11.2
Vectors in Engineering Applications
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…
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.
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Q1: How do you resolve cable tension into horizontal and vertical components?
Tension acts along the cable direction and can be separated into horizontal and vertical parts using unit vectors i and j. For a cable at 60 degrees to the horizontal, the horizontal component is tension times cosine of the angle, and the vertical component is tension times sine of the angle. This decomposition allows you to analyze forces in each direction independently, which is essential for vectors in space problem solving.
Q2: What does equilibrium mean for a suspended beam?
Equilibrium means the beam remains completely at rest with zero net horizontal and vertical force. The upward components of cable tensions must equal the beam's downward weight, while horizontal components from symmetric cables cancel each other. This balanced state prevents any motion or acceleration of the structure.
Q3: Why do horizontal components cancel in a symmetric cable system?
In a symmetric setup, identical cables pull from opposite sides at equal angles. One cable pulls upward and to the left while the other pulls upward and to the right. Their horizontal components act in opposite directions with equal magnitude, so they cancel completely, leaving only the upward vertical components to support the beam.
Q4: How do you calculate the tension magnitude in each cable?
For a 5000-newton beam supported by two identical cables at 60 degrees, the vertical components must sum to 5000 newtons. Since each cable contributes equally, each vertical component equals 2500 newtons. Dividing by sine of 60 degrees gives the tension magnitude of approximately 2887 newtons per cable.
Q5: What role do unit vectors play in analyzing cable forces?
Unit vectors i and j represent the horizontal and vertical directions respectively. Each cable tension is expressed as a vector sum of its horizontal component times i plus its vertical component times j. This notation allows precise mathematical representation and comparison of force components in each direction.
Q6: How does vector addition determine equilibrium conditions?
The vector sum of both cable tensions must equal the negative weight vector for equilibrium. By adding the tension vectors component-wise, horizontal components from both cables sum to zero, while vertical components sum to 5000 newtons upward. This vector equation ensures the beam experiences no net force.
Q7: Why is symmetry important in analyzing this cable-beam system?
Symmetry ensures both cables have identical tension magnitude and make equal angles with the horizontal. This simplifies calculations because horizontal components automatically cancel, and each cable contributes equally to vertical support. Without symmetry, you would need to solve a more complex system of equations.