[pontus]

Here's a fun thought experiment / apparent paradox.

In high school physics we learn that a 1kg mass accelerates as quickly as a 2kg mass when only subjected to the force of gravity. When I used to teach physics, the intuitive explanation I gave for this hinged on a thought experiment. Suppose that you have three 1kg masses falling side by side after being dropped from the same height. Clearly they are all going to fall at the same rate since they're equivalent. Now imagine redoing the experiment but this time taking two of the masses and placing them closer together. Does anything change? Clearly not, they're still all equivalent and ought to fall at the same rate. Now imagine doing this until those two masses are right next to each other, touching. Does anything change? Well no, all three should still fall at the same rate. But now, why not glue those two masses together and call it a 2kg mass? Once you do that you've shown that a 1kg mass and a 2kg mass fall at the same rate.

This usually convinces people, but there's actually a flaw in the argument that gets to the heart of why gravity is so different from the other forces.

To see the flaw, replace the above masses by three electrons falling next to each other in an electric field. Everything goes through in exactly the same way. You end up gluing together two of the electrons and these two electrons will accelerate at the same rate as the single electron. But if you're not careful you'd conclude that all electric charges fall at the same rate in an electric field, something we know is false.

Where's the flaw? Well, all of matter is built from some particles, and as long as you restrict yourself to particles that have the same "charge/mass ratio", the argument above works. It is true that one electron accelerates the same as 100 electrons tied together but that's just because e/m is the same for all those constituents.

So, the thing that's glossed over in my high school explanation for why 1kg and 2kg accelerate at the same rate is that the constituent particles all have the same "gravitational charge / inertial mass" ratio. Because this ratio is the same for all particles, we may as well absorbed that ratio into the gravitational constant and just use "m" in place of both of them. It's this "universal coupling" that's really responsible for the equivalence principle and what sets gravity apart from the other forces.

[splitwheel]

1kg mass and a 2kg mass do not fall at the same rate. The Gravitational force is (G*m1*m2)/r^2. You are observing that m1 (the earth) is much much greater than m2 (the 1 or 2 kg masses), and you are simplifying to (G*m1)/d^2 because of the precision of the measuring device. Also, d is the same for both masses.

[evanb]

They do fall at the same rate, even with Newtonian gravity. For,

    F = m a
    F_gravity = GMm/r^2

so that

    ma = GMm/r^2.

Now cancel m from both sides and get

    a = GM/r^2

If you plug in G=6.674e-11 m^3 kg^-1 s^-2, M = M_Earth = 5.972e+24 kg and r = R_earth = 6.378e+6 m you get

    a = 9.79... m/s^2

which ought to be familiar.

[splitwheel]

The force is on both objects at the same time. The force in F = ma is a function of the mass of both and their distance. If the mass is different in the two scenarios, then the force is different. On earth with small weights, they seem the same because of the precision of the measurement.

This is why you _weigh_ less on the moon.

[evanb]

Is what you're getting at the fact that the distance between the earth and the other object changes from two effects (the first being the ball falling towards the Earth and the second being the Earth falling towards the ball)? That's right, of course. But that distance's second derivative is not the acceleration a in F=ma. Indeed, in both Galilean and Einsteinian relativity acceleration is detectable locally without a needed reference to another object.

[splitwheel]

Yes - I was making a mistake. I was trying to describe the effect of both masses. When one is much smaller than the other, then the movement is mostly in one direction. When they are closer in mass or even equal, they move toward each other. For example, if you have a 1 liter water bottle filled with a material that gives it the same mass as the earth, then the two bodies will move toward each other, and the water bottle will seem to move toward the earth much faster that the 1 filled with water (1kg). If it is filled with a material, that gives it much grater mass than the earth, the earth will move toward it.

[hllooo]

what no they do fall at the same rate. acceleration is F/m so the mass of the object cancels out

[stbede]

The mass of the earth dictates the acceleration of the individual masses towards the earth. However the acceleration of the earth itself towards the masses are dependent on how much mass is falling towards the earth. When more mass is falling to the earth, the earth accelerates towards the masses faster. So the thought experiment is flawed because with only one 1 kg weight falling towards earth, the gap between the weight closes slower than when there are three 1 kg weights spaced 1 m apart and dropped simultaneously.

[balfebs]

If you define fall as the size of the gap. You could also take it as acceleration towards the barycenter, which would be the same. These are indistinguishable for everyday objects so could argue that the word “fall” could be interpreted either way.

[KineticLensman]

Sorry, but "Mr Galileo was correct" : https://moon.nasa.gov/resources/331/the-apollo-15-hammer-fea...

[thedailymail]

Nitpicky, but shouldn't that be "Mr. Galilei"?

[KineticLensman]

I was just quoting what the astronaut said

[lisper]

>You end up gluing together two of the electrons and these two electrons will accelerate at the same rate as the single electron.

Not really. You have swept under the rug the fact that it's really hard to glue electrons together. And if you were to actually find a way to do it, you would have to add so much potential energy to the system that it's inertial mass would increase dramatically. In fact, two electrons that were actually "together" (whatever that might actually mean for a quantum particle that obeys the Pauli exclusion principle) would have a mass orders of magnitude higher than two electrons separately.

You probably want to talk about ions in an electric field, which you can "glue together", but then it becomes rather obvious that they don't all accelerate at the same rate.

The equivalence principle is not the only thing that distinguishes gravity from other forces. There is also the fact that there is only one gravitational "charge" and it's mutually attractive. (Gravity is also many orders of magnitude weaker than all other forces.)