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Gravity

Posted by: SuperShuki - Tue Mar 22, 2005 9:07 pm
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Gravity 

If you drop two objects on earth, one more massive then the other, which will reach the earth first?
The more massive object. 25%  25%  [ 3 ]
The less massive object. 0%  0%  [ 0 ]
They never hit the ground. 0%  0%  [ 0 ]
Is this a trick question? 17%  17%  [ 2 ]
I don't like trick questions. 17%  17%  [ 2 ]
They both hit the ground at the same time 42%  42%  [ 5 ]
Who cares? 0%  0%  [ 0 ]
Total votes : 12

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Post Gravity   Posted on: Tue Mar 22, 2005 9:07 pm
Something else I learned in 'high school physics'.

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Post    Posted on: Tue Mar 22, 2005 9:37 pm
That IS a trick question.

Gravity is stronger for a larger mass, so you may be fooled into thinking it a more massive object would fall faster.
BUT, a more massive object requires more force to accelerate. The two opposing effects cancel each other out and objects of different mass fall at the same speed.

Here is the mathematical proof.

The force of gravity F is:
(1) F = GMm/R^2
where G is the universal gravitational constant, M is the mass of the object being dropped, m is the mass of the Earth and R is the distance from object being dropped to the center of the Earth.

Objects accelerate due to a force according to Newton’s law:
(2) F=MA
where F is the force acting on the object, M is the mass of the object and A is the acceleration of the object.

The acceleration of a falling object is just the force of gravity acting on it. So substituting the value of F from equation 1 into equation 2 gives:

(1) F = GMm/R^2
(2) F = MA
or
GMm/R^2 = MA

Now divide both sides of the equation by M to get

Gm/R^2 = A

Acceleration depends ONLY on the gravitational constant, mass of the Earth and distance from the center of the Earth. It does NOT depend on the mass of the falling object.

(Note that capital M is usually used as the mass of the Earth and m is usually the mass of the small falling object , but I choose to reverse that convention because capital M is usually used in F=MA and I couldn’t bring myself to write it as F=mA. But that is just a naming convention and does not change the Physics or Mathematics at all.)

(EDIT)
BUT THERE IS MORE!

The Earth is also being accelerated by the falling object. In reality the object does not fall to a stationary Earth, both the Earth and the falling object move toward a point in between them. And since the acceleration the Earth feels depends on the mass of the falling object, it will accelerate more toward the more massive falling object. If you dropped the two objects at different times and timed the fall VERY carefully, the larger object would take SLIGHTLY less time to hit. BUT, if you dropped them at the SAME time, then the Earth would be accelerated by the combined gravity of both objects at some larger value than it was by either object alone, so it would move toward the meeting point a little faster. But the two objects would move toward that same point at the same speed and arrive at the same time.

So the tricky bottom line is that when dropped at the same time, both objects hit the ground at the same time, but when dropped separately, the more massive object takes slightly less time to hit.

To get even more tricky, do we need to add the mass of the object already on the ground to the mass of the Earth when calculating the fall time of the second object? Now my head is spinning!


Last edited by campbelp2002 on Tue Mar 22, 2005 10:07 pm, edited 4 times in total.



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Post    Posted on: Tue Mar 22, 2005 9:47 pm
lol, I vote for the more massive one ;)
I'm sure a normal pingpong ball would fall a lot slower compared to an iron pingpong ball :P

And about all the math on "other types of objects".. oh well not my problem :P

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Post    Posted on: Tue Mar 22, 2005 10:05 pm
Sigurd wrote:
I'm sure a normal pingpong ball would fall a lot slower compared to an iron pingpong ball
Unless your iron pingpong ball has a parachute! :lol:


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Post    Posted on: Tue Mar 22, 2005 11:01 pm
what height are they dropped from?

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Post    Posted on: Tue Mar 22, 2005 11:31 pm
Quote:
BUT THERE IS MORE!

The Earth is also being accelerated by the falling object. In reality the object does not fall to a stationary Earth, both the Earth and the falling object move toward a point in between them. And since the acceleration the Earth feels depends on the mass of the falling object, it will accelerate more toward the more massive falling object. If you dropped the two objects at different times and timed the fall VERY carefully, the larger object would take SLIGHTLY less time to hit. BUT, if you dropped them at the SAME time, then the Earth would be accelerated by the combined gravity of both objects at some larger value than it was by either object alone, so it would move toward the meeting point a little faster. But the two objects would move toward that same point at the same speed and arrive at the same time.

So the tricky bottom line is that when dropped at the same time, both objects hit the ground at the same time, but when dropped separately, the more massive object takes slightly less time to hit.


How did you guess?! My physics teacher didn't get that!
:wink:

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Post    Posted on: Wed Mar 23, 2005 2:53 am
I don't guess. I know. 8)
And the strange choices in the poll indicated an obvious trap!


