Equilibrium Analysis for Concurrent Force Systems

If a body is in static equilibrium, then by definition that body is not accelerating. If we know that the body is not accelerating then we know that the sum of the forces acting on that body must be equal to zero. This is the basis for equilibrium analysis for a particle.

We must also remember that forces are vectors, they have both a magnitude and a direction. In order to add together the vectors we will use the vector components in the x, y and sometimes z directions. To ensure that the sum of the vectors is truly equal to zero, the sum of the magnitudes of the components in any one direction must be equal to zero. To solve the equilibrium equations we will break the single vector equation (sum of forces is equal to zero) down into two scalar equations for 2D problems (sum of the x components of the force vectors is equal to zero, and sum of the y components of the force vectors is equal to zero) or down into three scalar equations for 3D problems (sum of the x components of the force vectors is equal to zero, sum of the y components of the force vectors is equal to zero, and sum of the z components of the force vectors is equal to zero).

2D Equilibrium Equations
In two dimensional problems, the sum of forces in the x direction and the sum of forces in the y direction must be equal to zero
3D Equilibrium Equations
In three dimensional problems, the sum of forces in the x direction, the sum of forces in the y direction, and the sum of forces in the z direction must be equal to zero.

Once we have written out the equilibrium equations, we can solve the equations for any unknown forces.

Finding the Equilibrium Equations:

The first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. This diagram should show all the force vectors acting on the body. In the free body diagram, provide values for any of the know magnitudes or directions for the force vectors and provide variable names for any unknowns (either magnitudes or directions).

A free body diagram
The first step in equilibrium analysis is drawing a free body diagram. This is done by removing everything but the body and drawing in all forces acting on the body. It is also useful to label all forces, key dimensions, and angles.

Next you will need to chose the x, y, and z axes. These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis.

Once you have chosen axes, you need to break down all of the force vectors into components along the x, y and z directions (see the vector decomposition page for more details on this process). Your first equation will be the sum of the magnitudes of the components in the x direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the y direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the z direction being equal to zero.

Once you have your equilibrium equations, you can solve these formulas for unknowns. The number of unknowns that you will be able to solve for will be the number or equations that you have.

Particle vs. Extended Body Analysis:

In any body to be in equilibrium, the sum of the forces and the sum of the moments has to be equal to zero. For a particle, or for a concurrent force system, there will no moments acting on the body though. For this reason the moment equations will not be helpful for particles (the equation would end up being zero is equal to zero). Therefore in particle analysis will rely solely on the sum of the forces being equal to zero. When we start analyzing extended bodies with non-concurrent forces we will begin to encounter moments, and we will need to ensure the the sum of the moments as well as the sum of the forces are equal to zero.

Worked Problems:

Question 1:

The diagram below shows a 3lb box (Box A) sitting on top of a 5lb box (box B). Determine the magnitude and direction of all the forces acting on box B.

Problem 1 Diagram

Solution:



Question 2:

A 600 lb barrel rests in a trough as shown below. The barrel is supported by two normal forces (F2 and F3). Determine the magnitude of both of these normal forces.

Problem 2 Diagram

Solution:



Question 3:

A 6 kg traffic light is supported by two cables as shown below. Find the tension in each of the cables supporting the traffic light.

Problem 3 Diagram

Solution:



Question 4:

A 400 kg wrecking ball rests against a surface as shown below. Assuming the wrecking ball is currently in equilibrium, determine the tension force in the cable supporting the wrecking ball and the normal force that exists between the wrecking ball and the surface.

Problem 4 Diagram

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Question 5:

Barrels A and B are supported in a foot truck as seen below. Assuming the barrels are in equilibrium, determine all forces acting on barrel B.

Problem 5 Diagram

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Question 6:

Three soda cans, each weighing .75 lbs and having a dimeter of 4 inches, are stacked in a formation as shown below. Assuming no friction forces, determine the normal forces acting on can B.

Problem 6 Diagram

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Question 7:

The skycam shown below is supported by three cables. Assuming the skycam has a mass of 20kg and that it is currently in equilibrium find the tension in each of the three cables supporting the skycam.

Problem 7 Diagram

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Question 8:

A hot air balloon is tethered to the ground with three cables as shown below. If the balloon is pulling upwards with a force of 900lbs, what is the tension in each of the three cables?

Problem 8 Diagram

Solution: