# Two tensions in massless string

There should only be one tension in a massless string.

But imagine this situation. Let a massless rope hang from a ceiling and two monkeys of different weights are holding on to the rope at two different heights. Then there should be two different tensions which balance the two weights of the monkeys.

How do you account for this?

• In a situation, it is the same as having a block in the middle of two ropes, and tension is thus still uniform in each rope. Feb 17, 2014 at 4:28
• Can u please explain, sir Feb 17, 2014 at 5:23
• Effectively there are two ropes. A "rope" is a two force member with equal and opposite forces on each end. Feb 25, 2014 at 16:43

The natural tendency of a rope is to move to the lowest possible tension. When an object hangs onto a rope, it slides down to the lowest possible point, at which the tension is uniform.

Now, consider the case when something hangs onto the rope but does not slide. This is equivalent of having two ropes: one from the ceiling to the body, and one from the body downwards. This can be repeated with multiple bodies. Thus, the tension in the highest rope will be the largest, followed by each successive block. However, each rope section will have uniform tension.

Ruben is right. In order to extend his answer, Note that the monkeys will have an acceleration which shall change the Tension to be uniform on both sides.

Let's consider the rope hanging from a ceiling...I shall explain things from the bottom to the top where the rope meets the ceiling.

Now the bottom is plucked by one monkey (Say 30 Newtons). and another of weight (Say 20 Newtons) a little above. The total force that has to be provided near the ceiling is 50 N. This tension is acting on the rope between the ceiling and the monkey of weight 20 N. Clear ?

And tension of the rope in between the monkeys is equal in magnitude to the weight of the monkey weighing 30 N.

Then there should be two different tensions which balance the two weights of the monkeys.

If the monkeys are of different weights, the ropes are not balanced at all (unless the friction in the ceiling is great enough). This is the case, when the monkeys (and, of course, the rope with them) move with some acceleration. Because the rope does not extend during the process, the differences between tensions for each little segment of the rope are the same. Thus, the tension is uniform throughout the rope.

P.s.: yes, I do write "each little segment of the rope" by which I mean a segment of mass dm. You might say, that the rope is massless, but in reality, it does not have zero mass, it has a mass, which is negligible by the conditions of current problem. So this interpretation of breaking the rope in little sections is right.