For Statics:Simple Systems of Equations to Solve Equilibrium Problems

[jklops686]

Homework Statement

I have a picture attached with this static equilibrium problem.
(Solve system for Fac and Fab)

Homework Equations

Sum of x and y both=0

The Attempt at a Solution

I just don't have a systematic easy approach to solving these systems. Does anyone have one? From the picture, they used the system of equations and add them together cancelling variables, but I don't know how exactly they did that. This class would be so much easier if I could do this well!

 

[Simon Bridge]

I just don't have a systematic easy approach to solving these systems.

Nobody does. That is because this sort of problem can be arbitrarily complicated.
Instead you have to use your understanding of physics and resolve all the forces into appropriate components.

The heart of the matter is that the sum of all the vectors has to be zero.
Sometimes there is a shortcut using your knowledge of geometry - for instance, three forces have to form a triangle (if they are to sum to zero) and you know about the geometry of a triangle. This one can be solved using the sine rule for arbitrary triangles. However, the general approach to these problems has been demonstrated in the model answer.

Of course it helps to have some idea of which way the forces actually point - in fig2, for eg, the only way those forces cancel is if one of them points the opposite way... and sure enough...

 

[jklops686]

That does make sense, thanks for the reply. I understand how to set most of the problems up and I'm great with visualization and geometry it's just once they are set up, then I am having troubles solving for the unknowns. My algebra is gone out the window or something. From the picture, they used the system of equations and add them together cancelling variables, but I don't know how exactly they did that.

 

[Simon Bridge]

You don't always have to use the setup that is provided for you.

Do you understand how the simultanious equations were obtained?

 

[jklops686]

You don't always have to use the setup that is provided for you.

Do you understand how the simultanious equations were obtained?

I'm not sure what other way i'd use. It seems easy but I just can't remember how it's solved. And Yes I understand that the equations were obtained by adding up the x force components and then the same for the y components and setting them equal to zero.

 

[Ray Vickson]

Homework Statement

I have a picture attached with this static equilibrium problem.
(Solve system for Fac and Fab)

Homework Equations

Sum of x and y both=0

The Attempt at a Solution

I just don't have a systematic easy approach to solving these systems. Does anyone have one? From the picture, they used the system of equations and add them together cancelling variables, but I don't know how exactly they did that. This class would be so much easier if I could do this well!

There is a very standard way to solve such problems. It is not always the fastest way, but it is one that works every time. You just use one of the equations to solve for one variable in terms of the other; then you put that expression into the other equation.

In your case you have two equations of the form a*x + b*y = 0, c*x + d*y = e. The unknowns are x and y, and the given parameters are a, b, c, d, e. You can solve for x in terms of y from the first equation: a*x + b*y = 0 ==> x = -(b/a)*y (since a ≠ 0). Now, where you see x in the second equation, just substitute in that expression, so the second equation now becomes c*(-b/a)*y + d*y = e, or [d - (b*c/a)]*y = e. If d - (b*c/a) ≠ 0 we can solve for y by division: y = e/[d - (b*c/a)]. Once we know y, we can substitute that into our x-expression to find x.

Note that if d - (b*c/a) = 0 we will have an equation of the form 0*y = e. If e ≠ 0 this is an impossible equation, which indicates that we must have made an error in setting up the problem (or else were handed an inconsistent system by somebody evil). On the other hand, if e = 0 we have the equation 0*y = 0, which allows any value whatsoever for y.

Later, if and when you take linear algebra you will learn such methods as "row reduction", etc., but they are really nothing more than the above method made more streamlined.

 

[Simon Bridge]

I'm not sure what other way i'd use. It seems easy but I just can't remember how it's solved. And Yes I understand that the equations were obtained by adding up the x force components and then the same for the y components and setting them equal to zero.

So it is solving the simultanious equations that gives you trouble?

Ray has described the brute-force approach.
There are also a bunch of tricks to use for shortcuts ... with practise you get used to it.