Centre of Gravity & The Principle of Moments

Step 1: Identify the Pivot: Choose a point to take moments about; often, choosing a point where an unknown force acts simplifies the calculation because that force's moment becomes zero ($d = 0$).

Step 2: Identify All Forces: List every force acting on the system, including external loads, support reactions, and the weight of the object itself acting at its centre of gravity.

Step 3: Determine Perpendicular Distances: Ensure the distance used is the shortest distance from the pivot to the line of action of the force. If the force is at an angle $\theta$, use $d \sin(\theta)$ or $d \cos(\theta)$ as appropriate.

Step 4: Apply the Principle: Set up the equation $\sum M_{clockwise} = \sum M_{anticlockwise}$ and solve for the unknown variable.

Stable Equilibrium: If an object is slightly displaced, its centre of gravity rises, and the resulting moment returns it to its original position. This is common in objects with a low CG and a wide base.

Unstable Equilibrium: A slight displacement causes the centre of gravity to fall, creating a moment that moves the object further away from its original position until it topples.

Neutral Equilibrium: When displaced, the height of the centre of gravity remains constant, and the object stays in its new position without any restoring or toppling moments (e.g., a rolling ball).

The 'Zero Distance' Trick: Always look for an unknown force you don't want to calculate and place your pivot there. Since the distance from the pivot is zero, that force is eliminated from your moment equation.

Unit Consistency: Check that all distances are in the same units (metres or centimetres) before calculating. Mixing units is a frequent cause of incorrect magnitudes.

Weight of the Beam: Never forget to include the weight of the object itself if it is described as 'uniform' or has a given mass. It must be placed exactly at the geometric centre of the object.

Sanity Check: If a heavy weight is close to the pivot, a smaller weight must be much further away to balance it. If your calculated distance is very small for a light weight, re-check your algebra.