Suppose there's a stick whose length is one light year and I push it from one side by one centimeter. How much time would it take for its other side to move by one centimeter and why?
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2 Answers
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It depends on the material. When you push one end of the stick, you move the atoms at the very end of the stick. Those atoms push the atoms next to them, those atoms push the next atoms, and so on down the stick. This is a sound wave that travels down the stick, so the time you have to wait for the other end to move is the length of the stick divided by the speed of sound in the material of the stick. If the stick is wooden, the speed of sound is about 4000 m/s (compared with 330 m/s in air). It would take $\frac{9.5\cdot10^{15}\,m}{4000\,m/s} = 2.4\cdot10^{12}$ seconds (74 000 years) for the other end of the stick that is a light-year away to move.
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$\begingroup$ Could you set up a shock in the sick if you pushed hard enough? $\endgroup$– EL_DONCommented Sep 8, 2017 at 1:47
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$\begingroup$ @EL_DON You could, but shock waves (sonic booms, for example) travel at the same speed as sound waves. $\endgroup$– Mark HCommented Sep 8, 2017 at 1:50
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$\begingroup$ I think that there might be even more to this. In fact, I'm not even sure if this is the highest order effect. To get the stick to move, you'd need more than a compression wave - you'd actually need to displace the object's center of mass by 1 centimeter. That would be very challenging considering how massive the stick is. Surely, the speed of sound is the lower limit for an infinite applied force, but what if the applied force is finite? $\endgroup$– GeoffreyCommented Sep 8, 2017 at 21:21
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$\begingroup$ @Geoffrey Even if the force is a sustained finite force instead of an impulse, each part of the stick can only move in response to local forces. These forces propagate in a sound wave. As an alternative, you could imagine that the stick bends from the force instead of compressing, so now you have a transverse wave traveling down the stick and it would take somewhere around 74 000 years for the stick to straighten out, with the result that it has shifted to a new position. $\endgroup$– Mark HCommented Sep 8, 2017 at 22:01
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$\begingroup$ Yes, but that's missing the point. Imagine striking the end of the rod with a brief but comparatively weak force. The rod will not instantaneously move 1 cm. Even much smaller objects take finite time to accelerate and travel a finite distance. A sustained force is no better. The stiffness of the rod resists the push. It is not as though you can get the end of the rod near you to move 1 cm and the rest of the rod must catch up. Even if the rod were made of an elastic, non-dissipative gel, the part near you would rebound after the compression passes through it. Mass matters, movement takes time $\endgroup$– GeoffreyCommented Sep 8, 2017 at 23:58
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The answer to this depends on the material properties of the stick. If it were a mythical "rigid body," the entire stick would move all at once, but that "rigid body" is an approximation that isn't valid in situations like this. The rigid body assumptions assume that you can transmit information about movement instantaneously. In many cases, that's close enough to correct, but in situations like this, the speed at which you can propagate the force through the stick is not infinite, and matters.
Instead, you would need to set up a compression wave along the stick which transmits the information about you pushing it down the length of the stick. The speed of that compression wave depends on the material you use. Thus, in the real world, the "best case scenario" is dependent on the speed of sound in the medium you build the stick out of.
answered Sep 8, 2017 at 1:23