Rope tension question?

Question

If two ends of a rope are pulled with forces of equal magnitude and opposite direction, the tension at the center of the rope must be zero. True or false? The answer is false. I chose true though and I'm not understanding why. Forces act at the center of mass of the object, so if there are two forces of equal and opposite magnitude, then they should cancel out resulting in zero tension, no?

Answer 1

I think the problem is you are crossing up the RESULT of no net force with the presence of the forces themselves. Newton's 2nd law states that net force = mass * acceleration, relating the net force to the resulting acceleration of the system.

If we look at a small section of the rope at its center, it will be pulled into two directions by tension leading away one way along the rope and tension leading away in the other. As you state, these two forces will cancel out. This results in zero NET FORCE, and zero acceleration, as per Newton's 2nd law. However, the lack of a net force does not mean the individual forces are not present.

It is similar to imaging how you may be sitting or standing now. If you are stationary, the net force on you is zero, and that may be due to the force of gravity and another force (perhaps a normal force) on you cancelling out. However, the fact that these forces cancel out does NOT mean the force of gravity on you is zero.

Also, not all forces act at the center of mass of an object or system. That is only generally applied to forces exerted by a field, like gravity.

If you have any other questions, please do not hesitate to ask!

Answer 2

In the simplest case, assume that the rope is not moving.....more precisely, not accelerating. This means that there is no net force acting on a segment. If there were, it would be accelerating. This is no different than a block with 2 forces pushing on it. There is no net force, and therefor no acceleration.

Newton's 3rd law states that every action has an equal and opposite reaction. The common demonstration is to push on something stationary and notice how it appears to be pushing back. If a string is tied to a stationary object, there is an exact analog: Pull on the string, and it appears to be pulling on you.

Another analog: For simple problems (including an assumption that the string has no mass), we use a rule that the tension will be the same for any part of a string. An exact analog is that the compressive force in a solid block will be the same at all locations (on the axis of the applied force)

Bottom line: Even though forces often add up to zero, they are still there.

Answer 3

The diagram the rope with persons pulling on it might look like this

Force <-----------------------> Force

if you draw a free body diagram and cut the rope in the middle, the diagram might look like this

Force <--------------> x y <---------------------> Force

if there is no force x, there would be motion to the left and the same for the other end.... if there is no force y, the half rope would move to the right. I understand your reasoning about the mass center but this applies regardless where you cut the rope. There must be an equal and opposite force at x and y in order to remain in equilibrium & no motion. So, x & y are the same but in opposite directions!