1
The Catenary and Parabolic Cables
The Catenary
Assume that one has a cable which
hangs under its own weight with only
tangential forces acting along the length of
the cable.
The cable is assumed to have
uniform density,
w
0
, along its length.
Arrange the coordinate system so that the
y

axis passes through the lowest point of the
cable.
Consider a
freebody diagram
of that
portion of the cable above the interval
[0,
x
]
.
Suppose that the length of this portion of the
cable is
s
.
Then its weight is
w
0
s
.
In the
freebody diagram this is a force acting
downward on that portion of the cable.
There is also a tangential force
T
0
acting
horizontally to the left in the freebody diagram.
This force is due to the cable to the left
of the figure.
Similarly, there is a force
T
acting tangentially at the right in the freebody
diagram.
This force is due to the cable to the right of the figure.
Since the cable is
assumed static, the sum of the horizontal and vertical forces must be zero.
This leads to
two equations.
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 Spring '08
 BLOCK
 Calculus, freebody diagram, Christiaan Huygens, Parabolic Cables

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