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In this section we are going to submerge a vertical plate in
water and we want to know the force that is exerted on the plate due to the
pressure of the water. This force is
often called the hydrostatic force.
There are two basic formulas that we’ll be using here. First, if we are d meters below the surface then the hydrostatic pressure is given
by,
where, ρ is the density of the fluid and g is the gravitational
acceleration. We are going to assume
that the fluid in question is water and since we are going to be using the
metric system these quantities become,
The second formula that we need is the following. Assume that a constant pressure P is acting on a surface with area A.
Then the hydrostatic force that acts on the area is,
Note that we won’t be able to find the hydrostatic force on
a vertical plate using this formula since the pressure will vary with depth and
hence will not be constant as required by this formula. We will however need this for our work.
The best way to see how these problems work is to do an
example or two.
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Example 1 Determine
the hydrostatic force on the following triangular plate that is submerged in
water as shown.
Solution
The first thing to do here is set up an axis system. So, let’s redo the sketch above with the
following axis system added in.
So, we are going to orient the x-axis so that positive x
is downward, corresponds to the water surface and corresponds to the depth of the tip of the
triangle.
Next we are break up the triangle into n horizontal strips each of equal
width and in each interval choose any point . In order to make the computations easier we
are going to make two assumptions about these strips. First, we will ignore the fact that the
ends are actually going to be slanted and assume the strips are
rectangular. If is sufficiently small this will not affect
our computations much. Second, we will
assume that is small enough that the hydrostatic
pressure on each strip is essentially constant.
Below is a representative strip.
The height of this strip is and the width is 2a. We can use similar
triangles to determine a as
follows,
Now, since we are assuming the pressure on this strip is
constant, the pressure is given by,
and the hydrostatic force on each strip is,
The approximate hydrostatic force on the plate is then the
sum of the forces on all the strips or,
Taking the limit will get the exact hydrostatic force,
Using the definition
of the definite integral this is nothing more than,
The hydrostatic force is then,
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Let’s take a look at another example.
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Example 2 Find
the hydrostatic force on a circular plate of radius 2 that is submerged 6
meters in the water.
Solution
First, we’re going to assume that the top of the circular
plate is 6 meters under the water.
Next, we will set up the axis system so that the origin of the axis
system is at the center of the plate.
Setting the axis system up in this way will greatly simplify our work.
Finally, we will again split up the plate into n horizontal strips each of width and we’ll choose a point from each strip. We’ll also assume that the strips are
rectangular again to help with the computations. Here is a sketch of the setup.
The depth below the water surface of each strip is,
and that in turn gives us the pressure on the strip,
The area of each strip is,
The hydrostatic force on each strip is,
The total force on the plate is,
To do this integral we’ll need to split it up into two
integrals.
The first integral requires the trig substitution and the second integral needs the
substitution . After using these substitution we get,
Note that after the substitution we know the second
integral will be zero because the upper and lower limit is the same.
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