Moment of inertia
Moment of inertia quantifies the rotational inertia of an object, i.e. its inertia with respect to rotational motion, in a manner somewhat analogous to how mass quantifies the inertia of an object with respect to translational motion.
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 moment of inertia in place of the mass 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 (capital i).
It should not be confused with the second moment of area or area moment of inertia, which is a property of a shape that is used to predict its resistance to bending and deflection. Most structural engineers, however, do refer to the second moment of area or area moment of inertia as simply the moment of inertia.
Unit
The SI unit for moment of inertia is kilogram metre squared (kg m2)
Derivation for point mass
Say we wanted to develop a formula for rotational kinetic energy that is analogous to the linear kinetic energy formula for a point mass.
An object with mass <math>m</math> that is moving with velocity <math>v</math> has a kinetic energy (KE) given as:
- <math>KE = \frac{1}{2} m v^2</math>
If an object is rotating at an angular velocity <math>\omega</math> in a circle of radius <math>r</math>, its linear velocity, v is equal to <math>\omega r</math>. Substituting <math>v = \omega r</math> into the kinetic energy equation above, we have:
- <math>KE = \frac{1}{2} m r^2 \omega^2</math>
Drawing parallels to the linear kinetic energy equation, we see that, for a rotating point mass, we substitute angular velocity, <math>\omega</math> for linear velocity, <math>v</math>. That leaves us with the quantity <math>m r^2</math>, which takes the place of the object's mass in the linear kinetic energy equation. We give this quantity, a kind of "rotational mass", a name: moment of inertia.
Mathematical definition
For a small (pointlike) mass m located at distance r from axis of rotation the moment of inertia (versus this axis) is defined by:
- <math>I = mr^2\,</math>
For system of small masses moment of inertia is defined as the sum of moments of all parts:
- <math>I = \sum_i m_i r_i^2</math>
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 <math>dm \,\!</math> over all three-dimensional space involved:
- <math>I = \int r^2\,dm \,\!</math>
<math>dm \,\!</math> is defined by the spatial density distribution <math>\rho \,\!</math>.
- <math>dm=\rho\,dV \,\!</math>
Types of moment of inertia
There are an infinite number of moments of inertia for any object, one for every possible axis of rotation through the object's centroid. For convenience, the three moments of inertia typically used for an object are about axes parallel to the three Cartesian axes (X, Y, and Z):
- <math>I = \;</math> moment of inertia about the current axis of rotation
- <math>I_{xx} = \;</math> moment of inertia about the line through the centroid, parallel to the X-axis
- <math>I_{yy} = \;</math> moment of inertia about the line through the centroid, parallel to the Y-axis
- <math>I_{zz} = \;</math> moment of inertia about the line through the centroid, parallel to the Z-axis
If the origin of the axis system is positioned at the object's centroid, we can simplify the notation further:
- <math>I_{x} = \;</math> moment of inertia about the X-axis
- <math>I_{y} = \;</math> moment of inertia about the Y-axis
- <math>I_{z} = \;</math> moment of inertia about the Z-axis
Application of moment of inertia
A common equation which describes the relationship between the linear force applied to an object, the object's mass, and the object's linear acceleration, in a frictionless setting, is:
- <math>F = ma\,</math>
A similar equation can be used to describes the relationship between the rotational force (torque) applied to an object, the object's rotational mass (moment of inertia), and the object's rotational (angular) acceleration, in a frictionless setting:
- <math>T = I{\alpha}\,</math>
Where:
- <math>T = \,</math> torque
- <math>I = \,</math> moment of inertia
- <math>{\alpha} = \,</math> rotational (angular) acceleration
Inertia tensor
The moment of inertia can be used to describe the amount of angular momentum a rigid body possesses, via the relation:
- <math>\vec{L} = I \vec{\omega}\,</math>
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:
- <math>I = \sum_i m_i r_i^2 (E-P_i)</math>
where
- E is the identity matrix
- P is the projection operator.
Alternatively the elements of the inertia tensor can be expressed as:
- <math>I_{xx} = \sum_i m_i (y_i^2+z_i^2)</math>
- <math>I_{yy} = \sum_i m_i (x_i^2+z_i^2)</math>\
- <math>I_{zz} = \sum_i m_i (x_i^2+y_i^2)</math>
- <math>I_{xy} = I_{yx} = -\sum_i m_i x_i y_i\;</math>
- <math>I_{xz} = I_{zx} = -\sum_i m_i x_i z_i\;</math>
- <math>I_{yz} = I_{zy} = -\sum_i m_i y_i z_i\;</math>
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:
- <math>I\vec{\omega}=\lambda \vec{\omega}.\,</math>
In the case of rotation about a principal axis with constant angular velocity, the angular momentum is, like the angular velocity, along this axis, so it remains constant. Thus no torque is required to maintain this rotation.