Moments, Levers and Gears

(Separate Physics only) Moments, levers and gears are assessed in AQA GCSE Physics only. They are not in the Combined Science specification.

When a force acts on an object that can rotate about a fixed point, it can make that object turn. This turning effect is called the moment of the force. Pushing a door near its handle swings it easily; pushing near the hinge barely moves it, even with the same force.

The pivot (or fulcrum) is the fixed point the object turns about. The size of the turning effect depends on two things: how large the force is, and how far from the pivot it acts.

The moment of a force is the turning effect it produces about a pivot.

You meet moments every day: turning a spanner, opening a door, pressing a seesaw, or squeezing a pair of scissors. In each case a force at a distance from a pivot produces rotation. The next slide gives the equation that quantifies this.

The moment of a force is calculated from the force and the perpendicular distance from the pivot to the line of action of the force.

$M=Fd$

where $M$ is the moment in newton metres (Nm), $F$ is the force in newtons (N), and $d$ is the perpendicular distance from the pivot to the line of action of the force in metres (m).

You must recall and apply this equation. $M=Fd$ is not given on the equation sheet.

The word perpendicular is critical. The distance $d$ is measured at right angles to the direction of the force, from the pivot to the line along which the force acts. If the force is not at right angles to the arm, you cannot simply use the length of the arm.

Worked example — a mechanic pushes down with a force of 40 N on the end of a spanner, 0.25 m from the centre of the bolt. Find the moment.

$M=Fd=40\times 0.25=10\text{ Nm}$

The turning effect on the bolt is 10 Nm. Using a longer spanner, or pushing harder, would increase this moment.

When an object is balanced (in equilibrium and not turning) about a pivot, the turning effects on each side cancel out.

For a balanced object, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot.

This is the principle of moments. It lets you find an unknown force or distance whenever a beam, seesaw or lever is balanced.

Worked example — a uniform beam balances on a central pivot. A 6 N weight hangs 0.30 m to the left of the pivot. A weight $W$ hangs 0.20 m to the right and the beam is balanced. Find $W$.

Anticlockwise moment (left side): $6\times 0.30=1.8$ Nm

Clockwise moment (right side): $W\times 0.20$

Setting them equal:

$W\times 0.20=1.8$ $W=\frac{1.8}{0.20}=9\text{ N}$

The unknown weight is 9 N. The closer a weight is to the pivot, the larger it must be to balance a given moment on the other side.

A lever is a rigid bar that turns about a pivot. Levers let a small effort force move or lift a large load by increasing the distance at which the effort acts.

Because the moment depends on distance as well as force, applying the effort far from the pivot creates a large moment. That moment can balance the moment of a much larger load acting close to the pivot.

A lever acts as a force multiplier: a smaller effort force at a larger distance produces the same moment as a larger load force at a smaller distance.

Worked example — a crowbar pivots 0.10 m from a rock (the load) that needs a 300 N force to shift. The person pushes 0.60 m from the pivot. Find the effort needed.

Load moment (to be overcome): $300\times 0.10=30$ Nm

For the crowbar to just balance this, the effort moment must equal it:

$\text{effort}\times 0.60=30$ $\text{effort}=\frac{30}{0.60}=50\text{ N}$

An effort of only 50 N balances a 300 N load, because the effort acts six times further from the pivot. Everyday levers include scissors, spanners, wheelbarrows and bottle openers.

Gears are toothed wheels that mesh together so that turning one turns the other. They transmit the rotational effect of a force from one part of a machine to another and change the size of the moment.

When two gears mesh, the teeth push on each other with the same force at the edge of each wheel. Because the moment is force times distance from the axle, the larger gear (bigger radius) produces a larger moment than the smaller gear.

A gear with a larger radius turns more slowly but produces a greater moment than a smaller gear meshed with it. A smaller gear turns faster but produces a smaller moment.

This is why low gears on a bicycle (a large rear sprocket) give a big turning effect for climbing hills, while high gears (a small rear sprocket) turn the wheel faster for speed on the flat.

Gear	Radius	Turning speed	Moment produced
Small (few teeth)	small	faster	smaller
Large (many teeth)	large	slower	larger

Two meshed gears also turn in opposite directions: if the driving gear turns clockwise, the driven gear turns anticlockwise.

1. Using the wrong distance in M = F d

The distance is the perpendicular distance from the pivot to the line of action of the force, not the length of the arm or the distance measured along a tilted bar. Always measure at right angles to the force.

2. Forgetting the unit is the newton metre

The moment is in newton metres (Nm) because it is newtons multiplied by metres. Do not write joules; a joule is also N m but is the unit of energy, not moment.

3. Mixing up clockwise and anticlockwise sides

Label each moment clearly as clockwise or anticlockwise before applying the principle of moments. On a balanced beam, one side turns clockwise and the other anticlockwise about the pivot.

4. Converting centimetres incorrectly

If a distance is given in centimetres, convert to metres before using $M=Fd$. For example 25 cm = 0.25 m. Leaving it in cm gives a moment 100 times too large.

5. Assuming a bigger gear always turns faster

A larger gear turns more slowly than a smaller gear it drives, but produces a larger moment. Speed and turning effect trade off against each other.