I accidentally dropped a 1 kg object on the ground

[Count Iblis]

It was released from a height of 1 meter above the surface. Describe the motion of the Earth from the moment I released the object in a realistic way. Order of magnitude estimates are ok. but you have to justify those approximations in a reasonable way.

Hint: This problem may look rather trivial. If you think so, then the hint is that this problem is extremely difficult.

 

[xxChrisxx]

Easy peasy.

As I am an engineer, any movement of the Earth is too small to be reliably measured. Therefore I would totally ignore it.

 

[Count Iblis]

Yes, you can ignore it for all practical purposes. But then, that is not the question.

 

[Dadface]

The momentum gained by the Earth is equal and opposite to that gained by the object.Just chuck the numbers in and calculate.

 

[Count Iblis]

The momentum gained by the Earth is equal and opposite to that gained by the object.Just chuck the numbers in and calculate.

This will yield the center of mass motion of the Earth. If the whole Earth were to move as a rigid object, this is all there is to the problem. But it is easy to see that the Earth cannot respond to the release off the object as a completely rigid body, not even in a very rough first approximation.

 

[uart]

The momentum gained by the Earth is equal and opposite to that gained by the object.Just chuck the numbers in and calculate.

Yes but the net momentum gained is zero since the object stops after approx 0.45 seconds when it strikes the ground. So you would need more than this if you were to calculate the Earths net displacement for example.

Here's one "solution", I don't think it's very realistic however.

1. Before the object is dropped the gravitational force of attraction of the object on the Earth is exactly countered by the downward force of the weight of the object though the holders feet, so the net effect is zero. (That is, the object has no net pull on the Earth).

2. When the object is dropped, then while it is in free fall there is a net attraction and so the Earth accelerates upward (relative to the trajectory that it would have taken had the object not been dropped), with a = 1.6E-24 m/s^2 (approx).

3. After approx 0.45 seconds the object strikes the ground giving an impulse which exactly counters the momentum gained during the previous 0.45 sec acceleration, so no further displacement occurs.

4. The net displacement (again relative to the trajectory the Earth would otherwise have travelled) is approx 1.7E-25 meters.

 

[uart]

This will yield the center of mass motion of the Earth. If the whole Earth were to move as a rigid object, this is all there is to the problem. But it is easy to see that the Earth cannot respond to the release off the object as a completely rigid body, not even in a very rough first approximation.

Yes you beat me to it, but that's exactly what I was about to add. In my post I declared that "Here's one "solution", I don't think it's very realistic however.", and that was precisely the reason why (I said it was unrealistic). In any case I posted it so that at least we can get the "rigid body" solution out of the way.

Actually I like xxchrisxx's solution the best so far. Nothing happens to the Earth seems plenty "realistic" enough to me. :)

 

[Dale]

Describe the motion of the Earth from the moment I released the object in a realistic way.

Realistically it does not move. There is no experimental measurement you could realistically perform that would be able to detect the motion.

 

[Count Iblis]

Realistically it does not move. There is no experimental measurement you could realistically perform that would be able to detect the motion.

Motion is more than just the center of mass motion in this case; the Earth is not a rigid object, as uart also noted.

We can just decide to regard the object as part of the Earth, so that the center of mass does not move at all. Let's then discuss the motion of the whole system about the center of mass.

 

[SpectraCat]

Motion is more than just the center of mass motion in this case; the Earth is not a rigid object, as uart also noted.

We can just decide to regard the object as part of the Earth, so that the center of mass does not move at all. Let's then discuss the motion of the whole system about the center of mass.

Fine. The reduced mass in this case is 1 kg to any measurable precision ... all of the motion is concentrated in the 1kg mass, and nothing else moves to any measurable extent. I guess (like others) I don't see what you are getting at here?

 

[Count Iblis]

Part of the problem is indeed to see what else could be moving besides the center of mass of the Earth and the 1 kg object. If I were to eleborate on that in detail, I would be spoiling the problem.

 

[{~}]

I don't feel like working this one out but

http://en.wikipedia.org/wiki/Tidal_force"

When talking about small displacements you can't talk about rigid bodies because they don't stretch or flex yet even strong materials do just that. Small displacements caused by small forces on very large inertias result to a much greater extent a change in geometry of said inertia than any total displacement.

Hint if you didn't click my spoiler: What if you dropped the mass over board while you were out to sea?

 

[SpectraCat]

Hint if you didn't click my spoiler: What if you dropped the mass over board while you were out to sea?

It completely depends on the shape of the mass (which was unspecified). If it displaces more than 1kg of water, and does not "swamp" when it lands, then it will float. If it displaces less than 1kg of water, it will sink. In either case, it will create waves on the surface of the water, which (due to chaos theory), may contribute to storm development in the arctic. Also, if the 1kg happens to land in the correct orientation at the bottom of the ocean to block a thermal vent, it could cause pressure buildup that causes a massive undersea earthquake, propagating huge tidal waves around the globe, which further cause the Antarctic ice shelf to break off all at once and fall into the sea, causing an instantaneous rise of 3 meters in the global sea level, drowning the servers that host Physics Forums, mercifully putting this thread out of its misery.

Seriously, proper problem solving in physics requires details ... please provide them, or we can't make any meaningful progress on this "problem" you have posed. I love cleverly posed problems that make me think, but this one is way too open-ended. Isn't there a clearer way you could make the point you are trying to get across?

 

[russ_watters]

This is much too freeform/pointless to be a worthwile discussion. Locked.