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 First Mass Moment Center of Mass

First Mass Moment

The 1st mass moment of a particle is defined as the product of its mass and position vector (Figure 1):

Figure 1

[1]

where q = the first mass moment, m = the mass, and r = the position vector. The total mass moment of the system of particles shown in Figure 2 is then

Figure 2

[2]

where i = point number.

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Center of Mass

Center of mass is an imaginary point which suffices the following relationship:

[3]

where mtotal = the mass of the system, rCM = the position vector of the CM. In other words, the CM is a single point which effectively represents the whole system. From [3]:

[4]

Based on Figure 3, [2] can be rewritten as

Figure 3

[5]

The second term of [5] is the total 1st mass moment of the system with respect to its CM. From [3] and [5]:

[6]

In other words, the sum of the 1st mass moments of the particles with respect to their CM always becomes zero.

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