Forces and equilibrium: vectors, resultants, equilibrium of a particle

We add vectors component‑wise:

$\displaystyle \mathbf{R} = \sum \mathbf{F}_i = \left(\sum F_{ix}\right)\mathbf{i} + \left(\sum F_{iy}\right)\mathbf{j}$

Example: Two forces – $4\text{ N}$ at $0^\circ$ and $3\text{ N}$ at $60^\circ$.

$F_{1x}=4,\;F_{1y}=0$
$F_{2x}=3\cos60^\circ=1.5,\;F_{2y}=3\sin60^\circ=2.60$
$\displaystyle R_x=5.5,\;R_y=2.60$
$R=\sqrt{5.5^2+2.60^2}\approx 6.02\text{ N}$ at $\theta=\tan^{-1}\frac{2.60}{5.5}\approx 25^\circ$

🔹 Equilibrium of a Particle – The “Balanced” State

A particle is in equilibrium when the sum of all forces acting on it is zero:

$\displaystyle \sum \mathbf{F} = \mathbf{0}$

This means the particle is either at rest or moving at a constant speed in a straight line.

**Example: Block on an Incline**

• Weight: $\mathbf{W}=mg$ acting vertically downwards.
• Normal force: $\mathbf{N}$ perpendicular to the surface.
• Friction (if present): $\mathbf{f}$ parallel to the surface.

Resolve $\mathbf{W}$ into components:

$W_{\parallel}=mg\sin\alpha$ (down the slope)
$W_{\perp}=mg\cos\alpha$ (into the slope)

For equilibrium:

$N = W_{\perp}$
$f = W_{\parallel}$ (if friction exactly balances the component down the slope)

📚 Examination Tips – Ace Your Paper!

Tip 1: Always draw a free‑body diagram before doing maths. It helps you see all forces and their directions.
Tip 2: Use component form for addition. It’s quicker and less error‑prone.
Tip 3: Check units – Newtons (N) for forces, meters (m) for distances.
Tip 4: For equilibrium problems, set up two equations: one for horizontal, one for vertical forces.
Tip 5: Practice with different angles; the sine and cosine values are your best friends.
Tip 6: Remember: “Sum of forces = 0” is the key phrase for equilibrium.
Tip 7: Use colour or symbols to keep track of each force in your diagram.
Tip 8: Time‑management: Spend 5 min on diagram, 10 min on calculations, 5 min on checking.
Tip 9: For multiple-choice, eliminate the obviously wrong answers first.
Tip 10: Stay calm – you’ve got this! 🚀

📊 Quick Reference Table – Vector Addition