18.6 Moments of inertia of rigid bodies

In the module titled Rotation of rigid body , we derived expressions of moments of inertia (MI) for different object forms as :

$$\begin{array}{ll}\mathrm{1.\; For\; a\; particle\; :}& I=m{r}^2\end{array}$$

$$\begin{array}{ll}\mathrm{2.\; For\; a\; system\; of\; particles\; :}& I=\sum m_i{r_i}^2\end{array}$$

$$\begin{array}{ll}\mathrm{3.\; For\; a\; rigid\; body\; :}& I=\int r^2đm\end{array}$$

In this module, we shall evalaute MI of different regularly shaped rigid bodies.

Evaluation strategy

We evaluate right hand integral of the expression of moment of inertia for regularly shaped geometric bodies. The evaluation is basically an integration process, well suited to an axis of rotation for which mass distribution is symmetric. In other words, evaluation of the integral is easy in cases where mass of the body is evenly distributed about the axis. This axis of symmetry passes through "center of mass" of the regular body. Calculation of moment of inertia with respect to other axes is also possible, but then integration process becomes tedious.

There are two very useful theorems that enable us to calculate moment of inertia about certain other relevant axes as well. These theorems pertaining to calculation of moment of inertia with respect to other relevant axes are basically "short cuts" to avoid lengthy integration. We must, however, be aware that these theorems are valid for certain relevant axes only. If we are required to calculate moment of inertia about an axis which can not be addressed by these theorems, then we are left with no choice other than evaluating the integral or determining the same experimentally. As such, we limit ourselves in using integral method to cases, where moment of inertia is required to be calculated about the axis of symmetry.

In this module, we will discuss calculation of moment of inertia using basic integral method only, involving bodies having (i) regular geometric shape (ii) uniform mass distribution i.e uniform density and (iii) axis of rotation passing through center of mass (COM). Application of the theorems shall be discussed in a separate module titled " Theorems on moment of inertia ".

As far as integration method is concerned, it is always useful to have a well planned methodology to complete the evaluation. In general, we complete the integration in following steps :

1. Identify an infinitesimally small element of the body.
2. Identify applicable density type (linear, surface or volumetric). Calculate elemental mass "dm" in terms of appropriate density.
3. Write down the expression of moment of inertia (đI) for elemental mass.
4. Evaluate the integral of moment of inertia for an appropriate pair of limits and determine moment of inertia of the rigid body.

Identification of small element is crucial in the evaluation of the integral. We consider linear element in evaluating integral for a linear mass distribution as for a rod or a plate. On the other hand, we consider thin concentric ring as the element for a circular plate, because we can think circular plate being composed of infinite numbers of thin concentric rings. Similarly, we consider a spherical body, being composed of closely packed thin spherical shells.