Physics

Physics

Problem

A particle moves according to x(t)=Acos⁡(ωt+ϕ)x(t)=A\cos(\omega t+\phi). It is released from rest at its extreme positive displacement at t=0t=0. Which one of the following values of ϕ\phi is consistent with this condition? a) π\pi b) −π2-\dfrac{\pi}{2} c) 00 d) π2\dfrac{\pi}{2}
Intuitive explanation

The key is to translate the initial conditions into constraints on the phase constant ϕ\phi. At t=0t=0, the particle is at maximum displacement and has zero velocity—so you need both x(0)=Ax(0) = A and v(0)=0v(0) = 0. Use the position equation to find what ϕ\phi makes cos⁡(ϕ)=1\cos(\phi) = 1, then verify that the velocity v(t)=−Aωsin⁡(ωt+ϕ)v(t) = -A\omega\sin(\omega t+\phi) also equals zero at that same phase value.

Intuitive explanation

The key is to translate the initial conditions into constraints on the phase constant ϕ\phi. At t=0t=0, the particle is at maximum displacement and has zero velocity—so you need both x(0)=Ax(0) = A and v(0)=0v(0) = 0. Use the position equation to find what ϕ\phi makes cos⁡(ϕ)=1\cos(\phi) = 1, then verify that the velocity v(t)=−Aωsin⁡(ωt+ϕ)v(t) = -A\omega\sin(\omega t+\phi) also equals zero at that same phase value.

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