Net Force on a Dam
Consider a dam holding back an incompressible fluid such as water.
Let us suppose that
the
xy
–plane is parallel to the ground and the
z
–coordinate perpendicular to the ground.
The purpose of this exercise is to show that the total net force in the
y
–direction on this
dam can be calculated by using the projected area onto a plane parallel to the
xz
–plane
and centroid of this projection.
You can visualize this result in the following diagram.
Here is an argument that this calculation will yield the correct result.
Consider a thin
strip of the dam at a certain
z
height.
The intersection of the plane at this
z
height with
the dam will give a curve with
y
a function of
x
.
Consider a small segment of this curve
∆
!
with change
∆
!
and
∆
!
in the
z
–level plane parallel to the
xy
–plane.
This portion of the force,
∆
Force
, is perpendicular to the curve, that is, perpendicular to
∆
!
.
The slope of the curve
∆
!
is given by the derivative.
The derivative is approximately
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 Spring '08
 BLOCK
 Calculus, Derivative, net force

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