Question

The midpoint of a guitar string executes simple harmonic motion with motion following the
form x(t) = Asin(ωt + φ). It has an angular frequency of ω = 2.76 × 10-3 s−1 and an
amplitude of A = 1.60mm. Take the phase constant to be φ = π/2.
(a) What is the period of oscillation of the string?
(b) What is the maximum speed of the string during this motion?
(c) What is the maximum magnitude of the acceleration of the string?
(d) Determine the initial displacement, velocity and acceleration of the midpoint of the
string.

Answer 1

a)

$$T=\frac{2\pi}{\omega}=\frac{2\pi}{0.00276}=2280\ s$$

b)

$$V=A\omega=(0.0016)(0.00276)=4.42\cdot 10^{-6}\frac{m}{s}$$

c)

$$a=A\omega^2=(0.0016)(0.00276)^2=1.22\cdot 10^{-8}\frac{m}{s^2}$$

d)

$$x(0)=A\sin{(0 +0.5\pi)}=A=1.6\ mm$$

$$v(0)=0\frac{m}{s}$$

$$a(0)=-A\omega^2=-1.22\cdot 10^{-8}\frac{m}{s^2}$$