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Moment of Inertia

<aside> 📕 The moment of inertia of an object is a measure of its resistance to angular acceleration about a given axis.

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In linear dynamics, an unbalanced force produces a linear acceleration. The size of the linear acceleration produced by a given linear force will depend on the mass (or inertia) of the object.

The word inertia can be loosely described as resistance to change in motion of an object. (Objects with large mass are difficult to start moving, and once moving are difficult to stop).

The moment of inertia ($I$) of an object can be described as its resistance to change in its angular motion.

The moment of inertia of an object depends on:

• Mass: The greater mass the mass, the greater the moment of inertia.
• Distribution of mass about a given axis of rotation: The further away the mass is from the axis of ration, the higher the moment of inertia.

Point Mass

The moment of inertia for a point mass, or for objects at an equal distance ($r$), is given by:

$$ I = mr^2 $$

where:

• $I$ is the moment of inertia of the mass, in $kg\,m^2$.
• $m$ is the mass, in $kg$.
• $r$ is the radius, in $m$.

<aside> 📏 Units of moment of inertia: $kg\,m^2$

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Discrete Masses

Most objects do not have all their mass at a single distance ($r$) and hence the relationship $I = mr^2$ cannot be used. Consideration must be given to the distribution of the mass.

Consider a rigid body made up of many particles at different distances from the axis of ration. The moment of inertia for of the body is the sum of all of the individual particles (point masses) that make up the body:

$$ I = m_{1}r_{1}^2 + m_{2}r_{2}^2 + ... m_{n}r_{n}^2 $$

$$ \therefore I = \Sigma mr^2 $$

where:

• $I$ is the moment of inertia of the mass, in $kg\,m^2$.
• $m$ is the mass, in $kg$.
• $r$ is the radius, in $m$.