Three point masses each of same mass M are placed at the corners of an equilateral triangle as shown. A fourth point mass m is placed at the centroid O of the triangle. The gravitational constant is G. Choose correct statements. [Consider the mutual gravitational forces of the four given masses only]

Escape velocity of m is,

v_{e} = \sqrt{\frac{2 \sqrt{3} G M}{l}}

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Escape velocity of m is,

v_{e} = \sqrt{\frac{6 \sqrt{3} G M}{l}}

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At a point on the line passing through O and perpendicular to plane of triangle net force on m is along negative z-axis or positive z-axis

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Escape velocity of m is,

v_{e} = \sqrt{\frac{3 G M}{l}}

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Solution

The correct option is C At a point on the line passing through O and perpendicular to plane of triangle net force on m is along negative z-axis or positive z-axis
$\frac{1}{2} m v^{2} + m v_{0} \geq 0$
$v_{0} = (3) ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ - \frac{G M}{\frac{l}{\sqrt{3}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ = - \frac{3 \sqrt{3} G M}{l}$
$v \geq \sqrt{\frac{6 \sqrt{3} G M}{l}}$

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The correct option is C At a point on the line passing through O and perpendicular to plane of triangle net force on m is along negative z-axis or positive z-axis12mv2+mv0≥0 v0=(3)⎛⎜ ⎜ ⎜ ⎜⎝−GMl√3⎞⎟ ⎟ ⎟ ⎟⎠=−3√3GMl v≥√6√3GMl