# Moment of Inertia of a Rigid Body

## Important Formula

## What do the formula symbols mean?

## Moment of inertia

`$$ \class{brown}{I} $$`Unit

`$$ \mathrm{kg} \, \mathrm{m}^2 $$`

## Distance

`$$ r_{\perp} $$`Unit

`$$ \mathrm{m} $$`

## Volume

`$$ V $$`Unit

`$$ \mathrm{m}^3 $$`

## Mass Density

`$$ \rho(\boldsymbol{r}) $$`Unit

`$$ \frac{ \mathrm{kg} }{ \mathrm{m}^3 } $$`

We want to calculate the **moment of inertia** \( I \) of a rigid system that rotates around a fixed axis. You must, of course, define the axis of rotation. The moment of inertia counteracts the rotary motion and depends on the mass distribution of the system, which is distributed around the axis of rotation. Here we have to distinguish whether the system has a *discrete* mass distribution (e.g. planets orbiting the sun) or whether it has a *continuous* mass distribution (e.g. a rotating cylinder).

In the case of a **discrete mass distribution**, you have \(N\) masses that are distributed in space and have different distances \(r\) from the axis of rotation. To determine the **moment of inertia** \( I \) for this discrete rotating system, you need to know the individual **masses** \( \class{brown}{m_i } \) and their **distances** \( r_{\perp i} \) perpendicular to the axis of rotation:

~&=~ \underset{i~=~1}{\overset{N}{\boxed{+}}} \, \class{brown}{m_i} \, {r_{\perp i}}^2 \end{align} $$

In the case of a **continuous mass distribution**, you do not have individual masses, but the total mass is "smeared" over a certain area of space. A hollow cylinder, for example, has a continuous mass distribution in which its mass is smeared on its surface. To calculate the moment of inertia \( I \) of such a system, you have to replace discrete summation in 1

by an integral and the masses \(\class{brown}{m_i}\) by the **mass density** \(\rho(\boldsymbol{r})\). Then you integrate over the **volume** \(V\) of the rotating system:

The given mass density \(\rho(\boldsymbol{r})\) depends on the position vector \(\boldsymbol{r} \) in a three-dimensional system. The volume element \(\text{d}v\), in which the mass \(\rho(\boldsymbol{r}) \text{d}v\) is located, has the perpendicular distance \( r_{\perp} \) to the axis of rotation.

If the theoretical calculation using the integral is difficult (because the system has no symmetries), then the moment of inertia can be determined *experimentally*. For this purpose, the **total torque** \(M\) and the **angular acceleration** \(\alpha\) are measured and the moment of inertia \(I\) is calculated from their ratio: