A particle is executing simple harmonic motion with an amplitude A and a time period T . What is the minimum time taken by the particle, starting from the mean position, to reach a point where its kinetic energy becomes equal to its potential energy?

Step-by-step solution

Let the displacement of the particle from the mean position be x . Potential energy, U = 1 2 kx^2 Kinetic energy, K = 1 2 k(A^2 - x^2) Given that K = U : 1 2 k(A^2 - x^2) = 1 2 kx^2 A^2 - x^2 = x^2 2x^2 = A^2 x = A 2 The equation for displacement starting from the mean position is x = A ( t) . Substituting x = A 2 : A 2 = A ( t) ( t) = 1 2 t = 4 Since = 2 T : ( 2 T )t = 4 t = T 8 Answer: T 8