The equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At t = 0, they are given an initial velocity and . Here and ware unit vectors along the edges of the triangle. If the three mases interact gravitatio

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Solution

Q.

Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side $a$ in free space. At t = 0, they are given an initial velocity ${\vec{V}}_{A} = V_{0} \vec{A C}, {\vec{V}}_{B} = V_{0} \vec{B A}$ and ${\vec{V}}_{C} = V_{0} \vec{C B}$ . Here $\vec{A C}, \vec{C B}$ and $\vec{B A}$ ware unit vectors along the edges of the triangle. If the three mases interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is : [2025]

1 $\frac{1}{2} a m V_{0}$

2 $3 a m V_{0}$

3 $\frac{\sqrt{3}}{2} a m V_{0}$

4 $\frac{3}{2} a m V_{0}$

Ans.
(3)

$\tan 30 ° = \frac{2 r}{a} = \frac{1}{\sqrt{3}} \Rightarrow r = \frac{a}{2 \sqrt{3}}$

Net angular momentum about point O

$L = (m v r_{⊥}) \times 3 = m v_{0} \frac{a}{2 \sqrt{3}} \times 3 = \frac{\sqrt{3}}{2} m v_{0} a$

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