Rope over an un-ideal pulley in an Atwood's machine

Please consider a pulley with a taut rope over it and two objects (with different masses m1 and m2) hanging on each side. The pulley is not negligible - it has a mass Mp.

In the solution to a problem I'm working on (from An Introduction To Mechanics by Kleppner and Kolenkow), the free body diagram seems to show the following:

1. The tension of the string on each of the two sides of the pulley is different

2. However, on each side, the tension exerted on the pulley is equal (in magnitude) to the tension exerted on the hanging mass

The first point, I think, makes sense to me. I understand that since the pulley is not-idealized, i.e. it does have a mass, we can't just assume that the tension everywhere on the string is the same (which is possible in exercises with idealized pullies). So we assign a different variable - T1 and T2 - to each side of the string.

However, I'm not sure how we can justify the second point. Why should the tension on each end of the string (for each side of the pulley) by the same?

Trying to think it through, I realize that since the string is always taut, all of the little rope pieces move at the same speed. Thus they have equal accelerations. And thus the net force on each little piece of rope should be the same.

However, I was unable to prove algebraically that the tension at the top equals the one at the bottom. This is because, while each little piece of rope along the string is only connected to its other fellow rope-piece neighbors, the piece of rope at the top is touching the pully itself (otherwise it wouldn't exert a force on it). And then I get stuck. Or, I reach equations which claim that the tensions at the two ends (top and bottom) shall be equal only if the rope is not accelerating (which is not what's happening in the problem above).

So, is the 2nd point exhibited in the book's diagram correct? If so, please explain why.

Please note: I'm not looking for help with the exercise. The diagram from the problem made me run into this question, and I'm using it to illustrate my confusion.

• "Or, I reach equations which claim that the tensions at the two ends (top and bottom) shall be equal only if the rope is not accelerating": Without seeing the equations you reach, I suspect the tensions are also equal if the rope is massless. Mar 19 at 1:11
• Your 2nd point is valid because rope is massless, your 1st point is valid because pulley is not massless( it has moment of inertia , hence, can rotate) you may ask, why would it rotate? Well, read the question, it says rope doesn't slip on pulley, so it has to rotate Mar 19 at 3:20
• There's static friction in the rim of the pulley which makes the two tensions unequal (I believe this should be true, but in many books it is written pulley is frictionless, somebody can address this issue) Mar 19 at 3:45
• Voting to reopen. Clearly a conceptual question about the textbook example, and not a "do my homework for me" question. Mar 20 at 11:48
• Take a look at my answer here, particularly the third paragraph. Basically, if the rope has "zero mass" then the forces on it have to balance out ($\sigma F = m a = 0$), meaning that it exerts the same amount of force on the object on each of its ends. Mar 20 at 12:28