Tonight's Notes

Galileo Galilei

1609, telescope

·    Moon has craters

·    Way more stars than thought

·    Jupiter has 4 moons – Io, Europa, Ganymede, Callisto…. Jupiter actually has over 60 moons.

·    Saturn has rings!

·    Venus goes through phases (like the Moon)

·    Sun has spots!  (Sunspots)

·    Siderius Nuncius (Starry Messenger)

http://galileo.rice.edu/sci/observations/sunspot_drawings.html

See also Galileo’s book:  Dialogue on Two World Systems (1632)

Newton, 1642-1727

·    Laws of motion – inertia, F=ma, action/reaction

·    Calculus

·    Binomial theorem

·    Alchemy (oops)

·    Rules of optics

·    White light is made of colors (prism)

·    Reflecting telescope (used a mirror)

·    Explained tides

·    Universal gravity

·    Rules of reasoning in philosophy

Principia Mathematica Naturalis Philosophae, 1687

F_g = G m₁ m₂ / d²

Newton’s law of universal gravitation

m₁ = first mass

m₂ = second mass

G = universal constant of gravitation (a very tiny number, 6.67 x 10^-11)

d = distance between masses

This is an INVERSE SQUARE law – meaning that as the distance increases, the force gets smaller at a rate of 1 over the distance squared.  For example, if you double the distance, the new force is ¼ the original.  Triple the distance and the new force is 1/9 the original.

Local gravitation (g)

On Earth, near the surface:

g is approximately 9.8 m/s/s (or around 10 m/s/s).  This means that a freely-falling object increases its speed by roughly 10 m/s with every second of freefall.  Or conversely, if a body is projected upward, it loses roughly 10 m/s upward with each second of travel up.

This looks like this for a falling object:

After 1 second, its speed is 10 m/s.

After 2 seconds, its speed is 20 m/s.

After 3 seconds, 30 m/s

Or if you like equations:

v = g t

The distance that an object falls is a bit trickier to follow.  I will skip the derivation, but it is given by this formula:

d = ½ g t²

Or, d = 5t² , near the surface of the Earth.

So, after 1 second, an object falls 5 m.

After 2 seconds, d = 20 m.

After 3 seconds, d = 45 m.

After 4 seconds, d = 80 m.

Notice how the distance is climbing up exponentially.  If you graphed distance versus time, you’d get a parabola.

Now for Galileo’s odd number’s rule (just for mathematical fun) – see the earlier blog entry.

With increased altitude, g becomes progressively (but slowly) weaker.

On the Moon, local gravity (at the Moon’s surface) is approximately 1/6 that of Earth.

On Venus, it’s around 9/10 that of Earth.

On Jupiter, it’s around 2.5 times that of Earth.

Local gravity (g) can be calculated with this expression:

g = G M_planet / r²

Where G is the same constant as before, M_planet is the mass of your planet and r is the radius of your planet.

So, do all bodies experience the same local gravity?  Why?

FYI: 

http://en.wikipedia.org/wiki/Isaac_Newton

http://en.wikipedia.org/wiki/Galileo

http://en.wikipedia.org/wiki/Dialogue_Concerning_the_Two_Chief_World_Systems