1. What is a Moment?

According to the "697324-2026-2028-syllabus (3).pdf", you must be able to describe the moment of a force as a measure of its turning effect. When a force is applied to an object that is hinged or pivoted, it causes the object to rotate about that point. This rotational effect is called a moment.

Everyday examples include pushing open a door (the hinges act as the pivot), using a spanner to loosen a nut, or two children playing on a seesaw.

The quantitative definition is strictly defined as:

$$ \text{Moment} = \text{Force (F)} \times \text{perpendicular distance from pivot (d)} $$

The standard SI unit for a moment is the Newton-metre (Nm). Since force is a vector, a moment also has a directional property related to rotation: it is categorized as either Clockwise or Anticlockwise.

Diagram 1.0: Forces on opposite diagonals create opposite turning effects. Forces acting perpendicular to the Clockwise Diagonal result in a clockwise turn, while forces acting perpendicular to the Anticlockwise Diagonal result in an anticlockwise turn.

2. The Principle of Moments

When an object is perfectly balanced and not rotating, we say it is in equilibrium. The syllabus dictates you must "State and use the principle of moments for an object in equilibrium".

The Principle of Moments states: For an object in equilibrium, the total clockwise moment about any pivot equals the total anticlockwise moment about that same pivot.

$$ \sum \text{Clockwise Moments} = \sum \text{Anticlockwise Moments} $$

3. Centre of Gravity (Section 1.5.6)

The centre of gravity is defined as the point through which the entire weight of an object is considered to act. For a uniform object (like a perfectly straight wooden ruler), this point is at its exact geometric centre.

To determine the centre of gravity of an irregularly shaped plane lamina (a flat piece of card), you use a plumb line. Hang the lamina freely from a pin through one edge. Suspend a plumb line (a weight on a string) from the same pin. Draw a vertical line along the string. Repeat this from two other points on the lamina. The single point where all three lines intersect is the centre of gravity.

Stability: An object will topple if the vertical line drawn downwards from its centre of gravity falls completely outside its base. A low centre of gravity and a wide base increase an object's stability, reducing the chance that its weight will create a turning effect (moment) that causes it to tip over.

Level: Easy (Calculating a single moment)

Question: A mechanic uses a wrench to tighten a bolt. He applies a force of $40\text{ N}$ at a perpendicular distance of $25\text{ cm}$ from the center of the bolt. Calculate the moment of the force.

Step 1: Convert distance to meters.
$d = 25\text{ cm} = 0.25\text{ m}$

Step 2: State the formula.
$M = F \times d$

Step 3: Substitute and solve.
$M = 40\text{ N} \times 0.25\text{ m} = 10\text{ Nm}$

Level: Medium (Principle of Moments)

Question: Two children sit on a uniform seesaw balanced at its center. Child A weighs $300\text{ N}$ and sits $1.5\text{ m}$ from the pivot. Child B weighs $450\text{ N}$. How far from the pivot must Child B sit for the seesaw to be in equilibrium?

Step 1: Identify the moments.
Anticlockwise moment (Child A) = $F_A \times d_A = 300 \times 1.5$
Clockwise moment (Child B) = $F_B \times d_B = 450 \times d_B$

Step 2: Apply the Principle of Moments.
$\text{Total Anticlockwise} = \text{Total Clockwise}$
$300 \times 1.5 = 450 \times d_B$

Step 3: Rearrange and solve.
$450 = 450 \times d_B$
$d_B = \frac{450}{450} = 1.0\text{ m}$

Level: Hard (Involving Weight of the Beam)

Question: A uniform metre rule is balanced at the $30\text{ cm}$ mark when a mass of $100\text{ g}$ is hung from the $5\text{ cm}$ mark. The acceleration of free fall $g = 9.8\text{ m/s}^2$ (as per syllabus section 1.2). Calculate the mass of the metre rule.

Step 1: Find the distances from the pivot (at 30 cm) for all forces.
Distance for hanging mass ($d_1$): $30\text{ cm} - 5\text{ cm} = 25\text{ cm} = 0.25\text{ m}$
Distance for rule's weight ($d_2$): $50\text{ cm} - 30\text{ cm} = 20\text{ cm} = 0.20\text{ m}$

Step 2: Calculate the force of the hanging mass.
Mass $= 100\text{ g} = 0.1\text{ kg}$
$W = m \times g = 0.1 \times 9.8 = 0.98\text{ N}$

Step 3: Set up the Principle of Moments equation.
The weight of the mass creates an anticlockwise moment. The weight of the rule ($W_{rule}$) creates a clockwise moment.
$0.98\text{ N} \times 0.25\text{ m} = W_{rule} \times 0.20\text{ m}$
$0.245 = W_{rule} \times 0.20$
$W_{rule} = \frac{0.245}{0.20} = 1.225\text{ N}$

Step 4: Find the mass of the rule.
$m = \frac{W}{g} = \frac{1.225}{9.8} = 0.125\text{ kg} = 125\text{ g}$