For a simple harmonic motion with a time period T , determine the time required for the displacement to increase from half the amplitude to the full amplitude.

Step-by-step solution

Let the equation of simple harmonic motion starting from the mean position be x = A ( t) . Time taken to reach x = A/2 from the mean position is t₁ : A 2 = A ( t₁) ( t₁) = 1 2 t₁ = 6 t₁ = 6 = 6 T 2 = T 12 Time taken to reach x = A from the mean position is t₂ : A = A ( t₂) ( t₂) = 1 t₂ = 2 t₂ = 2 = 2 T 2 = T 4 Time required to increase the displacement from half the amplitude to the full amplitude is: t = t₂ - t₁ = T 4 - T 12 = 3T - T 12 = 2T 12 = T 6