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An earlier question was asked on what would happen if a rope was attached between Moon and Earth, but the question was more about the impacts in terms of geology.

We keep the same experiment setup :

Moon and Earth linked by a rope

Now let's add a guy holding the rope on Earth and an astronaut holding the other end of the rope, on the Moon. Since the distance between Earth and Moon is 384 000 km and that nothing can go faster than 300 000 km/s, what would happen if the Earth guy would pull on the rope. Would it take a little bit more than a second for the astronaut to feel the change in tension ? Does it mean that the rope has been somehow elastic during that 1 sec ?

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    $\begingroup$ Probably related: physics.stackexchange.com/q/2175 $\endgroup$ – Kyle Kanos Nov 23 '15 at 11:24
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    $\begingroup$ Actually, the tension on the rope would propagate only at the speed of sound in the material of the rope. Scale down the thought experiment to a smaller distance and work out the wave equation for deforming, say, a massless 1km long bar. $\endgroup$ – LLlAMnYP Nov 23 '15 at 12:16
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A tensile pulse travels through the rope at the speed of sound. This speed depends on the density and the bulk modulus of the material - a rope strong enough to support its own weight would probably have a very high bulk modulus. The equation is

$$v = \sqrt{\frac{K}{\rho}}$$

Where $K$ is the bulk modulus and $\rho$ the density.

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