Moment of inertia

The moment of inertia is a measure of a mass and its distance perpendicular to an axis of rotation. It describes an objects resistance to change in rotation.

The formula to calculate a moment of inertia between a point and a point mass is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = m r^2}

where r is the displacement between the two points, m is the mass of the point mass and I is the moment of inertia.

If you have a finite set of point mass's and want to find their moment of inertia around a point you simply add the moment of inertia of each object to get the following:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = \sum_{n=1}^{n} m_{n} r_{n}^2}

It is important to note that you can not simply take the centre of mass of an object and use it as a point mass. Instead you need to use integration:

Mathematically, the moment of inertia I for a mass m that is moving a perpendicular distance of r from an axis is expressed as:

$I=\int r^{2}\,dm$

The formula for a moment of inertia is the result of combining the equation for kinetic energy with the equation for angular velocity.