If the strength of gravity is determined by the size of an object,  how come Earth's gravity is overcome by a small magnet?

for example: if you throw a piece of metal, it will fall to the ground but if you hold a magnet over the piece of metal it will not fall.

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Just a bit of clarification. It is not the size of an object that determines its gravity; the object's mass determines the force of its gravity.

Presently it is believed that there are four fundamental forces in nature. Here they are in order of strongest to weakest:

The strong interaction (binds particle in the atom)

The electromagnetic force (electrical attraction and magnets)

The weak force (radioactive decay)

The gravitational force (gravity)

As you see, gravity is the weakest of all the fundamental forces. It seems so great because it is easiest for us to perceive, it acts at great distances, and it works with all masses. Nonetheless, a magnet is, by nature, much stronger than gravity.

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You are really talking about apples and oranges here.  You are implying that only something bigger than the Earth could overcome the Earth's gravity.  Obviously, this is not true because I can overcome the Earth's gravity by holding up the clothes I am wearing for an indefinite time.

Gravity is determined by the size of the objects and the distance between their cores, you are right.  But when the magnet picks up the piece of metal, it is not using gravity any more than I am using gravity to hold my shirt up off the floor.

The gravitational attraction of the magnet on the metal would be tiny.  But it is not gravity that is attracting the metal and overcoming the Earth's gravity.

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