If you drop a high-mass object and a low-mass object at the same time, in a vacuum, they both fall at exactly the same rate. I've heard this said so many times.
But why?
Wouldn't the gravity of the more massive object combine with the gravity of the planet to make the massive object fall half a nanometer per second faster?
Or is something else happening that cancels out the respective gravitational values of the two objects entirely?
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>Wouldn't the gravity of the more massive object combine with the gravity of the planet to make the massive object fall half a nanometer per second faster?
The "proof" people usually give for this doesn't even need to be TeXed up:
F = m1 a
F = m1 m2 G / r^2
by Newton 3 the forces must be equal so,
a = m2 G / r^2 (= g)
Which I guess makes sense in Newtonian mechanics, probably not in GR, but I don't know anything about GR so we'll wait for a physicist to pop in.
youtube.com
see for yourself
This doesn't show their acceleration values measured in the nanometer scale, though.
OP, the gravitational force that any object receives in the presence of Earth is:
F = GMm/r^2
where G is the gravitational constant, M is the mass of the Earth, m is the mass of the object (feather or bowling ball) and r is the distance between them.
If you crunch the numbers for the bowling ball and feather, you will find that the bowling ball receives a larger force.
However, when you go to calculate the acceleration of either body using F=ma, you will find that the acceleration (a) comes out to the exact same number for each object, 9.81 m/s^2 (at Earth's surface anyway).
TL;DR The bowling ball receives more force because it is heavier. They fall at the exact same rate, however, because the forces and masses "balance" and you get 9.81 m/s^2 for any mass object.
You forgot that Earth also accelerates at m1 G / r^2. Since m1 is bigger in case of the bowling ball, Earth will have higher acceleration in that case, therefore OP is right, and the bodies (ball and Earth) indeed touch sooner.
Though it would be an immeasurably small time difference.
It cancels out with inertia
If you drop them at the same time, then Earth's acceleration is irrelevant because wherever Earth is they will hit it at the same time.
Does someone want to calculate the acceleration of Earth? If you did two separate experiments taking Earth's acceleration into account, how much faster would the bowling ball hit? Is it even a Planck time faster?
What if the two objects are a pebble and the moon? Would the moon's gravity also be cancelled out and be effectively negligible in this case?
Yeah, why not. It is even better, because you dont have to assume the vacuum
They have actually done that experiment
No, I mean, what if the two objects you're dropping onto the Earth are a pebble and the moon.
Yeah this is correct. The acceleration of the objects towards earth is the same, but the acceleration of the Earth towards the objects is not. So the OP is only true if you consider a reference frame fixed on the Earth.
Oh, oops I cant read.
It would be also the same, I guess. Imagine releasing that pebble in the same orbit as the moon. If no other force disturbs it, it will continue its orbit the same way the moon does.
I had no idea such a huge vacuum chamber existed! That's fucking awesome. That's hilarious that they evacuated that entire chamber to do that one experiment. I wonder how expensive it was to get time in that room and run the pumps.
Anyway, to OPs question. Newton's 2nd Law states:
F=ma
For a gravity field g, we can model the force as
F=mg
Putting these equations together, you get
ma=mg
a=g
The mass terms cancel out!! This is something unique to gravity. If you, say, put a charge q in an electric field E, the force is
F=qE
so you get
qE=ma
which is clearly different.
It is true that more massive objects experience more force than lighter objects when put into a gravity field. So how is it that the feather and bowling ball fall at the same time? Well, it's because their masses are different! Yes, a larger force acts on the bowling ball, but at the same time the ball has more inertia, and these two precisely cancel out. Thus, although the force on the ball and feather are different, they fall at the same rate.
Make sense?
People have already written this, but I'll recap before making a point:
By Newton's second law,
F=ma=-mg
You cancel the mass and you get
a=-g, so all masses fall at exactly the same rate (without additional forces--damping).
I like to imagine negative mass. Let's say negative mass exists wrt Newton's second law. Our result says that the m's cancel, so a=-g regardless of mass (even if it's negative). This means that negative mass objects fall in exactly the same way as ordinary feathers and bowling balls (in a vacuum).
Next imagine I take a ball of negative mass (-1kg) and a ball of positive mass (1kg) and tie them together with a string and then hold them vertically in the air. I'll let the negative mass be on top. What happens when I let go? In the approximation that the gravitational field acting on both is equal (g is the same for both), the sum of the forces on the positive mass is:
Fp = T-m*g
where T is the tension force from the string. The equation for the negative mass (which has a mass -m) is
Fn = -T+m*g
We see that these forces cancel exactly which means that the entire system floats in the air. What happens when I cut the string? Well, we already went though this, they both fall to the ground at the same rate.
Think about what happens when you put negative mass in a jar. It's a waste of time, but it's kind of amusing to think about.
Holy shit, I've been there! It takes fucking forever to pump down.
Gravity is caused by the fabric of space warping to cause a well. Higher the mass of an object, the greater the well.
