Tension

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victorias

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A rope is attached to a wall, and the rope is tied onto a horse. The horse is able to walk away from the wall to generate a tension T in the rope. If the rope is tied onto two horses, and they each walk away from each other with the same force as the original horse walking away from the wall, what is the tension in the rope?

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The solutions say that the tension would still be T. Can someone show me the set up for this problem? WHat am I doing wrong?

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The horse exerts a force F on the wall and by Newton's third law, the wall must also exert a force of F on the horse - each acting through the rope. Whether you replace that wall with a horse doesn't change the tension. Because even if it's two horses, horse 1 exerts a force F on horse 2 and horse 2 exerts force F on horse 1, each acting through the rope.
 
Always start with a free body diagram!

Horse 1:

T1 - F1 = m1a1

Horse 2:

F2 - T2 = m2a2

So the obvious answer is the rope is pulling horse 1 with T1 and horse 2 with T2. And that is what actually happens with a real rope: each section of a real rope will have a different tension. However, since we are probably talking about a massless rope, the tension must be distributed evenly for the rope to be taut, and that even distribution is the assumption that makes T1 = T2 = T.

Now if the question involves a heavy and long rope, it would be obvious that the middle section will be saggy and each end will have different tension. You can easily verify this irl!
 

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