Question

four particles with mass M 2m 3M and 4m are placed at the four corners of a square of the gravitational force acting on a particle of mass M placed at the centre is​

Answer 1

In the given question we are asked to calculate only the magnitude of the force not the direction, so...

from fig.1 AC = √(a² + a²) ⇒ √2.a

AO = a÷√2 = r

∴ let F₁, F₂, F₃, F₄ be the forces

F₁ = (Gm.m) ÷ r²    ⇒ 2Gm²÷a²                                                [let Gm²÷a² = k]

⇒ 2k

F₂ = (Gm.2m) ÷ r² ⇒ 4Gm²÷a²

⇒ 4k

F₃ = (Gm.3m) ÷ r² ⇒ 6Gm²÷a²

⇒ 6k

F₄ = (Gm.4m) ÷ r² ⇒ 8Gm²÷a²        

⇒ 8k

Force along AC = F₃ - F₁ ⇒ 6k - 2k

⇒ 4k

Force along BD = F₄ - F₂ ⇒ 8k - 4k

⇒ 4k

∴ net Force = √[(4k)² + (4k)²] ⇒4√2.k

⇒ 4√2.Gm²÷a²                                                               ∵[k = Gm²÷a²]

Ans.