If you were to hold a string at both ends what shape would the string take(in earth's gravity). Obviously this depends on the length of string and the distance you hold the 2 ends apart so let's say that you can only change the distance of the ends from one another on a straight line parallel to the ground. What equation could you use to predict the shape of the string at any given length(of the string) and distance(of the ends)?
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Are you talking about the Catenary shape?
The general equation is
$$ y(x) = a \left( \cosh \left( \tfrac{x}{a} \right) -1 \right) $$
where $y(0)=0$ is the lowest point on the curve, and the parameter $a$ defines how much it bends.
What remains constant along the curve is the horizontal component of tension $H$, which can be used to find $a$
$$ a = \frac{H}{w} $$
where $w$ is the weight per unit length.
jalex
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