Donut Shaped Planet

[circa 1991]

Question:  What would the gravity be like on a donut shaped planet and could 
such a planet exist?
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That is an interesting question.  The problem can be done the 
hard way, by integrating the gravitational potential for such a system...  but 
a rough answer can be obtained just by looking at the way gravity works.  If 
you were standing on the planet, then each little piece of the planet is 
putting a little bit of force on you, attracting you towards it.  If you were 
standing on the outside, with the center of the donut hole exactly beneath 
your feet, then you would feel a strong gravitational force straight down, 
because all the parts of the donut are on the same side of you, and the 
average direction is straight down.  If you were standing on the inside, with 
your head pointing to the donut hole center, then you would feel a somewhat 
weaker gravitational force keeping your feet on the donut.  The near parts of 
the donut are exerting the stronger force, and so keeping you attached, but 
the other side of the donut is above your head, and is exerting a weak force 
in that direction so that the force keeping your feet down is partially 
counteracted.  If you stand at any other point on the rim, then the force on 
you is not quite vertical - the net force does not point through your feet, 
but there will be a small amount due to the other side of the donut pulling 
you towards it.  Such a planet certainly could exist.  However, if it were the 
size of the earth it would quickly collapse into something like a sphere, due 
to the gravitational action of one side of the donut on the other -for a 
really big mass (like the earth) the force is so strong that no material would 
be strong enough to oppose it.

A. Smith
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