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Force the piano exerts on the rope is excluded

I'm reading my physics book trying to understand how to create force equations. (This next test is going to be a flop.) In this example, it states that if we were trying to find the tension in the rope we would only consider the the weight of the piano and the tension in the rope and exclude the force the piano exerts on the rope in our free body diagram. I do not understand how this excluded force is from the outside world and not directly contributing to the tension in the rope.

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    $\begingroup$ Isn't the force that the piano exerts on the rope the same as the tension in that rope? $\endgroup$ – Steeven Oct 1 at 5:09
  • $\begingroup$ The force the rope exerts on the crane is external $\endgroup$ – Adrian Howard Oct 1 at 6:13
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This is because a free body diagram only shows the forces acting on that particular body. The force exerted by the piano is not relevant to the rope. The tension in the wire due to the force exerted by the piano is relevant to the rope. Hence it is included in the free body diagram. In this question, the tension and force by exerted by the piano are numerically equivalent but in opposite direction.

This may seem unnecessary for this question here but in more complex questions and setups, this motor of solving helps prevent confusion and mistakes.

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A free body diagram consists only of forces acting on that particular system/body because the system's motion is dictated only by those forces. In this particular situation, the motion of the piano is governed by tension in the rope and gravitational force (i.e. piano's weight) and not by the force that the piano exerts on the rope.

Imagine a person applying a force $\vec F$ on a box (of mass m) kept on a frictionless floor, now its acceleration, $\vec a$ is equal to $\vec F$/m. In this case, we don't consider the force that the box applies on the person (pushing the box). Similarly in the piano case, we don't consider the force exerted by the piano on the rope.

Now, if we were to find the motion of the rope we would consider the forces acting only on the rope i.e. force due to the piano on it (= -(Tension in the rope from the first case)) and the rope's weight.

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