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delirium happy

Just keep on trying till you run out of cake

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Mass
mandelbrot
rho
I was originally going to write an entry about how electricity works, and this was going to be an aside in the bit where I analogised against gravity, but then I decided that this was sufficiently cool and interesting to warrant an entry all of its own.

I never know how much a general audience of non-physicists will know about various physicsy topics, because there's a lot of things that I've known for sufficientl long now that I can't remember where or when I learned them. The concept of mass is one of these things. The idea of mass is very closely related to the everyday idea of weight. The kilogram is the SI unit mass of mass, and doesn't measure weight as it does in common usage. Instead, weight is used specifcially to refer to the force on the object due to gravity, whereas mass is, essentially, a matter of how much "stuff" something contains. So, if you go to the moon, which has lower gravity than Earth, then your weight will be less, but your mass will remain the same.

One of the interesting things about mass, though, is that there are two different kinds of mass, both of which happen to be exactly the same.

Firstly, we have gravitational mass. This measures how much an object is affected by gravity. Hold a cannonball in one hand and a ping-pong ball in the other hand and you observe this effect. The cannonball feels heavier, because gravity is exerting a greater force on it because it has a greater gravitational mass.

Then we have inertial mass. This is to do with how hard it is to change an object's motion (speed it up, slow it down, stop it, change its direction, or whatever). This is the type of mass that shows up in Newton's second law of motion, when it's written as F = m × a. F is the force, m is the mass, and a is the acceleration. Essentially, given the same force, an object with a low mass will accelerate much more than an object with a high mass. You can demonstrate this one by putting the cannonball and the ping-pong ball down on the ground and then trying to get them to move. Pushing the ping-pong ball requires a whole lot less force.

Throughout all of classical physics (everything before relativity and quantum mechanics), there was no known reason why these two types of mass should be the same. All the fundamental laws of physics would have been identical. And yet, they are the same. There's no such thing as an object which is really easy to hold up in the air, but really difficult to start or stop moving. And seeing as they weren't fools, the physicists knew this too.

The famous result, with the famous appocryphal experiment with the Leaning Tower of Pisa, is that if you drop any two objects, they will fall at the exact same speeds (neglecting air resistance). This is because the two types of mass are identical. If you drop the cannonball and ping-pong ball, then there will be a much bigger force on the cannonball due to gravity, but the cannonball is that much more difficult to get moving and it requires that bigger force to give it the same acceleration as the ping-pong ball.

(Actually, this is a convenient fiction. Just as the two balls are attracted by gravity to earth, so the earth is attracted by gravity to the balls. However, the force in question is so small compared to the huge mass of the earth, that that attraction is negligible, and certainly far far less than effects like air resistance. But if you managed to get a ball that had a mass close to the earth's mass, then you'd see it fall faster.)

The equation that determines the gravitational force between two objects is:

F = GMm/r2

F is (again) the force, M and m are the masses of the two objects, r (for radius) is the distance apart of the two objects, and G is the gravitational constant, which is a very small number in conventional units (6.67300 × 10-11 m3 kg-1 s-2).

Now, if we take M as the mass of the earth and m as the mass of the ball, and put this together with our Newton's second law equation, we get ma = GMm/r2 (since both sides are equal to F), then we can cancel the m and we have that a = a = GM/r2. If you put in the values of G, the mass of the earth and the radius of the earth, then you get the familiar 9.8 metres (32 feet) per second per second. In fact, I believe that's how they come up with figures for the mass of the earth in the first place.

(And similarly, if you want to work out how much the earth accelerates, it comes out as Gm/r2, where the m is the mass of the ball. If you put in the numbers, this turns out to be a very very tiny acceleration indeed (about 1.6 × 10-24 ms-2 for a 1kg ball. Now you can see why we don't tend to bother about this.)

What's even cooler is that Einstein thought, "OK, we have these two masses which should be different but are actually the same. What if that means that acceleration and gravity are actually the same thing as well?" It is not possible to devise any experiment that can tell the difference between a gravitational force and an acceleration. This idea is one of the basic principles behind his theory of general relativity. I'm not even going to try to explain that, because I certainly don't understand it. You might want to try the wikipedia articles on the equivalence principle and on general relativity if you're curious.

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This post rocks, thank you rho! It is so comprehensive that I don't have anything further to add, but I really enjoyed that explanation. I knew most of this already but I love the way you connect it all together into a story. <3

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