Compare the tension in the rope

[RedDanger]

Homework Statement

The ends of a rope are tied to (a) two horses, (b) a tree and a horse, and (c) a tree and two horses, as shown. Each horse pulls with the same magnitude force. Compare the tension in the rope for the three situations. Explain your reasoning.

Homework Equations

F = ma

The Attempt at a Solution

(a) The tension in the rope would be zero because the formula for the force of tension is $$\sum$$F_T = ma, then because the two tension forces are pulling in opposite directions, they cancel out.

(b) The tension force in the rope would equal the force applied by the horse because the horse is pulling the tree, which is attached by the rope.

(c) The tension force would be twice as much as in the previous situation because there are two horses pulling on the tree instead of one.

Is this correct?

 

[Doc Al]

(a) The tension in the rope would be zero because the formula for the force of tension is $$\sum$$F_T = ma, then because the two tension forces are pulling in opposite directions, they cancel out.

Incorrect. Don't confuse the net force on the rope with the tension in the rope. The net force on the rope is zero, but the tension is not. (Note that the net force on the rope is zero for all three examples.)

The tension in the rope is the amount of force that the rope exerts. It must equal the force exerted by the horse, since the horse is exerting a force on the rope.

(b) The tension force in the rope would equal the force applied by the horse because the horse is pulling the tree, which is attached by the rope.

(c) The tension force would be twice as much as in the previous situation because there are two horses pulling on the tree instead of one.

OK.

 

[tiny-tim]

Welcome to PF!

Hi RedDanger! Welcome to PF!

(a) The tension in the rope would be zero because the formula for the force of tension is $$\sum$$F_T = ma, then because the two tension forces are pulling in opposite directions, they cancel out.

No, if a (light) rope is stationary, then the tension force on the two ends of the rope always cancel out …

that doesn't mean there is no tension.

You can apply F_net = 0 either to the whole rope, or to any small part of it …

for any part, the external forces on that part will be the tension at the two ends, so the tension is the same throughout the rope.

To answer (a), apply F_net = 0 to one of the horses, not to the rope.

 

[I Like Pi]

Hey

This is a classic question for introductory physics:

If you tie a rope that has a breaking strength of 300 pounds between a wall and a horse, and the horse pulls on the rope with 200 pounds of force, will the rope break? Most people would answer no (which is correct), because the tension in the rope is 200 pounds, which is less than the breaking strength of the rope.

You got b) on the right track. If you untie the rope from the tree and instead tie that end to a second horse, and that second horse pulls with 200 pounds force in a tug-of-war against the first horse (so that both horses are pulling with the same force) will the rope break? Many people get this wrong. They incorrectly say yes, because they think that if two horses each pull with 200 pounds then the rope is under 400 pounds tension, which exceed its breaking strength. That is wrong, the proper way to look at this is to consider that it doesn't matter whether you tie that one end of the rope to a wall or a horse, in both cases the first horse is pulling against 200 pound of force and is in equilibrium, hence that's the tension in the rope in both cases.

hope this helps!

 

[RedDanger]

This really cleared it up for me! Thanks!