Say you have a weight tied to each side a a rope which is strung over a pulley with friction. Here's a really easy way to see why the tensions on each side of the rope can't be equal.

Imagine a really stiff pulley - in other words, ${\bf F}_\text{friction}$ is high. If that's the case, it'll be possible to balance unequal loads on this pulley system - i.e. a heavy weight on the right side and a lighter weight on th left - without the system moving. If the weights don't move, then we can say that the forces acting on each weight add up to zero:

For the heavy weight, there's the weight downward, ${\bf w}_\text{heavy}$ and there's the tension of the right side of the rope upward, ${\bf T}_\text{right}$. The tension pulls up and the weight down, and the system doesn't move, so

$$ {\bf T}_\text{right} - {\bf w}_\text{heavy} = 0
$$

or

$$ {\bf T}_\text{right} = {\bf w}_\text{heavy}
$$

Similarly for the left (light) side,

$$ {\bf T}_\text{left} - {\bf w}_\text{light} = 0 \quad \Rightarrow \quad{\bf T}_\text{left} = {\bf w}_\text{light}
$$

As you can see, the tension on the right, ${\bf T}_\text{right}$ is equal in magnitude to the heavy weight, while the tension on the left, ${\bf T}_\text{left}$ is equal to that of the lighter weight. The friction is introducing an extra force which changes the tensions on each side.

As far as your question about rope stretching goes, if you anchor a rope on one side and pull, the rope will pull back, creating a tension. This is indeed because of stretching in the rope. This is not really what Newton's 3rd law is referring to. Newton's third law, in this case, tells us that the force that we feel from the rope, tension, is exactly the force the rope feels from us pulling. The two are equal and opposite. You can change the tension by changing the stiffness of the rope, but whatever the tension, Newton's 3rd law will still be true - the rope will feel us pulling it as much as we feel it pulling us.

What we mean by a frictionless pulley is that the friction in the bearings of the pulley is negligible, and the pulley is free to rotate without any resistance. We don't mean that the friction between the string and the pulley surface is negligible. In fact, we assume that there is enough static friction between the string and pulley surface to prevent the string from slipping. But, in this case, if you do a moment balance on the pulley, you must then conclude that the tensions in the string on either side are equal (even if the pulley has angular acceleration), since the moment of inertia of the pulley is assumed to be zero.

## Best Answer

You cannot assume equal tension throughout if the pulley is not "massless" (assuming the rope does not slip over the pulley.

Then adding a heavy load in one end will be carried by the string all the way up (Newton's 3rd law states that for all crosssections of the rope on this side, the forces must be equal).

But if the pulley has inertia by having mass, then it "helps" by holding up the mass - as if someone grapped the rope and held it back. The tension on the other side is therefore lower, since the pulley "helps" so this side does not carry the whole weight alone.

Also, if the rope is not rigid (if it is an elastic rubber band e.g.), you will feel different tension in the rope when adding a load in one end over a pully as long as stretching happens.