Teach Fzx Tuesday - EZr Equilibrium - Simultaneous Equations

We were always taught to solve all static equilibrium problems using one particular method.  They were to look like this:

But that's complicated and annoying and off-putting so my solution will look different while depending on the same principles.

We will be using the same method from my last post of a sample problem solved: Ezr Equilibrium - Sample Problem.  First sketch a good Free Body Diagram with the two tensions and the weight.

Then find the components of the triangle using the same method proposed in that post.  It's the exact same thing but it doesn't feel like it since the tensions aren't numbers, they are variables called T.  Find the values of the components by multiplying T by the decimal values of the sine and cosine of the angles.  Seriously, just find the values of the sin and cos of those angles and compare to the picture.

That completes the FBD.

After labeling the diagrams with all the known force values, we then gather the x-forces and then the y-forces in the black equations shown below.  The only force we know is the weight so algebra is the only option.

We have two equations with two unknowns so there are a few options.  We could graph the two lines and find the intersection.  That ordered pair representing the intersection represents both answers.

Or we could use the algebra method shown in the image above.  Often called the substitution method for solving a system of two equations we

• Solve one equation for one variable, 
• substitute it into the other equation, 
• solve  that equation for the unknown variables,
• and substitute back into the first equation to solve for that variable.

That part of the solution is in green and red.

My solution is less visually intimidating than the solution shown in the first picture above but it uses the same principles and gets the job done well.