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Three identical uniform spheres, each of mass M and radius R, are arranged so that every sphere touches the other two. Find the magnitude of the net gravitational force on any one sphere due to the other two.

1. G*M²/(4R²)
2. 2*G*M²/R²
3. 2*G*M²/(4R²)
4. sqrt(3)*G*M²/(4R²)

Correct answer: sqrt(3)*G*M²/(4R²)

Solution

Centre separation = 2R. Force between a pair: F = G*M²/(2R)² = G*M²/(4R²). On one sphere, two equal forces of magnitude F act with 60 deg between them. Resultant = 2*F*cos(30 deg) = 2*F*(sqrt(3)/2) = sqrt(3)*F = sqrt(3)*G*M²/(4R²).

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