A particle is performing a linear simple harmonic motion of amplitude ‘A’. When it is midway between its mean and extreme position, the magnitudes of its velocity and acceleration are equal. What is the periodic time of the motion?

Step-by-step solution

In linear simple harmonic motion, the velocity of particle is given by v = ω A 2 - x 2 … (i) where, ω = angular frequency A = maximum displacement of amplitude and x= displacement from mean position. The acceleration of a particle in simple harmonic motion, (SHM) is given by a = ω 2 x …. (ii) Here, x = A 2 Also, v = a (given) ω A 2 - x 2 = ω 2 x [from Eqs. (i) and (ii), we get] ⇒ A 2 - A 2 4 = ω × A 2 ⇒ 3 A 2 = ω × A 2 ⇒ 2 π T = 3 ∵ ω = 2 π T ⇒ T = 2 π 3 s