# Simple Harmonic Motion Displacement

**1. The problem statement, all variables and given/known data**

A particle is in simple harmonic motion with period T and with position as a function of time given by x(t) = A cos(wt+ø).

At time t = 0 the particle is at x = A/2 with positive velocity. The next time it is at the same position is ___?

**2. Relevant equations**

w/(2pi)=1/T

**3. The attempt at a solution**

So I know the cosine of the argument has to = 1/2 for the position to be half the amplitude. The first time this happens is at t = 0. This implies the phase constant ø is pi/3.

The next time the cosine argument is a positive 1/2 is when the argument equals 300 degrees. Therefore:

wt + pi/3 = 5pi/3

wt = 4pi/3

t = 4pi/(3w)

w = 2pi/T

t = 4pi/[3(2pi/T] = 4pi/[6pi/T] = 2T/3.

The answer however is t = T/3.

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