(5)
In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ. The forces on this reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three forces will be concurrent, and their ratios will be given by a triangle of forces. In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have
T cos ψ = T0, T sinψ = W,
(6)
where T0 is the tension at A, and W is the weight of PA. The former equation expresses that the horizontal tension is constant.
| Fig. 55. |
If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0 tan ψ. If we write T0 = wa, so that a is the length of a portion of the chain whose weight would equal the horizontal tension, this becomes
s = a tan ψ.
(7)
This is the “intrinsic” equation of the curve. If the axes of x and y be taken horizontal and vertical (upwards), we derive