When does the earth act like a point mass?

Graphic - earthaspointmass
Treating the earth as a point mass seems to be a rather doubtful idealisation!. But there are two reasons why it is (or may be) legitimate to do so.
  1. As we move further away from the earth it begins to ‘look’ more and more like a ‘dot’ (the photographs of the earth, from a satellite and from the moon, make the point visually!). This suggests (and it is true) that the earth acts like a point mass if it is far enough away. You might like to think what ‘far enough’ means.
  2. Even if we are close to (standing on!) the earth it acts like a point mass to the extent that it is spherical 

    To show this we have to add up the forces -felt by a ‘test’ mass- due to all the ‘bits (point masses) that make up the earth.

    The upward pull of a bit like A cancels the downward bit of a piece like B.

    And though C pulls harder than if it were at the centre this is offest by the fact that D pulls less. The net result is the same as if we had a single point mass, at the centre.

    Graphic - No title - sphericalearth