Some Examples of Static Equilibrium — Question 9

Question 9

Two cable ratings restrict where a payload may be placed

A uniform beam of length $4\,\mathrm m$ and weight $100\,\mathrm N$ hangs horizontally from vertical end cables. The left cable is rated for $200\,\mathrm N$ and the right for $250\,\mathrm N$. A $300\,\mathrm N$ payload is placed at distance $x$ from the left end.

1. Find both cable tensions as functions of $x$.

2. Find the complete allowable interval of payload positions.

Question 9 - Solution

Two cable ratings restrict where a payload may be placed

(a) Derivation and calculation

For payload position $x$ measured from the left, $$T_L+T_R=100+300=400\ \mathrm N,$$ $$4T_R=100(2)+300x.$$ Thus, with $x$ in metres, $$\boxed{T_R=50+75x},\qquad\boxed{T_L=350-75x}\quad(\mathrm N).$$

(b) Result and interpretation

The cable capacities give $$350-75x\le200\Rightarrow x\ge2.00\ \mathrm m,$$ $$50+75x\le250\Rightarrow x\le\frac83\ \mathrm m.$$ Both tensions are positive throughout, so $$\boxed{2.00\le x\le2.67\ \mathrm m}.$$ The left cable controls the lower endpoint; the right cable controls the upper endpoint.

Physical check

At $x=2$ m the tensions are 200/200 N; at $x=8/3$ m they are 150/250 N.