Two masses $M_1$ and $M_2$ are separated by a distance r. Find the moment of inertia of this arrangement about an axis passing through the centre of mass and perpendicular to the line joining them.
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If COM is origin, $M_1r_1 = M_2r_2$
Also, $r_1 + r_2 = r$
Using the equations, $\text{r}_1=\frac{\text{M}_2\text{r}}{\text{M}_1+\text{M}_2}\text{ and }\text{r}_2=\frac{\text{M}_1\text{r}}{\text{M}_1+\text{M}_2}$
$\text{I}_{\text{cm}}=\text{M}_1\text{r}^2_1+\text{M}_2\text{r}^2_2$
$=\frac{1}{(\text{M}_1+\text{M}_2)}[\text{M}_1\text{ M}^2_2\text{ r}^2+\text{M}_2\text{ M}^2_1\text{ r}^2]$
$=\frac{\text{M}_1\text{ M}_2\text{ r}^2[\text{M}_2+\text{M}_1]}{(\text{M}_1+\text{M}_2)}$
$=\frac{\text{M}_1\text{M}_2\text{r}^2}{(\text{M}_1+\text{M}_2)}$
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