Two small bodies with masses of 2.00 kg and 4.00 kg are held at rest, separated by a distance of 2.0 m . At what location should a particle of mass 0.10 kg be positioned so that it experiences zero net gravitational force from these two bodies? If the particle is placed at this equilibrium point, determine the total gravitational potential energy of the three-particle system, taking the usual reference level.

Step-by-step solution

Let m₁ = 2.00 kg , m₂ = 4.00 kg , and m = 0.10 kg . Let the distance between m₁ and m₂ be d = 2.0 m . Let the particle m be placed at a distance x from m₁ towards m₂ . For the net gravitational force on m to be zero, the forces exerted by m₁ and m₂ must be equal in magnitude: G m₁ m x^2 = G m₂ m (d-x)^2 2.00 x^2 = 4.00 (2.0-x)^2 Taking the square root of both sides: 1 x = 2 2.0-x 2.0 - x = 2 x x = 2.0 2 +1 = 2.0( 2 -1) 0.83 m Thus, the particle should be placed 0.83 m from the 2.00 kg body towards the 4.00 kg body.