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Post    Posted on: Wed Mar 23, 2005 4:17 am
well also air resistance has to be taken into account, so really the object with the smallest drag:mass ratio will hit first, pretty much regardless of the relative masses. obviously i'm not counting something like the moon or a similarly large object as an "object" for this model lol.

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Post    Posted on: Wed Mar 23, 2005 10:40 am
campbelp2002 wrote:
That IS a trick question.


It all depends on what the starting conditions are. Do you factor in air-resisance. Should the mass of the two objects falling be considered to be neglibible compared to that of the Earth? Should the small fluctuations of the Earths gravitational field depending on where you are standing. Is the latitude at which you are standing the same?

They are all important questions. Of course, we should all (although sadly there are probably people here who don't) know what the answer is for a two different balls dropped in a vacuum, where their masses are considered to be neglibible compared to the size of the Earth.

In reality, the larger mass attracting the Earth towards it is going to be immeasurable. The position of the moon in the sky would make more difference :P

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Post    Posted on: Wed Mar 23, 2005 12:27 pm
If you may ignore airfriction and thus shape, both objects will hit the ground at the same time.

If you have 2 led balls, one 3 kg and one 1 kg, and drop them from the Big Ben, or Empires state building, and you would have completely wind-still conditions, they will hit the ground at the same time.

/edit/
If this question would come up at an examn, 99% of the time they mean you can ignore Fw (airfriction). Unless they state otherwise off course...


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Post    Posted on: Wed Mar 23, 2005 1:38 pm
If there WERE NO atmosphere both objects would hit the ground in the same moment - this has been proven when the first men were at the moon. They let fall a feather and a hammer to the moon's ground in parallel and they hit the moon's ground in the same moment. Their fall started at identical altitude.

So this would be valid on Earth too - if and ONLY if there were no atmosphere. But there IS atmosphere - because of this the hammer would hit the ground first.

My vote I based on the assumption that you are talking implicitly about the case that there were no atmosphere.



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Post    Posted on: Wed Mar 23, 2005 1:43 pm
I might point out that if you "drop two objects ON Earth" (emphasis added), that they're already (by your own statement) sitting on the ground, and therefore they both reach the ground at exactly the same time, at t=0.

So there.

And when playing with fluids, mass is irrelevant. Density, however is not.

Of course, we have yet to bring up Xeno's Paradox.....

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Post    Posted on: Wed Mar 23, 2005 2:23 pm
Ekkehard Augustin wrote:
If there WERE NO atmosphere both objects would hit the ground in the same moment - this has been proven when the first men were at the moon. They let fall a feather and a hammer to the moon's ground in parallel and they hit the moon's ground in the same moment.
Apollo 15 Dave Scott dropps a hammer and feather.
http://www.hq.nasa.gov/office/pao/Histo ... sout3.html
Look for the video clip at 167:22:02, about 1/4 of the way down the page.
Note how the 1/6 gravity makes both fall slowly enough to make it look somehow "wrong". Another reminder of how different the Moon is.

And another reminder of how much more amazing 1971 was than 2001 :cry:


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Post    Posted on: Tue May 03, 2005 3:12 pm
Trick question.

Okay, we've nearly covered the gamut here. We'll exclude the obvious, air resistance, and assume this takes place in a vacuum. Many good points have been brought up. Variation in earth's gravity, based on the underlying composition, or other factors, for that matter, such as the earth's rotation, or distance from the center of the earth, or the effect of the non-spherical nature of earth's shape (only perfect spheres of homogenous composition truly obey the inverse square law as measured from the center of gravity).

And of course, we've brought up the acceleration of the earth towards the object. If the objects are dropped at the same time, from essentially the same location, this is negligible. But what if the two objects are on opposite sides of the earth? Then the earth will preferentially accelerate towards the more massive object, and hence accelerate away from the less massive object. Of course, even this has a complication, because the less massive object is accelerated towards both the earth and the more massive object, so it will accelerate faster than the more massive object, and hence compensate for the fact that the earth is accelerating away from it (the smaller object).

And speaking of the earth's rotation, there would be a difference if the two objects were dropped ten feet apart, from exactly the same height. In one scenario, if the two objects are on the same latitute, but different longitudes, then the earth's gravity would affect them identically, but the object further east would accelerate ever so slightly towards the western object, thus slowing its rotation speed, and allowing it to fall faster. The western object would accelerate towards the eastern object, increasing its rotation speed, and hence slowing its decent. The effect would be nearly negligible, but if the objects are massive, and are dropped from a high enough altitute, the difference could be measured, or at the least, calculated. (Note: this effect is seen in Saturn's rings, where objects in "free fall" accelerate towards each other, and the trailing object is slung away from Saturn, and ironically loses orbital speed as it gains energy and a higher orbit.)

From the same longitude, but different latitudes, would be even more complex. Coriolis effects would come into play, and the coriolis effect on each object would be slightly different, altering their trajectories (on top of their gravitational attraction), so that one would fall slightly faster.


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Post    Posted on: Tue May 03, 2005 3:58 pm
There is only one missing bit of information required to properly answer this question: Who wants to know?


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