Benign paroxysmal positional vertigo $($BPPV$)$ is caused by the sudden movement of the head, especially when this movement changes the orientation of the head with respect to the direction of the gravitational force. BPPV occurs because of the accumulation of calcium carbonate crystals $($called otoconia$)$ that interfere with the flow of fluids in the inner ear, causing anomalous signals to be sent to the brain. Sitting up abruptly in bed can cause BPPV in $2\%$ of the population, especially those advanced in age. BPPV can also be caused by the rotational oscillations of a small boat on choppy seas about a horizontal axis passing laterally through the center of the boat. This motion causes the bow and stern of the boat to oscillate vertically.

$($a$)$

Comparing the change in the angle of a small boat with the horizontal as waves go by to the motion of sitting up in bed, estimate the period of oscillation $($in s$)$ of the boat that can cause the onset of BPPV$.($Assume a sine wave describes the up$-$and$-$down motion of the boat, and assume it takes about $1\mathrm{.}80$ s for a person to sit up in bed.$)$

How fast does a person sit up from bed? What is the angle through which the head rotates when sitting up in bed? What is the comparable situation for a boat, and what fraction of the wave's period is equivalent to sitting up in bed? s

In addition to the rotational motion, a person seated at the center of the boat will undergo translational oscillations in the vertical direction as the waves pass. If the amplitude of vertical oscillation of the center of the boat is $0\mathrm{.}700$ m$,$ what are the magnitudes of the maximum translational velocity and the maximum translational acceleration experienced by passengers seated at the center of the boat?

$($b$)$

What is the maximum translational velocity $($in m$/$s$)$ experienced by passengers seated at the center of the boat? $($Enter the magnitude.$)$

What is the relationship between the maximum speed, the amplitude, and the angular frequency of a simple harmonic oscillator? m$/$s

$($c$)$

What is the maximum translational acceleration $($in m$/$s$2)$ experienced by passengers seated at the center of the boat? $($Enter the magnitude.$)$

What is the relationship between the maximum acceleration, the amplitude, and the angular frequency of a simple harmonic oscillator? m$/$s$2$