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Mathematically, the position of the center of mass

To find the center of mass of a rigid body we simply treat it as a continuous system of particles, and as such we can use the above equations (taken as integrals).

Note that the mass of the rigid body is

Calculating the location of the center of mass is important because it allows you to analyze dynamics problems based on the motion of the center of mass. For example, if a hammer is thrown in the air, its center of mass will follow a parabolic path. It's the same as if a particle were thrown in the air. Even though the hammer will likely rotate as it flies through the air, its center of mass will behave as a particle under the influence of gravity, so the equations for projectile motion will apply for the center of mass of the hammer.

In addition, if we have a situation where every particle within a body is subjected to the same force (such as gravity), the resultant force will act through the mass center. This is useful because it allows you to determine the moment arm of the (resultant) gravity force, depending on where you calculate your moments.

The above three equations tell us that a rigid body is "perfectly balanced" about its mass center

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