Real Pulley and Newton's Third Law

Question

In a real pulley with two masses hanging from either side, the pulley rotates due to a non-zero torque, which I understand is caused by friction on the rope.

I understand that if each mass was wrapped around the pulley by a different string, the tension from one mass must be greater to cause this torque.

What bothers me, however, is that the two strings are attached to each other and thus, Newton's third law should apply, meaning the tensions are equal and opposite.

I have been solving such problems by taking into consideration "different tensions" to account for the torque-causing friction:
$$T_{12}-m_1g=m_1a_{y1}$$$$T_{21}-m_2g=m_2a_{y2}$$

I agree that this model accurately represents the system, but I feel that the tension is being misrepresented. How would I elaborate on this model to describe the tensions and the frictions differently?

Bob Jacobsen · Answer

“Newton's third law should apply, meaning the tensions are equal and opposite.”

If the pulley has mass, hence inertia, you’ve forgotten to include the non-zero force exerted on the pulley, hence by the pulley, in your calculation.

I.e. Tension on low mass side plus pulley force = tension high mass side

Steeven · Answer

What bothers me, however, is that the two strings are attached to each other and thus, Newton's third law should apply, meaning the tensions are equal and opposite.

No, the tensions in the rope are not equal on either side of the pulley. The tension in the rope is larger on the side with the heavier mass.

The heavier mass causes a large pull that pulls in (rotates) the pulley. That is tough because the pulley is heavy. It takes some force. But that is not all...

Because the mass on the other side causes a counter-torque, it is even harder for the heavy mass to rotate the pulley. It has to cause a tension in the rope that both makes the pulley rotate and counteracts the opposite pull.

Newton's 3rd law tells us that the tension must be larger on the side of the heavy mass, because that tension does more effort.

Real Pulley and Newton's Third Law

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