Planet gravitation represented as a single point

[colin9876]

Homework Statement

planet with mass m and uniform density and radius r, show by integration that the force this body generates on say a satelite is equivalent to a point mass m at the center of the planet

Homework Equations

i know that gravitational pull is inversley to r*r

The Attempt at a Solution

I tried breaking the sphere down into horizontal discs, then each disc into rings. in each ring the force on the satelite above it is mMsin(*)/d*d where * is the angle between the satelite and mass round the ring.
I tried integrating all the rings into a disc, then all the discs into the whole planet but couldn't do it??

There must be an easier way to show a planet can be represented by a point mass?

 

[Vanadium 50]

Your problem has spherical symmetry. When integrating, why are you disregarding this symmetry?

 

[colin9876]

ok, symmetry - so I could simplify it to be say half a sphere, or quater of a sphere but its still impossible to integrate?

 

[tiny-tim]

spherical symmetry

ok, symmetry - so I could simplify it to be say half a sphere, or quater of a sphere but its still impossible to integrate?

Nooo … whole spheres!

 

[colin9876]

can u explain a bit more please because vague comments are not much help!

 

[tiny-tim]

… oops!

Sorry … misread the question … ignore my last post.

Integrate over spherical caps of thickness dr, where r is the distance to the satellite …

in other words, every sphere of radius r intersects the planet in a "cap" whose angle, and therefore area, you can calculate.

 

[Vanadium 50]

can u explain a bit more please because vague comments are not much help!

Sure. I could write down the whole answer. Thing is, I already passed this class, and that wouldn't help me. It also wouldn't help you to simply write it down. Maybe someone else will do your work for you, but I won't be party to it.

What shapes have the same spherical symmetry as your problem? A hemisphere does not.

 

[colin9876]

ok i get the idea of integrating hollow sphere caps from 0 to r but its it seems very difficult to calculate what force each spear cap will produce as the mass points are ar different lengths from the satelite, and the force lines are at different angles?
If I could calc what a hollow sphere has as its combined gravity I could integrate them?

 

[Vanadium 50]

Colin, there's a theorem you can use. It's almost certainly covered in the same chapter as the problem. (As this problem is a classic example of the theorem).

 

[phagist_]

Wikipedia has a full explanation... but attempt it yourself first!
http://en.wikipedia.org/wiki/Shell_theorem

 

[colin9876]

Thanks!
I did have a go from first principles integrating from 0 to pi, rings of width d(theta) but it got complex so the Wikipedia post was very helpful.

In a way its quite amazing all the different forces on a sphere add up so niceley to be the same as a single point!