state and apply the principle of moments

Equilibrium of Forces – Principle of Moments

Analogy: The Balance Scale ⚖️

Imagine a balance scale with two plates. If the weights on both sides are equal, the scale stays level. The forces (weights) produce equal and opposite moments about the fulcrum, keeping the system in equilibrium.

Key Formula

Step‑by‑Step Application

1. Draw a clear diagram with the pivot point marked.
2. Identify all forces acting on the system.
3. Choose a pivot point (any point works, but pick one that simplifies calculations).
4. For each force, calculate its moment: $M = F \times d$ (use the perpendicular distance).
5. Assign a sign: moments that tend to rotate the system clockwise are negative; counter‑clockwise are positive.
6. Sum all moments and set the algebraic sum equal to zero: $$\sum M = 0$$.
7. Solve for the unknown quantity (force, distance, or angle).

Example: Lever with Two Weights

🔧 A lever of length 1.2 m has a fulcrum 0.4 m from the left end. A 10 N weight hangs 0.8 m to the right of the fulcrum. Find the weight that must hang 0.2 m to the left to keep the lever horizontal.

Solution:

1. Choose the fulcrum as pivot.
2. Right side moment: $M_{\text{right}} = 10\,\text{N} \times 0.8\,\text{m} = 8\,\text{N·m}$ (counter‑clockwise).
3. Let $W$ be the left weight. Left side moment: $M_{\text{left}} = W \times 0.2\,\text{m}$ (clockwise).
4. Set $\sum M = 0$: $8\,\text{N·m} - W \times 0.2\,\text{m} = 0$.
5. Solve: $W = \dfrac{8}{0.2} = 40\,\text{N}$.

So a 40 N weight on the left keeps the lever balanced.