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 12.5 h. How long does it take for the water to fall a distance 0.250d from its highest level?Number__________Unit____________.

Answer 1

2.08335 hours

Step-by-step explanation:

T = Time period = 12.5 h

Angular frequency is given by

`\omega=(2\pi)/(T)\\\Rightarrow \omega=(2\pi)/(12.5)\\\Rightarrow \omega=0.50265\ 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_0=x_mcos(\omega t+\phi)`

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

`0.25d=0.5dcos(0.50265t)\\\Rightarrow cos(0.50265t)=(0.25)/(0.5)\\\Rightarrow 0.50265t=cos^(-1)0.5\\\Rightarrow t=(1.047)/(0.50265)\\\Rightarrow t=2.08335\ h`

The time taken for the water to fall the distance is 2.08335 hours