# How does friction and mass of the string influence the tension force?

According to my book, "tension is the reaction of a rope when it is stressed". Then it also said when the string is massless, tension is same everywhere. However in a,let, pulley-rope system,when there is friction between the pulley & the rope, the tensions are not same. Why the tension is not same in this case? Does tension arise due to Newton's third law? When a block is attached to the rope ,the rope gets streched and can I say tension then arises due to Newton's third law or only to undo the deformation? If Newton's 3rd law is cause, then how can there be different tensions in a same rope? Plz explain.

• When two blocks of mass $M_1$ & $M_2$ are attached to a rope of a certain mass through a pulley (having friction between it and the rope) in either sides, the block $M_1$ stretches the rope with force $T_1$ and in order to undo the deformation ,rope exerts force $T_2$ in opposite direction. Which is tension $T_1$ or $T_2$ ? Are they equal(my book says they are not equal unless there is no friction between the pulley and the rope)? What is the cause? – user36790 Jul 22 '14 at 17:51
• No tension is the force needed to keep the rope from moving if it is sliced at a location. That is why we say "the tension at the end, or the middle." So it can vary with location in order to keep the balance. – John Alexiou Jul 22 '14 at 20:47
• @ ja72 : so tension force is varying over its length even if the rope is massless or having uniform density? Does it arise due to Newton's third law? – user36790 Jul 23 '14 at 7:23
• If it is massless and there is no other connection then it will not vary. Only when additional loading enters the picture the tension will change. The 3rd law describes the direction of tension mostly. The 2nd law really is used to find its magnitude. – John Alexiou Jul 23 '14 at 12:32

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.

• @ Melativity : Then tension is not due to Newton's third law? – user36790 Jul 23 '14 at 7:27
• @user36790 Tension comes from molecular interactions in the rope. Newton's third law just tells us that the tension force we feel from the rope is equal and opposite the force that the rope feels – Melativity Jul 23 '14 at 15:31