Q
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Superposition Principle
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 ( 2√(2) + 132) Gm²L², the length of the sides of the square is
Solution
Related Formula
F = (G m₁ m₂)/(r²)Core Logic
Let the side length of the square be a. Considering one corner mass, it experiences forces from the adjacent two masses (distance a) and the diagonally opposite mass (distance √(2)a).
The forces from the two adjacent masses are at 90^° to each other:
F = (Gm²)/(a²)The resultant of these two is √(2)F = √(2) (Gm²)/(a²), directed along the diagonal.
Step 2: Total Force Equation
The force from the diagonal mass is:
F' = Gm²(√(2)a)² = (Gm²)/(2a²)Total resultant force Fₙₑₜ = √(2)F + F':
Fₙₑₜ = √(2) (Gm²)/(a²) + (Gm²)/(2a²) = (Gm²)/(a²) ( √(2) + (1)/(2) ) Fₙₑₜ = (Gm²)/(a²) ( 2√(2) + 12 )Equating this to the given force value:
( 2√(2) + 132)(Gm²)/(L²) = (Gm²)/(a²) ( 2√(2) + 12 ) (1)/(32 L²) = (1)/(2 a²)a² = 16 L² a = 4L
Chapter Mix
Class 11 Physics: Gravitation
