Why does tension in a vacuum cause contradictory acceleration?

[PFuser1232]

Suppose two people, A and B, are pulling on a rope of mass $M_r$ in space. The force exerted by B is $F_B$ and the force exerted by A is $F_A$ (in magnitude). Suppose B pulls harder than A, we now have the equation of motion:
$$F_B - F_A = Ma_{rope}$$
We also have a constraint; since A, B and the rope are all connected, their accelerations must all be the same. That is $a_A = a_B = a_{rope}$. According to Newton's third law, the force exerted by the rope on B is equal in magnitude and opposite in direction to the force exerted by B on the rope. That is, it is not in the same direction as the acceleration of B, which is confusing because this force (tension) seems to be the only force acting directly on B, thus one would expect the acceleration of B to be in the same direction as the tension acting on B, in accordance with Newton's second law, except it isn't. I am really confused.

 

[mfb]

We also have a constraint; since A, B and the rope are all connected, their accelerations must all be the same.

Unless you fix A and B somewhere (like on a planet), they will accelerate towards each other, which gives them different accelerations. They move relative to the rope.
The rope will accelerate, too.

 

[Dale]

With the condition that the accelerations are equal the only solution is that the accelerations and forces are 0. That condition does not accurately represent the scenario I suspect that you are really envisioning.

 

[A.T.]

We also have a constraint; since A, B and the rope are all connected, their accelerations must all be the same

Nope, humans aren’t rigid bodies. The acceleration of the rope and your hand holding it can be different from the acceleration of your bodie's center of mass.

 

[russ_watters]

This is a bit of a mess with impossible assumptions:

mfb said it, but let me try to be more explicit:

Suppose B pulls harder than A...

This constraint is unacceptable. Because the tension has a single value, it is not possible for the two people to apply different forces to the rope, as per Newton's 3rd law.

Also, in a real game of tug-of-war, the forces arise because the players are pushing against the ground. In space, the only way to generate a force is for them to pull themselves toward each other, accelerating both in opposite directions by moving their arms (which was A.T.'s point).

There is a physics classroom demonstration of this issue (probably youtube videos) where the prof gets the biggest and smallest person in the class, puts them on skateboards, and tells each to push away from the other as hard as they can, with the assumption the bigger person can push harder. But that's impossible and what always happens is they move apart, with accelerations proportional to their different masses.

 

[A.T.]

...the tension has a single value...

Not in a massive accelerated rope.

it is not possible for the two people to apply different forces to the rope

It is possible.

as per Newton's 3rd law.

The probably most often misapplied law in physics.

 

[russ_watters]

Not in a massive accelerated rope.

Oops: rope with mass. I don't think I've ever seen a tug-of-war problem with the mass of the rope included, and I missed it

In either case we still have the undeveloped issue of how they apply the force (if neither are fixed).

The title says "tug-of-war", but the problem really isn't very similar to a tug-of-war.

 

[A.T.]

how they apply the force

By pulling with their hands.

(if neither are fixed).

So they will accelerate.

 

[PFuser1232]

Nope, humans aren’t rigid bodies. The acceleration of the rope and your hand holding it can be different from the acceleration of your bodie's center of mass.

So B will accelerate in an opposite direction to the acceleration of the rope, right?

 

[jbriggs444]

So B will accelerate in an opposite direction to the acceleration of the rope, right?

Right.

B pulls harder than A, so the rope accelerates toward B. The rope pulls on B so B accelerates toward the rope.

 

[russ_watters]

By pulling with their hands.

So they will accelerate.

Possibly. But since the OP's constraints/conclusions contradict each other, we'll have to let him tell us which is true and which is false.