Question

Three equal masses m are kept at vertices (A, B, C) of an equilateral triangle of side a in free space. At \(\mathrm{t}=0,\) they are given an initial velocity \(\vec{V}_{A}=V_{0} \overrightarrow{A C}, \quad \vec{V}_{B}=V_{0} \overrightarrow{B A}\) and \(\vec{V}_{C}=V_{0} \overrightarrow{C B}.\)

Here, \(\overrightarrow{A C}, \overrightarrow{C B}\) and \(\overrightarrow{B A}\) 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 :

(1) \(\frac{1}{2} \mathrm{amV}_{0}\)

(2) \(3\ a\ m\ V_0 \)

(3) \(\frac{\sqrt{3}}{2} a \mathrm{mV}_{0}\)

(4) \(\frac{3}{2} a m V_{0}\)

Answer 1

Correct option is (3) \(\frac{\sqrt{3}}{2} a \mathrm{mV}_{0}\) 

\(\tan 30^{\circ}=\frac{2 \mathrm{r}}{\mathrm{a}}=\frac{1}{\sqrt{3}}\)

\(r=\frac{a}{2 \sqrt{3}}\)

\(\mathrm{L}=\left(\operatorname{mvr}_{\perp}\right) \times 3\)

\(=m v_{0} \frac{a}{2 \sqrt{3}} \times 3\)

\(=\frac{\sqrt{3}}{2} \mathrm{mv}_{0} \mathrm{a}\)