Some Examples of Static Equilibrium — Question 2

Question 2

Cable capacity sets the shelf payload before hinge balance does

A horizontal shelf of length $2\,\mathrm m$ and weight $100\,\mathrm N$ is hinged to a wall at its left end. A cable from its right end pulls upward and leftward at $30^\circ$ above horizontal. A point payload $P$ is at the right end. Cable tension must not exceed $400\,\mathrm N$.

1. Find the maximum payload and the corresponding cable tension.

2. Find horizontal and vertical hinge reactions at that payload.

Question 2 - Solution

Cable capacity sets the shelf payload before hinge balance does

(a) Derivation and calculation

About the hinge, only the cable vertical component and the weights contribute: $$2T\sin30^\circ-100(1)-2P=0.$$ Hence, with forces in newtons, $$T=100+2P.$$ The tension limit gives $$100+2P\le400\quad\Longrightarrow\quad\boxed{P_{\max}=150\ \mathrm N}.$$

(b) Result and interpretation

At maximum payload, horizontal and vertical balance give $$H_x-T\cos30^\circ=0,\qquad H_y+T\sin30^\circ-100-P=0.$$ Thus $$\boxed{H_x=200\sqrt3=346\ \mathrm N\ \text{right}},$$ $$\boxed{H_y=100+150-400(.5)=50\ \mathrm N\ \text{up}}.$$

Physical check

The cable supports 200 N vertically and the hinge supplies the remaining 50 N.