An accelerating capstan

[etotheipi]

For a massless rope wrapped around a pulley with friction, it can be shown via integrating over string elements that the tensions on either side of the pulley are related by $T_2 = T_1 e^{\mu \theta}$, if $\mu$ is the coefficient of friction and $\theta$ is the angle subtended by the rope in contact with the pulley. Details on this derivation for the case where the system is in static equilibrium can be found here, under section 2.

It is subsequently possible to show that the torque on the pulley due to friction equals $R(T_2 - T_1)$ and the net force on the pulley due to the string equals the component of $\vec{T}_1 + \vec{T}_2$ in a direction along the axis of symmetry of these two forces.

I wondered if the relation $T_2 = T_1 e^{\mu \theta}$ could also be shown to be valid if the rope was instead accelerating around the pulley, or even if the pulley itself were undergoing translational acceleration (assuming it retains contact with the rope)? Or whether it no longer applies. Thank you!

 

[etotheipi]

I figured it out, the net force on any individual element of string must be zero regardless of the state of motion of the string, since the string element is massless. So this result holds always for massless strings, and no further proof is required.