While it's quite straightforward to solve problems with multiple planar vectors in equilibrium for 2 variables, I'm having issues with those where I'm asked to minimize or maximize a variable along with two others. For example:

All elements are connected at O We are asked to find:

  • $\theta$ such that $F_1$ is as small as possible
  • $F_1$ and $F_2$ such that our resulting vector $F_R = (3 kN, \angle 0°)$

I've gone ahead and established the equilibrium for the $x$ and $y$ axes but I'm at a loss as to how to continue from there, because I don't know $F_2$ to solve for $\theta$, and because I'm assuming this can be done without calculus, although I might be wrong.

$$\sum F_x=0=-5\cos(30°)+F_2 \cos(20°)+F_1 \cos(\theta) -7\cdot4/5$$ $$\sum F_y=0=5\sin(30°)+F_2 \sin(20°)+F_1 \sin(\theta) -7\cdot3/5$$

  • $\begingroup$ Could elaborate more on the question? $\endgroup$ – Manvendra Somvanshi Apr 29 at 0:08
  • $\begingroup$ How do you mean? $\endgroup$ – Ariel Arévalo Apr 29 at 3:48
  • $\begingroup$ You have a third constraint, don't forget $F_R$ in your analysis, this will help with the relationship between $F_1$ and $F_2$. $\endgroup$ – Triatticus Apr 29 at 8:16
  • $\begingroup$ That helps me with the second question but what about minimizing $F_1$? $\endgroup$ – Ariel Arévalo Apr 30 at 1:21

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