Friday's Physicsal Law - Gravity

Ok well yet again this is a catchup post of this series, it was due on Feb 30 March 2.

We all have a good daily experience with gravity, we can feel it pulling us down whatever we do. And our everyday experience on how it effects us is usually limited to the acceleration that we undergo or our weight

• F = mg

where g is the acceleration due to gravity and is approximately 9.8m s^-1. Now this is all well and good for describing the effects of gravity here at the surface of the Earth. But what it does not do is tell us anything about what gravity is or how we work out 'g' for other locations (ie surface of the Moon or Mars).

For our everyday experience, that is so well described by the above equation, we can derive myriad equations to describe the parabolic motion of projectiles, to determine time of flight, maximum height, distance traveled and velocity along the path but these equations are simply those that can be used for any acceleration (interestingly enough this equivalence between acceleration and gravity plays a role in the development of General Relativity too):

• d = v_it + ½at²
• v_f² = v_i² + 2ad

etc.

However to get the true experience of gravity we must leave this time and place and travel back to the time when Tycho Brahe was observing the motion of the planets. And since we are travelling back in time we might as well get our selves situated nicely above the plane of the solar system so we can see everything.

Johannes Kepler using Brahe's observations deduces three laws that govern the astronomical.

1. The orbit of every planet is an ellipse with the sun at one of the foci
2. A line joining a planet and the sun sweeps out equal areas during equal intervals of time
3. The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits

And it was upon these, in particular the Kepler's third law that Newton formulated his Law of Universal Gravity, basically by combining Kepler's law with his Laws of Motion.

Now to do this without resorting to inventing (or just using) calculus you and I will make a handwavey assumption (and one that isn't all that bad). The ellipse detailed in Kepler's first law are rather circular so to make the maths easier we will just use circles (note that a circle is a special case of an ellipse where the two foci are in the same place).

Now remember from last time that circular motion requires:

• a = v²/r

since the velocity around a circle depends on the circumference (2πr) and the period (T) (which probably should have been mentioned in the other post):

• v = 2πr/T

which gives us

• a ∝ r/T²

and combining this with Kepler's third law, which for a circle can be written:

• T² ∝ r³

then we get

• a ∝ 1/r²

So the acceleration and hence the force are inversely proportional to the square of the radius of the orbit (as the radius increases the force decreases). So this tells us how our weight (remember this is given by mg) varies as we change our position relative to the Earth, but what about on other planets?

Well if we were to go to the moon and weigh ourselves we would discover the scales read about 16% of what they did before we left Earth, since the Moon is smaller than the Earth then if nothing else was involved in gravity then our weight would go up, so something else must be involved and this turns out to be the mass of the object we are on (be it a planet or moon or whatever).

So this gives us Newton's Law of Universal Gravitation:

• F = G Mm/r²

where G is the gravitational constant and has a value of about 6.67×10^-11 N m² kg^-2 and M and m are the masses of the two objects (sometimes written m₁ and m₂).

Now many find it counter-intuitive that because of the M and m in the equation the force of gravity on me due to the Earth is the same as the force of gravity on the Earth due to me. Since clearly the Earth moves me and not the other way around. Of course we must remember that force is not the whole story, it is the acceleration that causes the motion and since the Earth ways more than me the movement of me is much more than that of the Earth.

This can be seen better in the case of binary stars, or Pluto and Charon, or any other objects that are orbiting a spot in between them. This consequence is really just an illustration of Newton's third law, equal and opposite action and reaction.