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Archive for March, 2013

The free body diagram

The free body diagram is a simple concept that most engineering students learn in their first year of college.  It’s a teaching tool, and the first step toward learning to analyze forces in a high rise building.  I’m going to get a little out of my water/drainage design wheelhouse here and go into an analytical tool used more for structural engineering.

Have I made it sound interesting enough that you’re willing to sit through this next part?  Because it really is an elegant concept.  I’m going to do my best to explain without getting deep enough to put anyone to sleep.  If you get bored, skip to the last two paragraphs. (Feel free to laugh at my MS Paint sourced drawings as well.  It was either that or hand drawn.)

A free body diagram is a conceptual drawing of a physical body with all the forces that are acting on it.  It’s isolated from anything in its environment that doesn’t cause a physical force on it.  The object is at equilibrium (either standing still or moving at a constant velocity).  It’s most widely used as a conceptual tool to teach students how to analyze forces, and generally you use known values to solve for unknown.

The classic teaching example is a block on a surface.

fbd1a
This version is simplistic enough to be of pretty much no use, but it does illustrate an important idea that most everyone instinctively knows but no one actually thinks about.  Gravity pulls your weight down (shown as W in the diagram) while the bulk of the earth holds you up by pushing back (shown as F1 on the diagram).  Since the block isn’t moving, F = W.

fbd2It starts to get more interesting when you incline the surface.  The block is still stationary, but it does introduce another force.  You still have W (weight of the object) and F1 (force holding it up), but now you also have a friction force (F2) keeping the block from sliding down the slope.  Friction is caused by two surfaces pushing against each other and it’s felt at the point where the block touches the surface.

This diagram is overly simplified, but still marginally useful as an example because it illustrates a couple of potential real world questions. (1) How much force do you need to apply to move the block up the incline (A) and (2) how much can you incline the surface before the object slides down it.

Solving either of those requires one last concept.  I’m going to quit for the day (or more likely quit for the week considering my posting frequency) once I get this one done, so stay with me just a little longer.

Notice the weight of the object is applied straight down, but the counteracting force is applied perpendicular to the surface.  Conceptually this means that only a portion of the entire weight of the block is trying to make it slide down the surface.  On a more practical level, we can’t solve the problem while the forces are going in so many directions.  The solution involves breaking the force down into its component vectors. *

* This concept of component vectors is the most basic to any engineering analysis, and it’s why the free body diagram is such a useful teaching tool.

Component vectors are essentially straight lines used to describe a diagonal.  The best example I’ve been able to come up with is directions on a city grid.  Let’s say we want to go from 1st and Broadway to 5th and Church.  As the crow flies that’s fairly straight forward.  But if you’re trying to tell your friend Maggie (who can’t fly) how to get there, you would tell her to go four blocks on Broadway, turn right, and go two blocks on 5th.

mapSo, back to our free body diagram.  W is ‘as the crow flies’ so we have to break it down into ‘four blocks on Broadway (Wx component) and four blocks on Church (Wy component)’.   I’m not going to get into how the sausage is made, suffice to say it involves principles of the geometry of right triangles that’s taught around the high school sophmore level.  I just want everyone to understand it has to be done.  Below is our revised free body diagram with all the forces on our grid system.

fbd3aThe take away here is this:

The weight acts in two directions.  It pushes back against the surface (Wy) and it wants to slide down the surface (Wx).  As you steepen the slope Wx goes up and Wy goes down.  (That’s pretty instinctive.  The steeper the slope, the more likely something is going to slide down it.)  Friction (F2) keeps the block from sliding down the slope meaning Wx is less than F2.  As you steepen the slope Wx gets larger while friction levels off.  When you get to where Wx is larger than friction, the block moves.

Now you have the same training as a freshman engineering student a month into her first Physics class.  From here on they start adding complications like springs, pulleys, diagonal forces, variable friction, and so on with the eventual goal of being able to analyze things like the forces at work in a high rise building frame.

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