Answer:
It takes 0.00127 seconds.
Explanation:
The equation its
[tex]y(x,t) = 4.7 \ mm \ sin ( kx + 675 \frac{rad}{s} t + \phi)[/tex].
We want the time for ANY POINT, so, for convenience, lets take x=0
[tex]y(0,t) = 4.7 \ mm \ sin ( k*0 + 675 \frac{rad}{s} t + \phi)[/tex].
[tex]y(0,t) = 4.7 \ mm \ sin ( 675 \frac{rad}{s} t + \phi)[/tex].
Now, we want the positions
[tex]y(0,t_1)=2 \ mm[/tex]
[tex]y(0,t_2)=-2 \ mm[/tex]
so, for the first position:
[tex]y(0,t_1)=2 \ mm[/tex]
[tex]2 \ mm = 4.7 \ mm \ sin ( 675 \frac{rad}{s} t_1 + \phi)[/tex].
[tex]\frac{2 \ mm} {4.7 \ mm }= sin ( 675 \frac{rad}{s} t_1 + \phi)[/tex].
[tex] 0.42 = sin ( 675 \frac{rad}{s} t_1 + \phi)[/tex].
[tex] sin^-1(0.42) = 675 \frac{rad}{s} t_1 + \phi)[/tex].
and for the second one:
[tex]y(0,t_2)=-2 \ mm[/tex]
[tex]-2 \ mm = 4.7 \ mm \ sin ( 675 \frac{rad}{s} t_2 + \phi)[/tex].
[tex]\frac{-2 \ mm} {4.7 \ mm }= sin ( 675 \frac{rad}{s} t_2 + \phi)[/tex].
[tex] -0.42 = sin ( 675 \frac{rad}{s} t_2 + \phi)[/tex].
[tex] sin^-1(0.42) = 675 \frac{rad}{s} t_2 + \phi)[/tex].
Now, we can subtract both:
[tex]sin^-1(0.42) -sin^-1(-0.42) = 675 \frac{rad}{s} t_1 + \phi - 675 \frac{rad}{s} t_2 + \phi[/tex]
[tex]sin^-1(0.42) -sin^-1(-0.42) = 675 \frac{rad}{s} t_1 - 675 \frac{rad}{s} t_2[/tex]
[tex]sin^-1(0.42) -sin^-1(-0.42) = 675 \frac{rad}{s} (t_1 - t_2)[/tex]
[tex]\frac{sin^-1(0.42) -sin^-1(-0.42)}{ 675 \frac{rad}{s}} = (t_1 - t_2)[/tex]
[tex]\frac{0.43 - (-0.43)}{ 675 \frac{rad}{s}} = (t_1 - t_2)[/tex]
[tex]\frac{0.86}{ 675 \frac{rad}{s}} = (t_1 - t_2)[/tex]
[tex] 0.00127 s = (t_1 - t_2)[/tex]
The strings take 0.00127 seconds.
