Practice Exercise S-1.2.1
Central Force Systems - Second Basic Task: Equilibrium
Problem Statement
Determine graphically and analytically whether the depicted system of forces is in equilibrium.
Before You Embark on This Quest: A Pre-Mission Briefing
Before you dive into this adventure, let's do a quick check to see if you're ready for the challenge.
What topic are we dealing with here? Can you explain why?
All the lines of action of the attacking forces intersect at one point. So, we're dealing with a central force system.
If you're not quite familiar with central force systems, no worries! Here are some refresher courses to get you up to speed:
Good luck!
Graphical Solution
Alright, let's figure out if the depicted force system is in equilibrium.
You're in luck! The system's diagram is already provided. You can see where all the forces are acting, but there's no physical object. Instead, they all act at the origin of an x,y coordinate system. It's like a meeting place for the forces!
Now it's getting exciting: Create a force diagram that tells us more about the forces.
- Choose a scale that fits the given forces. \(1~\mathrm{LU}~\widehat{=}~ 1~\mathrm{N}\) might work.
-
Convert the magnitudes of the forces into lengths. This will give us a feel for their strength.
$$ \begin{alignat}{3} \tag{1} F_1 &= 12~\mathrm{N}~ &&\widehat{=}~ 12~\mathrm{LU} \\[7pt] \tag{2} F_2 &= 9~\mathrm{N}~ &&\widehat{=}~ 9~\mathrm{LU} \\[7pt] \tag{3} F_3 &= 3~\mathrm{N}~ &&\widehat{=}~ 3~\mathrm{LU} \\[7pt] \tag{4} F_4 &= 7~\mathrm{N}~ &&\widehat{=}~ 7~\mathrm{LU} \\[7pt] \tag{5} F_5 &= 5~\mathrm{N}~ &&\widehat{=}~ 5~\mathrm{LU} \\[7pt] \tag{6} F_6 &= 6~\mathrm{N}~ &&\widehat{=}~ 6~\mathrm{LU} \end{alignat} $$
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Analytical Solution
To solve this task computationally, you need to decompose all forces into their horizontal (\(F_{i_x}\)) and vertical (\(F_{i_y}\)) components. To do this, you need the magnitude and angle (\(\alpha_i\)) for each force in the central force system. This can be the direction angle of the vector with respect to the positive x-axis. A better option would be one of the acute angles of the force triangle formed by the force vector and its components, e.g., \(\alpha_{i_{acute}}\).
You're in luck! The system's layout is already given to you with the problem statement. And with it, the required magnitudes and angles of the individual forces:
Given:
To decompose a force into its components, we use the Axiom of the Parallelogram of Forces. In this case, we are given the diagonal of the parallelogram of forces for each force and have to determine the two forces that form the sides of the parallelogram.
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