I vettori nelle applicazioni ingegneristiche

A five-thousand-newton steel beam is lifted by two identical, weightless cables. The goal is to find the tension in each cable that keeps the beam in equilibrium.

Equilibrium means the beam remains at rest, with zero net horizontal and vertical force. The setup is symmetric, and each cable makes a sixty-degree angle with the horizontal beam.

The tension in each cable is a vector that acts along the cable.

Since each cable pulls in the x and y directions, each tension is resolved into horizontal and vertical components using the unit vectors i and j.

The beam’s weight acts downward in the negative j direction. For equilibrium, the two cable tensions must balance this weight, so their vector sum equals the negative weight vector.

Comparing the horizontal components shows that the positive contribution from one cable cancels the negative contribution from the other. This means both cables have the same tension magnitude.

Comparing the vertical components shows that the sum of the upward components equals five thousand newtons.

Solving this equation gives the tension magnitude. Substituting it into the component expressions gives the full tension vector for each cable, which keeps the beam stable and at rest.