Question

Three point masses $m_1,$ $m_2$ and $m_3$ are located at the vertices of an equilateral triangle of length $a$. Determine the moment of inertia of the system anout an axis along the altitude of the triangle passing through $m_1$.

Answer 1

since, it can be seen in the figure given below that the area of totation passes through $m_1$. The distance of $m_1,m_2$ and $m_3$ from the axis of rotation are $O,a/2,a/2$ respectively.
Therefore, moment of inertia of the system about the axis throuh $m_1$ is
$I=m_1{r}_1^2+m_2{r}_2^2+m_3{r}_3^2=m_1(0{)}^2+m_2{\left(\frac{a}{2}\right)}^2+m_3{\left(\frac{a}{2}\right)}^2$
$\Rightarrow I=\frac{a^2}{4}(m_2+m_3)$