Why Power Lines Sag
Power lines droop between poles rather than running straight, and it is not carelessness. A wire that is nearly horizontal where it meets the pole has almost none of its pull pointing upward, so it needs an enormous tension to carry its own weight. Wind the sag out of a 45 m span here and the tension climbs from about 540 N to over 3500 N — the same equilibrium as a load on two cables, taken to its limit.
Como usar esta simulação
- Set the sag at mid-span anywhere from 0.15 m to 4 m
- Change the span between the poles from 25 m to 70 m
- Press Tighten the line to wind the sag out and watch the tension climb
- Use the camera bar for the roadside view or a close look at the pole top
O que observar
- The vertical pull on each pole barely changes — it is half the wire's own weight
- The horizontal pull is what runs away as the sag is reduced
- The force triangle at the pole grows longer and flatter as the line is tightened
- The wire is always a little longer than the gap it crosses
- The angle the wire leaves the pole at falls towards zero as the sag does
- At a realistic working sag the tension is already several times the wire's weight
A física por trás
A uniform wire hangs as a catenary, y = a cosh(x/a) − a, whose parameter a follows from the sag d over a span L through d = a(cosh(L/2a) − 1). At each support the wire pulls with a force whose horizontal component is H = wa and whose vertical component is half the weight of the wire, where w is the weight per metre. The tension along the wire is the resultant, T = √(H² + V²), and the catenary makes that equal to w(a + d). The vertical component is fixed by the wire's own weight, so taking out the sag cannot change it — only H grows, and T grows with it. In the idealised limit of a perfectly horizontal wire carrying any weight at all, the tension would tend to infinity, which is why a controlled amount of sag is designed into every span.