Three identical spheres of mass m, are placed at the vertices of an… | JEE Main 2025 (3 Apr, Shift 1)

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Newton's Law of Gravitation

Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length $a$ . When released, they interact only through gravitational force and collide after a time $T = 4$ seconds. If the sides of the triangle are increased to length $2a$ and also the masses of the spheres are made 2 m, then they will collide after__________seconds.

Numerical answer type. Work it out, then open the answer.

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Answer: 8

Acceleration scales as $\frac{Gm}{a^2}$ and distance as $a$ , so $T \propto \sqrt{\frac{a}{Gm/a^2}} = \sqrt{\frac{a^3}{Gm}}$ .
$\frac{T'}{T} = \sqrt{\frac{(2a)^3}{2m} \cdot \frac{m}{a^3}} = \sqrt{4} = 2$
$T' = 2 \times 4 = 8$ s

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