Galileo, Hammer, Feather II: The Electric Boogaloo

Cecil's Columns/Staff Reports
1

My original thread, Galileo, A Hammer, & A Feather (incidentally the first thread I ever started on the SDMB, so I feel sort of attached to it) was really interesting, but got really far off topic. My point in the OP was that a more massive body will, in fact, fall faster due to the fact that it exerts a larger gravitational effect on the earth than will a less massive one.

The previous thread degraded into discussion about relativity, Heisenberg, quantum effects, etc. All very interesting and enlightening stuff, but not having anything to do with the OP.

So once again, I’m making the same argument, but with the original values changed a little bit to make the petty details about relativity and such irrelevant.

Given:

Object A: A mythical, perfectly stationary point mass of 5.976e24 Kg, surrounded by a perfect vacuum.

This mythical object exists in a mythical universe where the Gravitational constant is exactly 6.673e-11 m^3 Kg^-1 s^-2 and all measurements are correct to infinite precision.

Object B: An imaginary perfect point mass of 0.001 Kg (1 gram).

So far everything is the same, but here is where we diverge.

Object C: Another perfect point mass of 5.976e24 Kg, also with no atmosphere.

Both of the dropped objects will be dropped a distance of 200 meters, but the target (Point T, where we will define the drop to have ended) is a point precisely 6.378e6 m above object A.

At any given time, only the following objects exist in this universe:
Object A.
Either Object B or Object C, depending on which is being timed.
A massless intelligent observer.
A massless timing device.

B and C originate at Point O, perfectly collinear and inertially stationary with A and T, and precisely 200 meters distant from T, and instantaneously begin to accelerate toward A due to mutual gravitational attraction.

Once again, I’m ignoring the change in acceleration over the 200m drop (it would support my statement even more strongly, but is small enough in this example to be negligible).

Here are my numbers:
Acceleration towards A: 9.80246…
Acceleration towards B: 1.64000e-27 (we all agreed in the previous thread that this is small enough to be overshadowed by quantum & relativistic effects, so can be considered zero)
Acceleration towards C: 9.80246… (exactly the same as A)

So…Additive accelerations:
Between A and B: 9.80246… (effectively the same as acceleration of A since B’s gravity is negligible)
Between A and C: 19.6049…

Total time (in sec.) to reach point T:
B: 6.38796…
C: 4.51697…

With a difference of 1.87099 seconds. Which IS significant.

Comments from you physics guys? Have I made any mistakes here? Lemme have it. :smiley:

2

Crap. I knew I’d forget to post the link.
The mailbag item being discussed is If a feather and a hammer are dropped together, won’t the hammer hit the ground first?

3

Hmm, I thought we covered that in the previous thread, as well as all of the fun tangents… To sum up, yes, your results there are correct, for the initial conditions you specified. The two problems are that first, a real hammer isn’t that massive, and in fact, for a reasonable-mass hammer, as for the feather, the effect is small enough to be utterly negligible; and second, the fact that you’re removing the objects from the Universe when you’re not using them. Given, however, a hammer of that extraordinary mass, and a means of isolating it from the rest of the system as needed, the hammer does, indeed, hit first.

Thanks for bringing up memories, by the way… That old thread was one of the first threads to which I posted, and the one which made me realize I was right at home here :slight_smile:

4

So, what does it mean, that A is “perfectly stationary?”

5

I don’t think I understand your arguement. Are you saying because A is moving toward C and C is moving toward A they will meet somewhere in the middle, and that this faster than A moving toward B because B excerts a negligible acceleration upon A? Well this is true, but it is in violation of your intial conditions which say A is perfectly stationary. You can only add the accelerations if A is allowed to move.

Of course, this is all far removed from the hammer and feather problem where the acceleration of the Earth (or the Moon) is negligible.

6

I think that he just meant that mass A was perfectly stationary at the start of the experiment, not that it necessarily stayed that way.

7

In other words, totally divorce the example from reality. Hmmm, then, what does that have to do with the column?

8

Oh, and by the way, just what is a “boogaloo”? I mean, aside from an embarassing dance from back in freshman orientation. :slight_smile:

9

[boogaloo hijack] Yes, boogaloo is the dance. The reference however is to the naming of sequels. The OP refers to Breakdance II: Electric Boogaloo. It could have easily been Gal/ Hammer/ Feather II: The Wrath of Khan. [/hijack]

10

What it has to do with the original column is the fact that a larger mass will, in fact, fall slightly faster than a smaller one. The point of the column, in other words. How much larger the mass has to be, how much farther it has to fall, and how much faster it will fall are all secondary to the question. The difference in mass and the distance dropped have to be large for the effect to be noticeable, but the effect does exist.

And by the way, it is common practice to ignore (theoretically) other effects that influence the outcome of an experiment in order to observe the results of another. I’m assuming you’ve taken a physics class or two, and I further assume you didn’t argue with your professors when they told you to disregard friction. Am I correct? Well I’m disregarding everything except the force that we are discussing. In order to avoid having people say “well THIS force is bigger and will make that difference negligible.” We’re discussing a concept, not a real world situation. Yes, from six feet above the surface a 2 pound hammer will not fall noticeably faster than a 1 ounce feather. But so what? If circumstances were different, it could be a big difference. See what I’m saying?

Exactly. Except it was “Breakin’ 2: The Electric Boogaloo”. :smiley:

11

I thought our previous thread established that this is not true if a) the hammer and feather are dropped together (the earth will be attracted to both as they fall), or b) if the hammer is dropped while the feather is on the ground, then the feather is dropped while the hammer is on the ground (the extra hammer mass will affect the fall time of the feather).

The only way around that is to make some of the mass disappear from one drop to the next–which is how you’ve structured the OP. Of course if you calculate using two different masses, you’re going to get two different answers.

12

You’re still missing the point. if you drop them together, all 3 objects fall to the center of mass, which will be closer to the larger mass than to the smaller, therefore the larger mass and the earth will collide first.

All you need to do is make the difference in mass and the distance between the objects large enough, and the drop high enough.

This is exactly the kind of quibbling over details I was hoping to avoid, and why I structured the OP as I did. Thanks for illustrating my point so well.

Want a better example? Go out into space, find a fist-sized object of such density that it has mass equivalent to the earth. I’m sure it exists somewhere. Cause it to hover 200 meters above the North rotational pole. How you do it is not my problem.

In the meantime, find a fist-sized rock that weighs about 5kg. Since differences between a hammer & feather are negligible, we want something massive enough that it will overcome atmospheric friction to the point that it, too, will be negligible. Cause this one to hover 200 meters above South rotational pole.

Now cause them to drop simultaneously. Again, how you do this is not my problem, though I’d suggest doing it electrically, with signal wires of precisely the same length, just to account for the speed of electricity.

Now, which mass hits the earth first? The 5,976,000,000,000,000,000,000,000 kg mass 200 meters above the north pole, or the 5 kg mass 200 meters above the south pole? They’re dropped together, aren’t they? But one hits first.

Yeah, I know. the surface at the north pole would be lifted up due to the fact that d is only 200m so A is huge. Ignore that. the earth moves as a unit.

13

No problem, you’re welcome. But I think you’re doing fine by yourself.