Moments, Levers and Gears

A moment (turning effect) is:

$M = F \, d$

• $F$ = force (N).
• $d$ = perpendicular distance from pivot (m).
• $M$ in N·m.

The perpendicular distance is the SHORTEST distance from the pivot to the LINE of the force — not the diagonal length of the lever.

Worked example. A 60 N push 0.4 m from a door's hinge. $M = 60 \times 0.4 = 24\,\text{N·m}$

Closer to the hinge = less turning effect (smaller d). That's why door handles are at the EDGE, not next to the hinge.

For an object in rotational equilibrium (not rotating):

$\sum M_{\text{clockwise}} = \sum M_{\text{anticlockwise}}$

Worked example (see-saw). A 30 kg child sits 2 m from the pivot. A 40 kg child wants to balance. How far from the pivot must they sit on the other side?

• Anticlockwise moment (30 kg): $W = mg = 30 \times 9.8 = 294\,\text{N}$; $M = 294 \times 2 = 588\,\text{N·m}$.
• Clockwise moment needed: 588 N·m.
• 40 kg weight: $W = 40 \times 9.8 = 392\,\text{N}$.
• Distance: $d = M/W = 588 / 392 = 1.5\,\text{m}$.

So the heavier child sits closer to the pivot to balance.

Levers. A lever consists of a rigid bar pivoted at a fulcrum. By applying a small force far from the pivot, you can balance a large force close to the pivot. Examples: crowbar, scissors, wheelbarrow, bottle opener.

The ratio is set by the distances: $F_{\text{load}} \times d_{\text{load}} = F_{\text{effort}} \times d_{\text{effort}}$

Gears. A gear is a wheel with teeth. When a small gear drives a larger one:

• The larger gear turns more SLOWLY but with more TORQUE (turning force).
• Used in bicycles, cars, wind turbines, drills.

Bicycle example. Low gear: small front cog driving large back cog → more torque (easier to pedal uphill) but slower.

Conservation of work. Levers and gears don't create energy. They trade force for distance (or torque for speed). Work in = work out (ignoring friction).