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

1.869 hours

Step-by-step explanation:

`T` = Time period = 11.1 h

Angular frequency is given by

`\omega=(2\pi)/(T)\\\Rightarrow \omega=(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=(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)=(0.25)/(0.5)\\\Rightarrow 0.56t=cos^(-1)0.5\\\Rightarrow t=(1.047)/(0.56)\\\Rightarrow t=1.869\ h`

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