Some Examples of Static Equilibrium — Question 5

Question 5

A shallow suspension requires a large tension even for a modest load

A weight $500\,\mathrm N$ hangs at the midpoint of two equal straight cable segments whose level supports are $6\,\mathrm m$ apart. Each segment has tension limit $1000\,\mathrm N$. Neglect cable weight; let the midpoint sag be $h$.

1. Find the minimum segment angle above horizontal.

2. Find the minimum sag compatible with the tension limit.

Question 5 - Solution

A shallow suspension requires a large tension even for a modest load

(a) Derivation and calculation

For equal cable tensions, $$2T\sin\theta=W=500\ \mathrm N.$$ The tension cap requires $$\sin\theta\ge\frac{500}{2(1000)}=\frac14, \qquad\theta_{\min}=14.5^\circ.$$ Using the 3 m half-span, $$\boxed{h_{\min}=3\tan\theta_{\min}=\frac3{\sqrt{15}}=.775\ \mathrm m}.$$

(b) Result and interpretation

Geometry gives $\sin\theta=h/\sqrt{3^2+h^2}$, so $$\boxed{T(h)=\frac{500\sqrt{9+h^2}}{2h}}.$$ As $h\to0^+$, $$T\sim\frac{750}{h}\longrightarrow\infty.$$ A perfectly horizontal ideal cable cannot supply a finite upward resultant with finite tension.

Physical check

At minimum sag each cable supplies 250 N vertically, while its total tension is 1000 N.