Problem Statement

A large sailboat has a centerboard extending $1\mathrm{.}5$ meters below its keel. The centerboard is essentially a thin, smooth plate

with a vertical back edge. The front edge tapers linearly such that the length of the top is $1$ meter and the length of the

bottom is $0\mathrm{.}5$ meters. The centerboard is pivoted at the top front edge so that it can be raised in shallow water.

Figure $1$: Centerboard Geometry. $($Remember: do NOT copy this image

into your report without a reference citation.$)$

As a result of both the boats motion and the local current, the horizontal velocity of the water passing over the centerboard

is defined by the following function:

$V(y)=7y^2+5$

where V is the velocity in knots $($from right to left in Figure $1)$ and y is the vertical location in meters measured from the

bottom edge of the centerboard. You may assume that the transverse velocity is negligible. Calculate the moment acting on

the hinge pin due to the drag force on the centerboard. $[$HINT: You will NOT get the correct answer by using the average

velocity to calculate a point force applied at the midpoint of the centerboard $-$ although this incorrect answer may be useful

for validation.$]$

Validation

Bracket your solution by solving for $($a$)$ a uniform velocity equal to $V_{\text{m}\text{a}\text{x}}$ acting on a rectangular surface with width

of $1$ m and $($b$)$ a uniform velocity equal to $V_{\text{m}\text{i}\text{n}}$ acting on a rectangular surface with width of $0\mathrm{.}5$ m $.$

Use one additional validation approach.

Hint

Finding the moment caused by a uniform velocity on a rectangular surface is fairly easy. We can use this to our advantage

by "discretizing" the domain $($the centerboard$)$ into a number of small horizontal slices. If the height of each slice $y$ is

small enough, the velocity and width are nearly constant for that segment, even though they have different values from one

segment to the next. Using $V(y)$ and $W(y)$ for each segment to calculate the moment acting on that segment, we can add up

all of the segments to determine the total moment. Taken to the extreme $ydy,$ instead of adding we integrate to

determine the total moment $(M={\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}dM).$