Two point bodies of masses m and 3m are connected by a massless spring of spring constant k = m ₀^2 and kept on a frictionless horizontal surface. The spring is extended by a small distance l over its natural length at time t = 0 and then released so that the masses execute simple harmonic motion. The maximum speed of the particle with mass m is given by

Step-by-step solution

By the principle of conservation of linear momentum, the center of mass of the system remains at rest. Let v₁ and v₂ be the maximum speeds of mass m and 3m respectively when the spring passes through its natural length. m v₁ = 3m v₂ v₂ = v₁ 3 By the principle of conservation of mechanical energy, the maximum potential energy of the spring is converted into the maximum kinetic energy of the system: 1 2 k l^2 = 1 2 m v₁^2 + 1 2 (3m) v₂^2 Substituting v₂ = v₁ 3 and k = m ₀^2 into the energy equation: 1 2 m ₀^2 l^2 = 1