Gravity Equation and Immersed Objects: Validity and Proofs

[AStaunton]

the equation for grav accel. is:

g = GM/r^2 (1)

where M is the total mass of the body in question and r is the distance from the C.O.M.

Is this equation still valid for an object that is immersed in the body?
For example if talking about a particle midway between the sun's surface and it's center of mass is g still described by the above equation.
Intuitively it feels like this equation would not longer hold as now there is a certain amount of mass on the other side of the object and this will exert a force in the opposite direction and so the actual value for g will be less than expected from the above equation..
However, I think there is an equation (that is most commonly associated with electromagnetism - possible Gauss' equation...or one of the others!) that proves that the above equation - equation (1) holds...

 

[tiny-tim]

Hi AStaunton!

(try using the X² icon just above the Reply box )

It's the same formula, except that M is now only the mass inside the sphere of radius r (for essentially the same reason as applies to an electric charge).

 

[Doc Al]

the equation for grav accel. is:

g = GM/r^2 (1)

where M is the total mass of the body in question and r is the distance from the C.O.M.

That's only true when r ≥ radius of the body and only under the assumption of spherical symmetry.

Is this equation still valid for an object that is immersed in the body?
For example if talking about a particle midway between the sun's surface and it's center of mass is g still described by the above equation.

Only if you reinterpret it like so: For a radius r inside the body, the value of g is given by that equation if M stands for the portion of the mass within a distance r of the center. (Again assuming spherical symmetry.)

Intuitively it feels like this equation would not longer hold as now there is a certain amount of mass on the other side of the object and this will exert a force in the opposite direction and so the actual value for g will be less than expected from the above equation..

Your intuition is correct.

(Tiny-tim snuck in there while I was dozing...)

 

[yogi]

"Intuitively it feels like this equation would not longer hold as now there is a certain amount of mass on the other side of the object and this will exert a force in the opposite direction and so the actual value for g will be less than expected from the above equation.."

"Your intuition is correct."

For a non-accelerating mass surrounded by a uniform density spherical shell of matter, the internal force from classical physics is net zero. The G field from All matter outside the radius r cancels

 

[Doc Al]

For a non-accelerating mass surrounded by a uniform density spherical shell of matter, the internal force from classical physics is net zero. The G field from All matter outside the radius r cancels

Good point. I should have more careful. I was responding to the conclusion, not the entire statement:

"Intuitively it feels like this equation would not longer hold as now there is a certain amount of mass on the other side of the object [STRIKE]and this will exert a force in the opposite direction[/STRIKE] and so the actual value for g will be less than expected from the above equation.."

 

[A_B]

the equation for grav accel. is:

g = GM/r^2 (1)

where M is the total mass of the body in question and r is the distance from the C.O.M.

This is not entirely correct, the formula is valid for point masses only, to calculate the gravitational field due to a body, you must integrate over the volume of that body. The result will not generally be in the direction of the center of mass of the body. This can easily be seen for a bowl for example, clearly the gravitational pull does not get infinity strong at its center of mass.