Question

O₁ and O₂ are centers two identical spheres, each of mass M. O₁ and O₂ are a distance 2d apart. A mass m is released from rest at point A as shown in Fig. m moves under gravitational force of O₁ and O₂ . The initial acceleration of mass m is a₀ . Its speed when it reach point O is v. Then

(1) \(a_0=\frac{GM}{4d^2};v=\sqrt{\frac{2GM}{d}}\)

(2) \(a_0=\frac{\sqrt{3}GM}{4d^2};v=\sqrt{\frac{2GM}{d}}\)

(3) \(a_0=\frac{GM}{4d^2};v=\sqrt{\frac{2GM}{2d}}\)

(4) \(a_0=\frac{GM}{2d^2};v=\sqrt{\frac{2GM}{d}}\)

Answer 1

 (2) \(a_0=\frac{\sqrt{3}GM}{4d^2};v=\sqrt{\frac{2GM}{d}}\)

f = gravitational force on mass m; at a due to O₁ or O₂

F = net gravitational force on mass m at a = 2f cosθ  = 2 (Gmm/4d²) x √3/2

\(=\frac{\sqrt{3}GMm}{4d^2}\)

a₀ = initial acceleration of mass m at A = F/m

\(=\frac{\sqrt{3}GM}{4d^2}\)

E_e = initial total energy of mass m at A

= \(-2[\frac{GMm}{2d}]\)

E_f = Total final energy of mass m at O

\(=\frac{1}{2}mv^2+(-\frac{2GMm}{d})\)

From law of conservation of energy, E_e = E_f