Question

Find the moment of inertia of a pair of spheres, each having a mass mass m and radius r, kept in contact about the tangent passing through the point of contact.

Answer 1

It is given that two bodies of mass m and radius r are moving along a common tangent.



Moment of inertia of the first body about XY tangent, $I\text{'}=I_{com}+m{r}^2$
$\mathrm{So},I\text{'}=\frac{2}{5}m{r}^2+m{r}^2=\frac{7}{5}m{r}^2$
Moment of inertia of the second body about XY tangent,
$I\text{'}\text{'}=\frac{2}{5}m{r}^2+m{r}^2=\frac{7}{5}m{r}^2$
Therefore, net moment of inertia,
$$\begin{gathered} I=I\text{'}+I\text{'}\text{'} \\ I=\frac{7}{5}m{r}^2+\frac{7}{5}m{r}^2 \end{gathered}$$
$=\frac{14}{5}m{r}^2\mathrm{units}$