Three equal masses $m$ are kept at the vertices $(A, B, C)$ of an equilateral triangle of side $a$ in free space. At $t = 0$, they are given initial velocities $\vec V_A = V_0\,\widehat{AC}$, $\vec V_B = V_0\,\widehat{BA}$ and $\vec V_C = V_0\,\widehat{CB}$, where $\widehat{AC}$, $\widehat{CB}$ and $\widehat{BA}$ are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is
Answer: (C) $\dfrac{\sqrt3}{2}amV_0$
Gravity between the masses is internal, so the total angular momentum is conserved; it keeps its initial value up to the collision. (The total momentum is zero, since the three unit vectors round the triangle add to zero, so the angular momentum is the same about any point.)
About the centroid: each velocity runs along a side of the triangle, whose perpendicular distance from the centroid is the inradius $\dfrac{a}{2\sqrt3}$. All three turn the same way (A → C → B → A), so
$$L = 3\cdot mV_0\cdot\frac{a}{2\sqrt3} = \frac{\sqrt3}{2}amV_0$$
Solution by Sreeraj P, M.Sc Physics