How Do You Calculate Moment of Inertia for a System of Point Masses?

For a system of point masses, calculate the moment of inertia by multiplying each mass by the square of its perpendicular distance from the chosen axis, then adding the results:

$$I=\sum_i m_i r_i^2$$

This is Additional Higher Level (AHL) content from A.4 Rigid body mechanics.

Before calculating, identify the rotation axis clearly. For every mass, draw or imagine the shortest line from the axis to the mass; that line's length is $r_i$. If coordinates are given, determine this perpendicular distance geometrically before substituting. Distances must be converted to metres.

The Reasoning

The moment of inertia measures how strongly a system resists angular acceleration about a particular axis. For each point mass, its contribution is

$$I_i=m_i r_i^2$$

where $m_i$ is the mass and $r_i$ is the shortest perpendicular distance from that mass to the axis of rotation. The total moment of inertia is the sum of all contributions.

The distance is squared, so moving a mass farther from the axis produces a large increase in moment of inertia. A mass located directly on the axis has $r=0$, so it contributes zero.

For example, doubling $r$ while keeping $m$ constant makes that mass's contribution four times larger. This is why placing material farther from an axis increases rotational inertia even when total mass is unchanged. Conversely, moving the same mass inward reduces its contribution very rapidly.

Worked Example

Three point masses rotate about a fixed axis:

Therefore,

$$I=(2.0)(0.30)^2+(1.5)(0.40)^2+(0.50)(0.20)^2$$ $$I=0.440\,\text{kg m}^2$$

A common misconception is that moment of inertia depends only on total mass. It depends on both the masses and their distribution relative to the specified axis. The same system can therefore have different moments of inertia about different axes.

Exam Technique

For an IB Calculate question, state $I=\sum m_i r_i^2$, show every substitution, and give the answer in $\text{kg m}^2$. Check that each is measured from the axis, not from another mass or from an arbitrary origin.