Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they can just touch each other is :
(Take k = 1/(4πε₀) and assume kQ² > Gm² where G is the Gravitational constant)

Question

Consider two identical metallic spheres of radius R each having charge Q and mass m. Their centers have an initial separation of 4R. Both the spheres are given an initial speed of u towards each other. The minimum value of u, so that they can just touch each other is :

(Take k = 1/(4πε₀) and assume kQ² > Gm² where G is the Gravitational constant)

G1 Physics · Accepted Answer

Use conservation of total mechanical energy for the two-sphere system. Initial centre-to-centre separation: rᵢ = 4R. When the spheres just touch: r𝒇 = 2R. The total initial kinetic energy is Kᵢ = 2(½ μ \mu μ ²) = μ \mu μ ². The electrostatic potential energy of the two charges is U(electric) = kQ²/r. The gravitational potential energy is U(gravity) = -Gm²/r. Hence the total interaction potential energy is U(r) = (kQ² - Gm²)/r. For the minimum speed required to just touch, the relative motion m…