Does Uniform Gravity Equate the Center of Mass with the Center of Gravity?

[Steve Cox]

Homework Statement

I wanted a proof that an object's center of gravity is the same as the center of mass by breaking the object into tiny pieces and then integrating over them.

Homework Equations

Well, the gravitational equation g=Gm1m2/r^2

The Attempt at a Solution

I tried using some calc three to integrate a uniform sphere using spherical shells. However, my answer wasn't working out and I would like a much more general proof.

Well, I just learned that the center of mass is different from the center of gravity of Earth because the gravitational field isn't uniform. Assuming gravity is uniform, how can we prove that the sum of all of the gravities are as gravity as a whole were acting on its center of mass?
coolbob13579@gmail.com

 

[haruspex]

the center of mass is different from the center of gravity of earth

The centre of mass of a body is different from its centre of gravity in a non-uniform field. I don't know why you wrote "of earth" at the end. If Earth is the body, is it the sun's field?

Assuming gravity is uniform, how can we prove that the sum of all of the gravities are as gravity as a whole were acting on its center of mass?

You would need to show that there is no net torque about the mass centre. It is not difficult.

 

[lychette]

²

Homework Statement

I wanted a proof that an object's center of gravity is the same as the center of mass by breaking the object into tiny pieces and then integrating over them.

Homework Equations

Well, the gravitational equation g=Gm1m2/r^2

The Attempt at a Solution

I tried using some calc three to integrate a uniform sphere using spherical shells. However, my answer wasn't working out and I would like a much more general proof.

Well, I just learned that the center of mass is different from the center of gravity of Earth because the gravitational field isn't uniform. Assuming gravity is uniform, how can we prove that the sum of all of the gravities are as gravity as a whole were acting on its center of mass?
coolbob13579@gmail.com

The 'gravitational equation is F = Gm1m2/r^2