# Moment of inertia

Moment of inertia quantifies the resistance of a physical object to angular acceleration. Moment of inertia is to rotational motion as mass is to linear motion.

It should not be confused with the second moment of inertia, also known as the second moment of area and area moment of inertia, which is a property of a shape that is used to predict its resistance to bending.

In general, an object's moment of inertia depends on its shape and the distribution of mass within that shape: the greater the concentration of material away from the object's centroid, the larger the moment of inertia. It also varies depending upon the axis of rotation specified; values relative to the object's centroid are typically taken as baseline values. See the list of moments of inertia for specific examples. The parallel axes theorem can be used to determine moments of inertia relative to displaced axes of rotation.

Rotational versions of Newton's second law and the formulas for momentum and kinetic energy, use the moment of inertia of an object (with torque, angular velocity and angular acceleration replacing force, velocity and acceleration, respectively).

Moment of inertia is often represented by the letter I.

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## Unit

The SI unit for moment of inertia is kilogram metre squared (kg m2)

## Mathematical derivation

A rigid body can be considered an infinite number of infinitely small particles, each with mass [itex]m_i[itex]. If each particle is a distance [itex]r_i[itex] from a particular axis of rotation, then the moment of inertia of the rigid body about that axis is given by:

[itex]I = \sum_i m_i r_i^2[itex]

Continuous mass distributions require an infinite sum over all the point mass moments which make up the whole. This is accomplished by integrating all the masses [itex]dm \,\![itex] over all three-dimensional space involved:

[itex]I = \int r^2\,dm \,\![itex]

[itex]dm \,\![itex] is defined by the spatial density distribution [itex]\rho \,\![itex].

[itex]dm=\rho\,dV \,\![itex]

## Inertia tensor

The moment of inertia can be used to describe the amount of angular momentum a rigid body possesses, via the relation:

[itex]\vec{L} = I \vec{\omega}.\,[itex]

For the case where the angular momentum is parallel to the angular velocity, the moment of inertia is simply a scalar.

However, in the general case of an object being rotated about an arbitrary axis, the moment of inertia becomes a tensor, such that the angular momentum need not be parallel to the angular velocity. The definition of the moment of inertia tensor is very similar to that above, except that it is now expressed as a matrix:

[itex]I = \sum_i m_i r_i^2 (E-P_i)[itex]

where

E is the identity matrix
P is the projection operator.

Alternatively the elements of the inertia tensor can be expressed as:

[itex]I_{xx} = \sum_i m_i (y_i^2+z_i^2),\; I_{yy} = \sum_i m_i (x_i^2+z_i^2),\; I_{zz} = \sum_i m_i (x_i^2+y_i^2)[itex]
[itex]I_{xy} = I_{yx} = \sum_i m_i x_i y_i[itex]
[itex]I_{xz} = I_{zx} = \sum_i m_i x_i z_i[itex]
[itex]I_{yz} = I_{zy} = \sum_i m_i y_i z_i[itex]

It is notable that (because it is symmetric), it is always possible to diagonalize the inertia tensor to find the principal axes of the rigid body, those which satisfy the eigenvalue problem:

[itex]I\vec{\omega}=\lambda \vec{\omega}.\,[itex]

In the case of the principal axes, the angular momentum is parallel to the angular velocity, so the object can be rotated in free space without an external torque applied.

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