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# Proof of formula for centre of mass relative to point Q?

Prove the following formula for the position of the centre of mass relative to point Q for a system of i masses, where G represents the centre of mass and M represents the total mass.

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- nyphdinmdLv 74 weeks agoFavorite Answer
Let r0i = vector point from some origin of coordinates to mass i. Let r0Q be the vector from the origin of coordinates to a point Q. Then the position of a mass i to the point Q is given by the vector:

ri/Q = r0i - r0Q

Now the center of mass for the collection of n masses relative to the origin is given by:

r0CM = 1/(sum[i=1,n] mi) * sum([i=1,n] mi r0i)

To change the reference point to Q use ri/Q in the above expression

rG/Q = 1/(sum[i=1,n] mi) * sum([i=1,n] mi (r0i - r0Q)) = 1/(sum[i=1,n] mi) * sum([i=1,n] mi ri/Q)

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