Torque — Question 7

Question 7

An angled cable must supply the required perpendicular moment

A horizontal uniform beam of length $2.0\,\mathrm m$ and weight $100\,\mathrm N$ is hinged at its left end. A cable at the right end pulls up and left at $30^\circ$ above the beam toward its support. The beam is at rest.

1. Find cable tension from torque balance about the hinge.

2. Find the hinge force components and check vertical force balance.

Question 7 - Solution

An angled cable must supply the required perpendicular moment

(a) Derivation and calculation

The hinge reaction has zero moment about the hinge. The cable’s horizontal component also has zero moment there: $$\sum\tau_z=L T\sin30^\circ-W\frac L2=0.$$ $$\boxed{T=\frac{W}{2\sin30^\circ}=100\ \mathrm N}.$$

(b) Result and interpretation

Resolve the cable and impose force balance: $$\vec T=(-T\cos30^\circ,T\sin30^\circ)=(-50\sqrt3,50)\ \mathrm N.$$ $$\boxed{H_x=50\sqrt3=86.60\ \mathrm N},\qquad \boxed{H_y=100-50=50.0\ \mathrm N}.$$ Vertically $H_y+T_y-W=50+50-100=0$; horizontally $H_x+T_x=0$.

Physical check

The cable balances the weight moment with half the vertical weight force because its lever arm is twice as long.