Four identical particles of mass m are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is the length of the sides of the square is

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Solution

Q.
Four identical particles of mass m are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is $(\frac{2 \sqrt{2} + 1}{32}) \frac{G m^{2}}{L^{2}},$ the length of the sides of the square is [2024]

1 2 L

2 4 L

3 3 L

4 L/2

Ans.
(2)

$F_{n e t} = \sqrt{2} F + F'$

$F = \frac{G m^{2}}{a^{2}} and F' = \frac{G m^{2}}{{(\sqrt{2} a)}^{2}}$

$F_{net} = \frac{\sqrt{2} G m^{2}}{a^{2}} + \frac{G m^{2}}{2 a^{2}}$

$(\frac{2 \sqrt{2} + 1}{32}) \frac{G m^{2}}{L^{2}} = \frac{G m^{2}}{a^{2}} (\frac{2 \sqrt{2} + 1}{2}) \Rightarrow a = 4 L$

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