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:

$I=mr^{2}$

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:

$I=\sum _{i=1}^{n}m_{i}r_{i}^{2}$

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:

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

or in terms of density as:

$I=\iiint _{v}\rho (x,y,z)\,r^{2}\,dV$

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

Parallel Axis Theorem

The parallel axis theorem states that the moment of inertia of a body is smallest when the axis of rotation passes through the centre of mass of that body. Mathematically, it can be stated as:

$I=I_{cm}+Mh^{2}$

where I_cm is the moment of inertia for an axis through the centre of mass, M is the mass of the body and h is the perpendicular distance from the centre of mass to an axis parallel to the one through the centre of mass. This theorem is useful as once one moment of inertia is calculated, any other moment of inertia can be calculated easily (so long as the axes are parallel).

Perpendicular Axis Theorem

The perpendicular axis theorem states that the moment of inertia around an axis at right angles to the plane of a planar body, I_z is equal to the sum of any two moments of inertia in the plane of the body so long as these two moments of inertia are perpendicular to each other. This can be written as:

$I_{z}=I_{x}+I_{y}$

where I_x and I_y are in the plane of the object. This theorem is particularly useful as means three dimensional objects can be broken down into a sum of two dimensional objects.

Examples

Discrete example

Suppose a massless string of length L has two particles, both of mass m, attached to it. One is attached halfway along the string, the other at the end (opposite end from the pivot). The string is now swung around so that it is taut. As there are a finite number of discrete masses, we must use the sum formula:

$I=\sum m_{i}r_{i}^{2}=m\left({\frac {L}{2}}\right)^{2}+mL^{2}={\frac {5mL^{2}}{4}}$

Continious Example

Suppose a rod of mass m and length L is rotating about one end. As the rod is continuous, we use the integration form. Therefore in terms of differentials, dI=r²dm. Defining the density of the rod as λ=m/L, dI can be written as dI=r²λdr. Integrating gives moment of inertia:

$I=\int dI=\int _{0}^{L}\lambda r^{2}\,dr$

This is integrated between 0 and L as the rod pivots about r=0 and has length L. If it pivoted about the middle, the boundaries would be -L/2 and +L/2. Performing the integration and substituting back in for λ:

$I={\frac {m}{3L}}\left(L^{3}-0\right)={\frac {mL^{2}}{3}}$

See also

• Rotational mechanics