Some Examples of Static Equilibrium — Question 3

Question 3

Unequal cable angles make the steeper cable carry more tension

A small ring supports a vertical weight $W$. Two massless cables pull from the ring upward-left at $30^\circ$ and upward-right at $60^\circ$ to horizontal. Their tension limits are respectively $80$ and $120\,\mathrm N$.

1. Express each tension in terms of $W$.

2. Find the maximum supported $W$ and identify the limiting cable.

Question 3 - Solution

Unequal cable angles make the steeper cable carry more tension

(a) Derivation and calculation

The ring balance equations are $$T_R\cos60^\circ=T_L\cos30^\circ,\qquad T_L\sin30^\circ+T_R\sin60^\circ=W.$$ Therefore $$T_R=\sqrt3T_L,\qquad\boxed{T_L=W/2},\quad\boxed{T_R=\sqrt3W/2}.$$

(b) Result and interpretation

Apply both independent cable limits: $$W/2\le80\Rightarrow W\le160\ \mathrm N,$$ $$\sqrt3W/2\le120\Rightarrow W\le80\sqrt3\ \mathrm N.$$ Thus $$\boxed{W_{\max}=80\sqrt3=139\ \mathrm N}.$$ The right cable controls; at this load, $T_L=40\sqrt3=69.3$ N and $T_R=120$ N.

Physical check

The steeper cable carries the larger tension, as required by horizontal balance.