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Three identical point masses each of m are placed at the corners of an equilateral triangle of side' a' . Choose the correct statement(s)

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The gravitational field intensity at centroid of the triangle is zero.

The gravitational potential at centroid of the triangle is zero.

The gravitational field intensity at the mid point of any side is $\frac{4 Gm}{3 a^{2}}$

The potential energy of the system is $- \frac{3 {Gm}^{2}}{a}$

answer is A, C, D.

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Detailed Solution

$\vec{E} = {\vec{E}}_{1} + {\vec{E}}_{2} + {\vec{E}}_{3}; \vec{E} = 0$

${\vec{E}}_{p} = {\vec{E}}_{1} + {\vec{E}}_{2} + {\vec{E}}_{3}, E_{1} = E_{2} & {\vec{E}}_{1} = - {\vec{E}}_{2}$

$∴ E_{P} = E_{3} = \frac{Gm}{{(\frac{\sqrt{3}}{2} a)}^{2}} = \frac{4 Gm}{3 a^{2}}$

$U = \frac{- Gmm}{a} \times 3 = \frac{- 3 {Gm}^{2}}{a}$

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