The bowling ball and feather are at the same location on the gravity well so are the same. If either one is moved closer or further from the wells center point, then the speed changes. One part of the well can be steeper or level then the other part.
But doesn't the bowling ball create it's own gravity well you might saying. Yes. But does the speed at which a gravity well swallows another gravity well increase or decrease as the size of the swallowed gravity well increases? Iunno.
I could see it taking more effort, hence a longer time to "swallow" a higher mass object. It has to swallow a more pronounced gravity well, meaning it might "drag" more in the fabric of space. So a bowling ball would fall slower than the feather.
...
Eh, the moon would attract the pebble. But if you measure them separately and make sure the ground doesn't move, they should fall at the same rate
thanks gran, university is going well thank you! love you!
gravity is just the electromagnetic resonance of condensed energy, when an object is attracted by a massive body, the proportion of that object's energy to the body is so incalculably small that the effect it's resonance has on the larger body is negligible, hence why dropped objects have a seemingly constant "gravity"
>gravity is just the electromagnetic resonance of condensed energy
This theory is so widespread that you'd almost think it's been tested before.
A lot of the maths above hinges on the huge assumption that gravitational mass is identical to inertial mass. Intuitively it is so but certainly is not there, to the point it has been measured. These are the same to within 1E14 but there are theories these should differ. Also there are theories that these are identical, and thanks to lack of certainty a lot of theories are growing like weed.
There is nothing intuitive about it. Inertial mass and gravitation mass being apparently the same thing is what inspired Einstein to draw strong parallels between gravitation and acceleration. In fact, this is integral to the theory of relativity.
It's astonishing the precision we've measured the two types of masses to be equal. While it is important to consider the math and its consequences in case they are not, our universe very much seems like the two are equivalent, and that is a fascinating observation with deep implications.
the problem being more that mass of the moon >>>>> mass of the pebble. So what's likely to happen is the moon and earth either attract each other and collide quicker, or as happens now the moon orbits the earth.
this is right and it is the part that gets forgotten the most
the only reason the ball and feather fall at the same rate is the earth is orders of magnitude more massive, making the difference in mass of the projectiles irrelevent
Relative to the center of gravity of the most massive object at the beginning of the fall, the two objects will fall at the same speed. It doesn't matter if the most massive object also moves, since the reference point stays the same (as in the objects move, the reference point is no longer centered on the center of mass, but remains in the same position relative to itself).
In short :
Scenario 1 : Object A falls toward object B. Relative to point K, object A falls at X speed
Scenario 2 : Massive Object C falls toward object B. Relative to point K, Object C falls at X speed. Relative to point K, object B moves a bit toward Object C.
If your point of reference is the center of mass of the most massive object (the earth for instance), and moves with it, then yes, a more massive object will move faster toward it.
Gravity is a constant acceleration. Think about F = ma. A larger object (say 100 kg) will have a larger force (980 N) than a smaller object (say 1 kg, so 9.8 N). However, that larger item needs a larger force to overcome it's inertia (caused by it's mass), so you end back up with a constant acceleration for gravity (9.8 m/s^2), which indicates that any two objects at rest or at the same velocity will fall at the same time if dropped simultaneously.
Because inertial mass is equal to gravitational mass.
> but why?
I don't fucking know.
In this case I dare to say that a nanosecond is many orders of magnitude above the reality.
Also, these objects are so small that you can assume that their weigh sums up when you crush the numbers to calculate this:
IF and only if you drop those objects alone, you will be able to measure any diference.
Actually, what I just said doesn't implies the invalidation about the idea that planck time
is the smallest frame of time possible?
I can't say it, I'm not a physics guy.
>If you drop a high-mass object and a low-mass object at the same time, in a vacuum, they both fall at exactly the same rate. I've heard this said so many times.
This is true only if the masses of the object are negligible compared to the mass of the Earth. This thread is full of retards.
> the Earth moves differently in the area below the feather than the area below the bowling ball
your retarded
>Gravity is a constant acceleration.
Please tell me you are joking. Or show how you defeated the inverse square law.
Obviously he was talking about the acceleration an object feels at a fixed distance.
This post made my head hurt
>acceleration an object feels
>acceleration
>feels
>fixed
The moon approximatelt weighs 7.35x10^22 kg
A bowling ball approximately weighs 4.54 kg
A feather approximately weighs .074 kg
4.54-0.74 / 7.35x10^22 = 6.07x10^-23
So the bowling ball accelerates faster than the feather by approximately 6.07x10^-23 m/(s^2)
The bowling ball moves faster than the feather at a fixed radius, but over the course of a few seconds of freefall, the change is so miniscule that an incredibly high speed camera couldnt pick up the difference.
No. Write down the formula for the gravitational force of the earth on a bowling ball. Divide that by the mass of the bowling ball.
Repeat the above process with the feather.
Is your formula the same? Good. Now shut up.
The result in slightly greater. The bowling ball actually moves slightly faster than i predicted, but still ×10^-23
If hes in a vacuum chamber, then how come I hear him speak?
3 hours is not THAT long