A hanging cable or chain with both ends anchored at the same height form a shape called a "catenary." An example is the shape made by telephone pole or electric tower wires. You can solve for the vertical deflection, or sag, at any point along the cable if you know its density and its tension at either end.
Denote the distance measured horizontally from the support to the right with the letter z, ranging from 0 to L. Denote the linear density of the cable (mass per unit length) with the letter w.
Solve for the horizontal component of the tension throughout the wire. Fortunately, this is a constant across the entire cable. Therefore, if you are able to measure the tension at the supports, as well as the angle θ from horizontal, then the horizontal component of tension is cos θ times the measured tension. Denote the constant horizontal component of tension with the letter T.
Solve for the deflection at point z from the left by plugging T and w into the equation y(z) = (T/w) [cosh(wL/2T) - cosh(w/Tx(L/2-z))]. Note that cosh(x) = [e^x + e^-x]/2, where e is the base of the natural logarithm and the caret ^ indicates exponentiation.
For example, if you want to find the deflection at z=L/2, then the second cosh function becomes 1, since cosh(0)= [1+1]/2.
Note that the above equation presume the cable does not stretch, and therefore that w is constant along its length.
Tips and warnings
- Note that the above equation presume the cable does not stretch, and therefore that w is constant along its length.