Assume all surfaces depicted in the diagram lack friction. The car has a mass M , the block has a mass m , and the spring possesses a force constant k . The system is released from rest with the spring initially stretched by a distance x₀ . Determine the time period of the resulting simple harmonic motions.

Step-by-step solution

Let x₁ and x₂ be the displacements of the block of mass m and the car of mass M from their respective equilibrium positions. Since no external horizontal force acts on the system, the center of mass remains stationary. Therefore: m x₁ + M x₂ = 0 x₂ = - m M x₁ The net extension or compression in the spring is the relative displacement x = x₁ - x₂ . Substituting the value of x₂ , we get: x = x₁ - (- m M x₁ ) = x₁ (1 + m M ) = x₁ ( M + m M ) The restoring force acting on the block of mass m is F = -kx . F = -k ( M + m