use a vector triangle to represent coplanar forces in equilibrium

⚖️ Equilibrium of Forces – Vector Triangle

📌 Objective

Use a vector triangle to represent coplanar forces in equilibrium and solve for unknown magnitudes or directions.

🤔 What is Equilibrium?

When all forces acting on a body cancel out, the body remains at rest or moves at constant velocity.

Mathematically: \$\displaystyle \sum \vec{F} = 0\$.

For two forces: \$\vec{F}1 + \vec{F}2 = 0\$ → they are equal in magnitude and opposite in direction.

🔺 Vector Triangle for Three Forces

When three coplanar forces \$\vec{F}1\$, \$\vec{F}2\$, and \$\vec{F}_3\$ are in equilibrium, they can be represented as the sides of a closed triangle:

\$\displaystyle \vec{F}1 + \vec{F}2 + \vec{F}_3 = 0\$

Think of it as a “force chain” where each force is a side of the triangle, and the chain loops back to its starting point.

🛠️ How to Construct a Vector Triangle

  1. Draw the first force \$\vec{F}_1\$ as a directed arrow.

  2. From the head of \$\vec{F}1\$, draw the second force \$\vec{F}2\$.

  3. From the head of \$\vec{F}2\$, draw the third force \$\vec{F}3\$.

  4. If the head of \$\vec{F}3\$ meets the tail of \$\vec{F}1\$, the forces are in equilibrium.

  5. Use the law of sines or law of cosines to find unknown magnitudes or angles if necessary.

📐 Example Problem

Three forces act on a point: \$\vec{F}1 = 10\,\text{N}\$ at \$30^\circ\$ above the horizontal, \$\vec{F}2 = 15\,\text{N}\$ at \$120^\circ\$ above the horizontal, and \$\vec{F}3\$ is unknown. Find the magnitude and direction of \$\vec{F}3\$ if the forces are in equilibrium.

ForceMagnitude (N)Direction (°)\$F_x\$ (N)\$F_y\$ (N)
\$F_1\$1030\$10\cos30^\circ \approx 8.66\$\$10\sin30^\circ = 5\$
\$F_2\$15120\$15\cos120^\circ = -7.5\$\$15\sin120^\circ \approx 12.99\$
\$F_3\$??\$-(8.66-7.5) = -1.16\$\$-(5+12.99) = -17.99\$

From the components: \$F{3x} = -1.16\$ N, \$F{3y} = -17.99\$ N.

Magnitude: \$|F_3| = \sqrt{(-1.16)^2 + (-17.99)^2} \approx 18.0\$ N.

Direction: \$\theta = \tan^{-1}\!\left(\frac{-17.99}{-1.16}\right) \approx 88.5^\circ\$ below the horizontal (or \$271.5^\circ\$ from the positive x‑axis).

📝 Examination Tips

  • Always check the vector sum – if it equals zero, equilibrium is satisfied.

  • Use the law of sines when you know two angles and one side, or the law of cosines when you know two sides and the included angle.

  • When drawing the vector triangle, keep the head-to-tail rule to avoid mistakes.

  • Show all calculations and unit conversions clearly; partial credit is often awarded.

  • Remember that angles are measured from the positive x‑axis, counter‑clockwise being positive.

🔚 Summary

Equilibrium of coplanar forces can be visualised as a closed vector triangle. By constructing the triangle and applying trigonometric laws, you can find any missing force magnitude or direction. Keep your diagrams neat, check your signs carefully, and practice with different configurations to master the technique.

Good luck, and keep practising! 🎯