Question

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]

• A. Escape velocity of m is, $v_e=\sqrt{\frac{2\sqrt{3}GM}{l}}$
• B. Escape velocity of m is, $v_e=\sqrt{\frac{6\sqrt{3}GM}{l}}$
• 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
• D. Escape velocity of m is, $v_e=\sqrt{\frac{3GM}{l}}$

Answer 1

Answer: (B), (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

$\frac{1}{2}m{v}^2+m{v}_0\ge 0$
$v_0=\left(3\right)(-\frac{GM}{\frac{l}{\sqrt{3}}})=-\frac{3\sqrt{3}GM}{l}$
$v\ge \sqrt{\frac{6\sqrt{3}GM}{l}}$