Specific learning outcomes

• Determine the centre of gravity (C.G.) of regular and irregular objects.
• Identify the states of equilibrium: stable, unstable and neutral.
• Determine the moment of a force about a point.
• Verify the principle of moments when objects turn (balanced torques).
• Describe torque and a couple in turning objects.
• Appreciate everyday applications of moments and stability.

1. What is a moment (or torque)?

A moment (also called torque) is the turning effect of a force about a point (pivot). It depends on the size of the force and how far from the pivot the force acts.

Formula: Moment = Force × Perpendicular distance from pivot
M = F × d (units: newton metre, N·m)

2. Calculating moments — worked examples

1. Find the moment of a 20 N force applied 0.3 m from a pivot.
   M = F × d = 20 N × 0.3 m = 6 N·m
2. Seesaw balance: a 30 N child sits 0.5 m to the left of pivot. How far must a 20 N child sit to the right to balance?
   Principle of moments (balanced): clockwise moment = anticlockwise moment
   30 × 0.5 = 20 × x → 15 = 20x → x = 0.75 m

3. Centre of gravity (C.G.)

The centre of gravity of a body is the single point through which the weight of the body may be considered to act. For uniform thin flat shapes the C.G. can be found by symmetry or simple rules.

• Rod or rectangle (uniform): C.G. at the centre (midpoint).
• Triangle (uniform plate): C.G. at intersection of medians — 1/3 of height from base (or 2/3 from vertex) along a median.
• Semicircular plate: C.G. about 0.424 × radius from centre along axis (more advanced; not required to integrate at this level).

Irregular shapes: Use the suspension (plumb-line) method.

4. States of equilibrium

A body is in equilibrium when the resultant force and resultant moment on it are zero (no translation, no rotation). There are three types of equilibrium:

• Stable equilibrium: If the body is slightly displaced it returns to the original position. Condition: the C.G. rises when disturbed (or C.G. remains over base).
• Unstable equilibrium: A small displacement makes the body move away. Condition: C.G. falls when disturbed (or C.G. moves outside the base).
• Neutral equilibrium: A small displacement neither returns nor moves further — C.G. stays at same height (e.g., a perfect sphere on a flat surface).

5. Principle of moments (Law of the lever)

If a body is in rotational equilibrium (not turning), sum of clockwise moments about any point equals sum of anticlockwise moments about that point.

Mathematical statement: ΣM(clockwise) = ΣM(anticlockwise)

Simple classroom verification: Place a metre rule on a knife edge near centre, put a known weight on one side and move a hanging weight on the other side until balanced. Check that weight × distance (one side) = weight × distance (other side).

6. Couple

A couple is a pair of equal and opposite forces whose lines of action are separated by a perpendicular distance. A couple produces pure rotation (no net force).

Magnitude of the couple = one force × perpendicular distance between forces.

7. Everyday applications (Kenyan context)

• Seesaw (playground): balancing children illustrate principle of moments.
• Carrying jerrycans on one side: raises C.G. and can make you unstable; carrying weight close to body lowers moment and reduces strain.
• Using a spanner or crowbar: a longer spanner increases moment for the same force (makes turning nuts easier).
• Lifting a laden wheelbarrow: position of load relative to wheel (pivot) changes effort needed.
• Design of wide-base market stalls, tents (stability) to avoid tipping in winds.
• Tipper trucks and buses: high centre of gravity can cause roll-over on slopes.

8. Suggested learning experiences (practical activities)

1. Find C.G. of a metre rule: balance it on a knife edge (regular uniform object) — C.G. at 0.5 m from end.
2. Suspension method for irregular shapes: cut an irregular cardboard, hang from two different points, draw verticals and find C.G.
3. Verify principle of moments: use a metre rule, pivot near centre, place known weights and measure distances until balanced; calculate moments and compare.
4. Investigate stability: push blocks of different heights/bases slightly and observe return or tipping. Experiment with lowering C.G. by adding weight at the bottom.
5. Measure torque with a spring balance on a spanner: compare turning moment for different lever lengths.

9. Practice questions

1. A wrench 0.25 m long has a force of 80 N applied perpendicular to the handle. Find the moment produced.
2. A metre rule balances at 42 cm when a 0.2 kg mass is hung at 10 cm from left end. Where should a 0.15 kg mass be hung on the right to balance? (Take g same both sides. Useful: weight = mass × g — g cancels in moments.)
3. Describe how you would find the C.G. of an irregular cardboard shape using only pins and a piece of string.

Key points to remember

• Moment = force × perpendicular distance (M = Fd). Units: N·m.
• Balanced object: clockwise moments = anticlockwise moments.
• Couple: pair of equal and opposite forces separated by distance; produces pure rotation (M = F × d).
• Stability depends on position of the C.G. relative to base of support — lower C.G. and wider base increase stability.
• Use simple practicals (metre rule, suspension) to find C.G. and verify principle of moments.