Questions about Universal Gravitation

[Blood Junkie]

A couple of questions about Universal Gravitation..

When satellites are launched with velocities greater or lesser than the required launch velocity [Underroot(G*Mass of earth/Radius of Earth)], their orbits become elliptical with the Earth as the near or far focus. Why?

Also, 'The Earth and moon must rotate about their common centre of mass, rather than the moon about the centre of mass of the earth'. Why?

This is A Level physics, incase that is of any use.

 

[HallsofIvy]

Gosh that's hard to answer! What really made Isaac Newton's reputation (the reason why he is more famous that Leibniz, the other "founder" of calculus was that he was able to answer that question! The answer would still take several pages of calculation. Here, I can only say that one can show that, with an "inverse square" law force, like gravity, paths of motion MUST be "conic sections" (circle, ellipse, parabola, hyperbola- with straight line as a "degenerate conic") with the center of force at a focus. The circle (which is really a special ellipse) and ellipse are the only closed "orbits". By the way, if the velocity of a satellite is greater than "escape velocity" the motions is a parabola or hyperbola, not an orbit.

The reason it is, strictly speaking, the centre of mass rather than one of the two objects at the focus is that gravity is symmetric- I pull on you, you pull on me. Imagine two objects of equal mass: which would orbit around which?

 

[chroot]

Originally posted by HallsofIvy
By the way, if the velocity of a satellite is greater than "escape velocity" the motions is a parabola or hyperbola, not an orbit.

Not to quibble, as I know you know this, but a parabolic orbit is one with exactly zero energy (in other words, the object is moving at exactly escape velocity). Orbits with E > 0 are always hyperbolae.

- Warren

 

[HallsofIvy]

"Not to quibble"! Hah! You know you enjoy quibbling!

 

[Dr.Brain]

What my prof. taught me was that:::

There is a gravitational field surrounding the Earth (generally taken to be circular in numerical problems) .The launch velocity you are talking about is root of GM/R.This is the required velocity which will provide necessary cent. force for the satellite to revolve around the earth.

The grav. field around Earth can be considered to be a type of energy barrier which has to be overcome to escape into the space (better known as escape velocity).CASES:

If V is the vel. provided to a satellite
Ve is the escape velocity
Vo is the launch velocity

then Vo<V -------------
falls to earth-----like a projectile ...cannot overcome enegry barrier...can be known by type on conic section it makes..

when V=Vo

Moves around Earth in a circle

when V is almost greater than Vo

Approaches elliptical path while revolving around earth

when V is greater than escape velocity

Goes outta the field of earth...breaks the energy barrier and becaomes par/hyperbolic

^^^^^

This eventually shows that as u give the satellite more velocity...thus more KE ...and thus with more energy ..it has more ability of overcoming the energy barrier...which is indicated by increase in shapes of the path taken by the satellite on providing diff. velocities.



About yr second question:

Consider earth-moon sys ... as two ends of a rod rotating about their c.m.It is not possible for one end to rotate about the other end ..only possible when other end is stationary..
Moreover...two cellestial bodies if seen rotating ...rotate around their common centre of mass under each other's grav. influence.

This was what I read in One of my Books.
Dr.Brain

 

[devilmech]

The second question was answered pretty well by Dr. Brain, but I'd like to add one thing.

Two objects will rotate around their barycentric center of mass, which is the point at which the mass on one side of the point of rotation equals the mass on the opposite side of the point of rotation.

When applied to the Earth and the moon, the barycentric center of mass is somewhere in between the Earth's core and it's crust