Question

The midpoint of a guitar string executes simple harmonic motion with motion following the form x(t) = A sin(ωt + φ).

Amplitude (A)=1.60mm, Angular Velocity (w)=2760 and phase constant (o) = pi/2.

How do I find the initial displacement, velocity and acceleration of the midpoint of the string?

Answer 1

x(t) = A sin(ωt + φ)
v(t) = A ω cos(ωt + φ)
a(t) = - A ω2 sin (ωt + φ) = -ω^2 x(t)

so
x(0) = A sin(φ)
v(0) = A ω cos( φ)
a(0) = - A ω2 sin ( φ) = -ω^2 x(0)

at t = 0
x(0) = 160 sin(φ)
v(0) = 160*2760 cos( φ)
a(0) = -2760^2 160 sin(φ)



if φ = pi/2
x(0) = A = 160
v(0) = A ω * 0 = 0
a(0) = -A ω^2 = -160*2760^2

Answer 2

By the way
2760/2 pi is about 440 Hz
to hear a 440 Hz test tone use:
https://www.youtube.com/watch?v=Awx8-mq8g6g