Lesson 3

Free-Body Diagrams and the Equations of Equilibrium

🟦  3.1  Free-Body Diagrams (FBDs)

The free-body diagram or FBD is the most important concept in all of Statics.

We begin with the problem statement, or context. We will typically call this the loading diagram.

In order to create a FBD, start with the loading diagram, and then fully free the body from the context. Remove supports, connections, elements, and members until you have isolated whatever it is you want to study.

Everything that is removed (cut, disassembled, not shown, etc.) is replaced with its effect on the body.

FBDs are trickier than you think, and they are the foundation of this entire course, so inspect the flipbook very carefully. It contains 1 loading diagram (image 1) and 5 FBDs (images 2 through 6).

Pay special attention to slide 6, which is the FBD of the monkey. You'll notice that the normal force was not drawn coincident with the center (centroid) of the monkey's mass. It is offset a bit to the right. This is because the FBD of the monkey is not symmetric. Don't worry about this too much right now. The key point is that each vector must be drawn in the exactly correct location -- and it is not necessarily at the center (centroid) of the object.

In Physics, your professors would lump all of the mass of an object (a baseball, a planet, a box on a ramp, etc.) into a single point. Graphically they would draw that as a solid dot, and then apply forces to that dot. This may have been called "particle equilibrium." In Statics, only the simplest systems can be modeled as a particle. We have to draw the actual geometry of the object or assembly of objects.

🟦  3.2  Visual communication and drawing

Drawing as a mode of thinking

Statics is certainly based in principles from physics (mostly Newton's Laws), but its strongest connection may actually be to geometry. If you think back to your high school geometry class, you'll (hopefully) remember doing a lot of drawings.

In order to tackle Statics problems, we need to create good quality drawings. This takes time, but it is time well spent. The act of making good quality drawings, wherein the geometry and the vectors are drawn to scale, will help you think through the problem.

In Statics, the diagrams, figures, and plots are so important that many people value them more highly than a correct numeric answer.

❏ Some friendly advice

The more precisely you draw free-bodies, the more likely you'll be able to work problems quickly and accurately. I understand that some engineering students have internalized the perception that "they can't draw" or the stereotype that "engineers are people that are good at math, not drawing." 

I want to challenge you to keep a positive attitude about drawing. We can even rebrand it -- let's call it "visual communication."

You do not need to be an artist to succeed in Statics, but this course does present a valuable opportunity for honing your visual communication skills.

Please get in the habit of drawing the pictures/diagrams LARGER THAN YOU THINK IS NECESSARY. Your calculations will complement the information shown in the figure or diagram or plot.

🟦  3.3  The equations of equilibrium (E.o.E)

We can use a FBD to determine whether a body is in motion or in static equilibrium.

If there is a net force in the x-direction, the body will translate in the x-direction.

If there is a net force in the y-direction, the body will translate in the y-direction.

If there is a net moment, the body will rotate accordingly. 

If the sum of the forces is equal to zero and the sum of the moments is equal to zero, then the body is in static equilibrium.

When a body is in static equilibrium, we use the equations of equilibrium (E.o.E.) to solve for unknowns.

For the early part of this course, we will focus on Statics problems that can be modeled in 2D. We will tackle 3D after mastering 2D (or planar) models.

Here is how we could express the (2D) E.o.E. in casual conversation:

"The sum of the forces in the x-direction must equal zero. The sum of the forces in the y-direction must equal zero. The sum of moments about any z-axis must equal zero. That is the definition of static equilibrium."

Key points (for a problem that can be modeled in 2D):

🟦  3.4  A pile of books (a collinear force system)

In this flipbook, you'll see various FBDs of one or more books.

Since the stack of books are supported by a table, and therefore in static equilibrium, we can sum forces in the vertical direction in order to solve for unknown forces. Specifically, we are interested in solving for the normal forces transferred between the books. 

We will use one of the equations of equilibrium (the summation of forces in the vertical direction must equal zero) to solve for these unknowns.

Work this problem carefully, following each step from the flipbook, and drawing each FBD.

❏ Flipbook: A stack of books is supported by a table

I want to draw your attention to one important detail from the flipbook. You can only draw a vector on a FBD when the body causing that force is not depicted as part of the free-body. 

In other words, if you were asked to depict the normal force between books B and C and tried to draw that on a FBD that consisted of the entire stack of books (A, B, and C), it would be a concept error. In order to reveal the force transferred between bodies B and C, you must disassemble the stack at that interface to reveal the normal force between B and C.

This problem can be modeled as a collinear force system. That means that all of the forces in the system share the same line of action.

🟦  3.5  Two cables support a weight (a concurrent force system)

In this flipbook, you'll see an example of how we can use the moment equilibrium equation to solve for unknowns.

Remember that the moment equilibrium equation can be applied with respect to rotation about any axis. Be sure to apply the sign conventions we learned:

🟦  3.6  Qualitative equilibrium

This section will be added late Summer 2024.

➜ Practice Problems