Question

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?

Answer 1

Answer:

1.869 hours

Explanation:

$T$ = Time period = 11.1 h

Angular frequency is given by

$\omega=\frac{2\pi}{T}\\\Rightarrow \omega=\frac{2\pi}{11.1}\\\Rightarrow \omega=0.566\ rad/h$

The distance moved from highest to lowest level is given by

$d=2x_{m}\\\Rightarrow x_m=\frac{d}{2}\\\Rightarrow x_m=0.5d$

At the ocean surface $x_m=0.25d$

$x=x_mcos(\omega t+\phi)$

$\phi$ = Phase constant = 0 as clock is started at $x_0=x_m$

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

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