Need help understanding this concept on field intensity

[aeromat]

Homework Statement

We are looking at gravitational field intensity in particular here, and using the Earth as an example. The question states: Calculate the gravitational field intensity at a height of 300.0 km from Earth's surface.

Why is it that: Since the point in question is outside of the sphere of Earth, the gravitational field there is the same as it would be if Earth's mass was concentrated at a point in Earth's centre. Therefore the equation for the gravitational field intensity near a point mass applies?

I don't understand how you could have the same gravitational field concentrated at a point, and it being the same even if you are 300km FAR FROM EARTH? Is this implying that the intensity is the same, because that doesn't make sense to me.

 

[ideasrule]

It's not the gravitational field that's concentrated at a point; it's the mass. A perfect sphere behaves gravitationally as if all of its mass were concentrated at its center, but only for points outside the sphere. As for why, you can use calculus to verify the theorem by calculating the gravitational field due to one small mass of volume dV, then integrating over the volume of the sphere.