At a certain harbor, the tides cause the ocean surface to rise and fall a distance d (from highest level to lowest level) in simple harmonic motion, with a period of 11.1 h. How long does it take for the water to fall a distance 0.250d from its highest level?

Respuesta :

Answer:

1.869 hours

Explanation:

[tex]T[/tex] = Time period = 11.1 h

Angular frequency is given by

[tex]\omega=\frac{2\pi}{T}\\\Rightarrow \omega=\frac{2\pi}{11.1}\\\Rightarrow \omega=0.566\ rad/h[/tex]

The distance moved from highest to lowest level is given by

[tex]d=2x_{m}\\\Rightarrow x_m=\frac{d}{2}\\\Rightarrow x_m=0.5d[/tex]

At the ocean surface [tex]x_m=0.25d[/tex]

[tex]x=x_mcos(\omega t+\phi)[/tex]

[tex]\phi[/tex] = Phase constant = 0 as clock is started at [tex]x_0=x_m[/tex]

[tex]0.25d=0.5dcos(0.56t)\\\Rightarrow cos(0.56t)=\frac{0.25}{0.5}\\\Rightarrow 0.56t=cos^{-1}0.5\\\Rightarrow t=\frac{1.047}{0.56}\\\Rightarrow t=1.869\ h[/tex]

The time taken for the water to fall the distance is 1.869 hours

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