# Problems with understanding the force of tension

If I had a object with a rope tied to it, and I began to pull the object on a surface of zero friction with a force of 4 N, both the object and the rope would start to move at the same acceleration. But I deduced that by intuition, not physics. When regarding all the forces acting on both the rope and the box, I always end up with the deduction that the object would move, and the rope would not: There are two forces acting on the rope, one exerted by me, and another one, equal and opposite to the first one, exerted by the object on the rope while the object is acted upon by a force of 4 N. Why isn't my analysis consistent with what really happens? Where does my misunderstanding lie?

Last edited:

Chestermiller
Mentor
You are right that the two equal but opposite forces on the rope produce no net force on the rope. But the usual assumption is that the mass of the rope is also virtually zero. How much net force does it take to accelerate a body of zero mass? If the rope actually does have some small mass, then the force on one end has to be a little higher than the force on the other end in order to accelerate it.

Inspiron
How much net force does it take to accelerate a body of zero mass?
Zero, I think. But, then, what would be the acceleration of the rope?
If the rope actually does have some small mass, then the force on one end has to be a little higher than the force on the other end in order to accelerate it.
What does make the force at one end smaller than the other end when the rope has mass?

A.T.
the rope would start to move at the same acceleration. But I deduced that by intuition, not physics.
You didn't deduce that, you introduced it as a boundary condition: If the rope doesn't snap, then the object must move with the rope.

What does make the force at one end smaller than the other end when the rope has mass?
What makes them equal for a massless rope?

Chestermiller
Mentor
Zero, I think. But, then, what would be the acceleration of the rope?
It would be whatever you want it to be. The net force you need to give any acceleration you want is tiny if its mass is tiny.
What does make the force at one end smaller than the other end when the rope has mass?
To understand that, consider the case of a rope having a small mass attached to the object in your original problem description. Draw a free body diagram of the rope, and a separate free body diagram of the mass. Show all the horizontal forces acting on each. Then write down a horizontal force balance equation for the rope, and a horizontal force balance equation for the object. Please show your equations.

Chet

Think about it that way. Consider the rope and the box as part of the same system, then apply Newton's law: you would have one force applied by you on the rope-box system: So the whole system would accelerate.

Here's the point: The forces you show in those diagrams are applied forces and the reaction forces. You included inertia force, so simply remove it, there's no force on either side of the massive body. Now it's alright. The inertia forces cancel out internally, the forces in the diagram are at-point-of-application, which are external. The difference is one can be felt externally, the other can't.

Last edited:
To understand that, consider the case of a rope having a small mass attached to the object in your original problem description. Draw a free body diagram of the rope, and a separate free body diagram of the mass. Show all the horizontal forces acting on each. Then write down a horizontal force balance equation for the rope, and a horizontal force balance equation for the object. Please show your equations.

Trying to do this, but not sure what you mean by a "horizontal force balance equation."

Chestermiller
Mentor
Then drop the word "horizontal". I'm still waiting to see two free body diagrams. You have been taught to draw free body diagrams, correct?

Last edited:
Using the original problem with frictionless surface:
Rope
∑Fx=FpullONrope-FmassONrope=ma

Mass
∑Fx=FropeONmass=ma

So the rope has a very small mass, and it is definitely accelerating. The only way for there to be acceleration is if FmassONrope is less than FpullONrope but how can that be since they have to be equal and opposite? Back to the original question...

What does make the force at one end smaller than the other end when the rope has mass?

Chestermiller
Mentor
Who says they have to be equal and opposite? What you have convincingly shown is that they are not equal and opposite if the rope has mass and it is accelerating .

Ah...
The force that the hand exerts on the rope must be equal to the force the rope exerts on the hand.
The force that the rope exerts on the mass must be equal to the force that the mass exerts on the rope.
But the force that the hand exerts on the rope does not have to equal the force that the mass exerts on the rope.
So energy is lost in the rope? Only when the rope has mass though.

Chestermiller
Mentor
H
Ah...
The force that the hand exerts on the rope must be equal to the force the rope exerts on the hand.
The force that the rope exerts on the mass must be equal to the force that the mass exerts on the rope.
But the force that the hand exerts on the rope does not have to equal the force that the mass exerts on the rope.
So energy is lost in the rope? Only when the rope has mass though.
No way. The rope gains kinetic energy, so no energy is lost.

Last edited: