or one over the radius of the pulley). Obviously, the total force on the pulley will depend on the total angle, which can be calculated easily by integrating the projection of $p$ over the angle bisector along the total arc along which there is contact with the rope.