# Pulley with mass and two blocks [closed]

The problem is

A 2.00-kg textbook rests on a frictionless, hori- zontal surface. A cord attached to the book passes over a pulley whose diameter is 0.150 m, to a hanging book with mass 3.00 kg. The system is released from rest, and the books are observed to move 1.20 m in 0.800 s. (a) What is the tension in each part of the cord? (b) What is the moment of inertia of the pulley about its rotation axis?

Once I write down the equations, I can solve this so that's all good. What is confusing to me here is why exactly are there two tensions in the cord? In problems with massless pulleys (and massless cord), the tension would be the same at all points of the cord. Why exactly does that break down here?

One explanation for this is that there has to be a net torque on the pulley. I understand this but I still don't quite get, fundamentally, why we have two tensions. Why don't we have a different tension at different parts of the cord? In particular, what exactly is the tension at points that are on the semicircle where the cord is touching the pulley?

Thanks!

The relation between the tensions and friction is: $$T_1=T_2e^{μθ}$$ (the Capstan equation, where $$\mu$$ is the coefficient of friction, and $$\theta$$ the angle of the rope).