Laws of Rotationary Motion?

[perm]

Having just started revising the Advanced Higher Physics course, I came upon the section on Rotationary Motion, and found myself a bit confused!

The derivations that we had been taught all involved using bodies travelling a perfectly circular path, however these same laws can apparently be applied to bodies travelling eliptical paths, such as the planetary orbits.

Why is this? I was under the impression that circles and elipses could not be interchanged with each other...am I just wrong?

[CT]

Not quite. A circle is just a perfect ellipse. In a planetary system Kepler's Laws apply. Law1-All planets move in ellpitical orbit with the sun at one focus. The forces acting on the systems will vary with time, however Kepler's laws will not. Sorry if this isn't too straight forward. Look here for some info on ellipses http://www.wordiq.com/definition/Eccentricity_%28mathematics%29

[CT]

Also use wordiq to find other information you're not too clear about. It has quite good science information.

[perm]

So...are the basic laws of rotation (like omega = omega (0) + alpha.time) only approximations of planetary motion? They all seem to rely on properties of circles, like C = 2.pi.r, but this can't be applied to elipses can it?

But basically what you're saying is, Kepler's Laws still hold for elipses? (that's the one when the period of the motion (squared) is directly proportional to the radius (cubed)?)

Sorry to drag this painful question on even more...but I've always wanted to know...why don't the planets just spiral into the sun, rather than orbit around it? My puny little mind has always just thought of it being like a satellite in orbit around earth, like "freefall", but this doesn't seem right...is it something to do with Newton's third law? *weep*

[CT]

Ahh, the freefall concept of satellites. That's not easy to get your head around to begin with. Yes, planetary motion is similar to this. They don't spiral inwards due to the lack of resisting forces on the planets. Obviously there are some forces acting, but they are fairly tiny. Imagine swinging an object on a string round your head, but without air resistance and the movement of your arm. p.s. I suppose that this isn't a perfect circle either, as you have to move your hand to keep the object moving.

[Peter Pedant]

If they go in a circle it is at constant speed, (not velocity, that is a vector and the direction is always changing) and in an ellipse the speed varies - quickest near the focus and slowest furthest away.

so no, w(o) and alpha.time does not apply to ellipses - because the angular velocity is varying.

Newton was the first to realise that both situations could be explained by an inverse square law.

[Peter Pedant]

and in terms of the free fall of satellites,

Newton was first there too as well! He saw the apple fall and thought it would fall from the top of the house and then the top of the hill, and the highest mountain, and so why would not the moon fall in? he realised that this was because that it was in a circular motion. And so althgouht gravity was acting on the moon as well as the apple, gravity was acting to keep the moon in a circular orbit. And he thought his while he was down from cambridge in 1665 to escap e the plague and was at woolsthorpe manor in lincolnshire. He only wrote it up in 1685.