11.2
Vectors in Engineering Applications
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.
A steel beam supported by two identical cables provides a practical example of static equilibrium. The beam has a downward weight of 5000 N, while the two cables support it from opposite sides. Because the arrangement is symmetric, each cable makes the same angle of 60° with the horizontal beam and carries the same tension.
In equilibrium, the beam remains completely at rest. This means that the total horizontal and vertical forces must both be zero. Each cable pulls along its own direction, so the tension force in each cable can be separated into horizontal and vertical parts. The horizontal parts act in opposite directions, canceling one another because the cables are identical and symmetrically positioned.
The vertical components of the cable tensions act upward and, together, balance the beam's downward weight. Since the beam weighs 5000 N, each cable must provide half the required upward support, accounting for the cable angle. Solving the force balance shows that the tension in each cable is approximately 2887 N.
The tension vectors have equal upward components and horizontal components that act in opposite directions. One cable pulls upward and to the left, while the other pulls upward and to the right. Their combined effect exactly balances the beam's weight and prevents any motion.
This example shows how vector components and equilibrium conditions are applied in mechanics to analyze forces acting on suspended structures such as beams, bridges, and support systems.
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.
From Chapter 11:
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11.2 : Vectors in Engineering Applications
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