Article 32
Catenary of
Uniform Strength.
If the area of the normal section at any point be made proportional to the tension at
that point, there will then be a constant tension per unit of area, and the
tendency to break will be the same at all points. To find the equation of the
curve of equilibrium under gravity, consider the equilibrium of an element
whose length
is , and whose
weight is ,
where is the
section at ,
and the
uniform density. This weight is balanced by the difference of the vertical components of the
tensions at
and ,
hence
therefore , the tension at
the lowest point, and .
Again, if
be the section at the lowest point, then by hypothesis
, and
the first equation becomes
or
where stands for the
constant , the length
of string (of section )
whose weight is equal to the tension at the lowest point; hence,
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the intrinsic equation of the catenary of uniform strength.
Also
hence
and thus the Cartesian equation is
in which the axis of
is the tangent at the lowest point.
-
Prob. 95.
- Using the same data as in Art. 31, find the parameter
and the depth of the lowest point. (The equation
gives ,
which, by putting ,
becomes .
From the graph it is seen that
is nearly .
If ,
then, from the tables of the gudermanian at the end of this chapter,
whence, by interpolation,
and .
Again, ;
but ;
and ;
hence ,
the required depth.)
-
Prob. 96.
- Find the inclination of the terminal tangent.
-
Prob. 97.
- Show that the curve has two vertical asymptotes.
-
Prob. 98.
- Prove that the law of the tension ,
and of the section ,
at a distance ,
measured from the lowest point along the curve, is
and show that in the above numerical example the terminal section is
times the minimum section.
-
Prob. 99.
- Prove that the radius of curvature is given by
. Also that the
weight of the arc
is given by ,
in which
is measured from the vertex.