Balancing Torques Ranking Task

A sign is to be hung from the end of a thin pole, and the pole supported by a single cable. Your design firm brainstorms the six scenarios shown below. In scenarios A, B, and D, the cable is attached halfway between the midpoint and end of the pole. In C, the cable is attached to the mid-point of the pole. In E and F, the cable is attached to the end of the pole.

A) Rank the design scenarios (A through F) on the basis of the tension in the supporting cable.

=> The torque due to the cable is given by

τ = τ F sin θ ,

where τ  is the distance from the hinge to the force and  θ  is the angle of the force relative to the pole. Since the pole is in equilibrium, the torque due to the cable is equal in magnitude to the net torque due to the sign and the weight of the pole. Thus,

 τ_cable = τ_sign + τ_pole ,

or

τ F sinθ = τ_sign+τ_pole,

and therefore

F = (τ_sign+τ_pole)/r sinθ

Substituting different values of τ and θ into this relationship can give you the relative sizes of the required cable forces.