Question

Four particles having masses m, 2m, 3m and 4m are placed at four corners of a square edge a. Find the gravitational force acting on a particle of mass m placed at the center.

Answer 1

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ANSWER::

Force in OD = (G x m x 4m) / (a/r²)² = 8Gm²/a²

Force in OC = (G x m x 3m) / (a/r²)² = 6Gm²/a²

Force in OB = (G x m x 2m) / (a/r²)² = 4Gm²/a²

Force in OD = (G x m x m) / (a/r²)² = 2Gm²/a²

Resultant of force in OF = √[64(Gm²/a²)² + 36(Gm²/a²)²] = 10Gm²/a²

Resultant of force in OE = √[64(Gm²/a²)² + 4(Gm²/a²)²] = 2√5Gm²/a²

Net Resultant ,

F = √[100(Gm²/a²)² + 20(Gm²/a²)² - 2(Gm²/a²)² x 20√5]

F = √[(Gm²/a²)²(120 - 40√5)]

F = √[(Gm²/a²)²(120 - 89.6)]

F = √40.4Gm²/a²

F = 4√2Gm²/a²

Hope it helps!