The mass of an object is an intrinsic property of that object.
- The inertial mass is the object’s resistance to being accelerated by a force.
- The active gravitational mass is the strength of the gravitational field generated by an object.
- The passive gravitational mass measure the gravitational force exerted on an object in a known gravitational field.

Center of Mass

Consider a system of points, where each point has mass , located with coordinates . The center of mass is the point on space such that
If the distribution of mass is continuous with density function within solid where , then the center of mass is given by

Where denotes an iterated integral over the solid .

In three dimensions this gives
The center of mass is the average position weighted by the mass at each point.

Moment of Inertia

Consider a system of points where each point has mass and distance from the origin point . The moment of inertia is defined as

If the distribution of mass is continuous with density function within solid where , then the center of mass is given by

Where denotes an iterated integral over the solid .

In three dimensions this gives
It is an analogue for center of mass but for rotations.
The Parallel Axis Theorem states that the moment of inertia of a body about an axis through its center of mass and the moment of inertia about any other axis parallel to the original axis, denoted is related by

Where is the distance between the two parallel axes.
- Intuition: We can recalculate the moment of inertia using an integration over slices about the axis through point . Recalculating and regrouping terms gives us in terms of
- This allows us to reframe rotation problems with an axis of rotation passing through the moment of inertia.

At high speeds, the mass changes with the velocity. Let be the mass at rest. When the mass is travelling at velocity , we have