Resource Lesson Discrete Masses: Center of Mass and Moment of Inertia
The center of mass is defined as the point where all of the mass of an object can be considered to be concentrated. When the gravitational field is uniform over the body, the center of mass and the center of gravity are coincident. For the purpose of calculating torques, the center of gravity is operationally defined as the point where all of the weight of an object can be considered to be concentrated.

The center of mass can be calculated for a collection of "point masses" or discrete masses using the following formulas.

Let's practice.

 For the collection of masses shown below, where is the center of mass?

 For the collection of masses shown below, where is the center of mass?

 For the collection of masses shown below, where is the center of mass?

Moment of Inertia

The equation used to calculate the moment of inertia of a collection of discrete masses about an arbitrary axis of rotation is

where m is the mass of each piece and r is its moment arm, or perpendicular distance, from the line of action of the axis of rotation. Recall that the moment of inertia is a quantity that determines the collection's rotational inertia, or resistance to a change in its state of rotation.

 If the mass of each connecting rod is negligible, what is the moment of inertia about an axis perpendicular to the paper and passing through the central, 2M, mass? Treat the masses as point masses.

 If the mass of each connecting rod is negligible, what is the moment of inertia about an axis perpendicular to the paper and through the center of mass? Treat the masses as point masses.