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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!

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We need to first look at why the case you've mentioned is any different from the case where tensions are equal. A rotating pulley has mass and friction. The mass does the obvious, it gives the pulley a moment of inertia.

However, let's look at why we need friction. When the rope wants to move on the pulley, the friction opposes relative motion. This is why the correct condition for the situation you outline is sufficient friction.

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).

This friction opposes the relative motion between the pulley and the rope. This means that the two tensions must come out to be unequal.

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For this problem I would assume static friction between the cord and the pulley, and no friction at the axle of the pulley. Two different tensions result in a net torque on the pulley, giving it an angular acceleration. Use a kinematics equation to get the linear acceleration. Then write force equations for each book, and a torque equation for the pulley. Your unknowns are the tensions and the rotational inertia.

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