Simple Machines - Chapter 4 Complete Revision Guide

Definitions · symbols and units · formulas · derivations · diagrams · worked examples

Big idea: A machine does not create energy. It makes work convenient by multiplying force, changing the direction or point of application of force, or changing speed. Ideally, input work equals output work.

Jump to: Definitions · Units · Formula sheet · Levers · Derivations · Diagrams · Examples · Exam checklist

1. Essential Definitions

Parallel forces: Forces whose lines of action are parallel. They are like parallel forces when they act in the same direction and unlike parallel forces when they act in opposite directions.

Resultant: A single force having the same external effect as all the forces acting together.

Equilibrant: The force that balances the resultant. It has the same magnitude as the resultant, acts along the same line, but points in the opposite direction.

Simple machine: A device that makes work easier or more convenient by multiplying force, changing its direction or point of application, or changing speed. It cannot create work or energy.

Load (L): The resisting force overcome by a machine. Effort (E): The force applied to operate the machine.

Mechanical advantage: The number of times a machine multiplies effort: M.A. = Load/Effort. It has no unit.

Velocity ratio: Distance moved by effort divided by distance moved by load in the same time. It depends on the geometry of the machine and has no unit.

Efficiency: Useful output work divided by input work, expressed as a percentage. Real machines have efficiency below 100% because of friction and deformation.

Lever: A rigid straight or bent bar that can rotate about a fixed point called the fulcrum. The perpendicular distances from fulcrum to the lines of action of effort and load are the effort arm and load arm.

Inclined plane: A smooth rigid plane at an angle to the horizontal, used to raise a load with less force over a greater distance.

Moment of force / torque: The turning effect of a force about a point or axis: force × perpendicular distance from the pivot to the force's line of action.

Couple: Two equal, opposite, parallel and coplanar forces acting along different lines. Their net force is zero, but they produce pure rotation.

Equilibrium: A body is in equilibrium when the resultant force and resultant moment are both zero, so there is no change in translational or rotational motion.

Wheel and axle: A large wheel of radius R fixed to a smaller axle of radius r so both rotate together about one axis.

Screw jack: A screw-and-lever machine that lifts a large load. The pitch is the axial distance moved by the screw in one complete turn.

Gear: A toothed wheel that transmits rotation and power. The gear supplying motion is the driver; the gear receiving it is the driven gear.

2. Symbols, Quantities and Units

Quantity	Symbol	SI unit	Note
Load / effort / force	L, E, F	newton (N)	1 N = 1 kg m s^-2
Length, arm, radius, pitch	l, d, R, r, p	metre (m)	Convert cm to m for N m answers
Moment / torque / moment of couple	τ or M	newton metre (N m)	CGS unit: dyne cm; dimensions [ML²T^-2]
Work	W	joule (J)	1 J = 1 N m, but work and torque are different quantities
Mechanical advantage	M.A.	no unit	Ratio of two forces
Velocity ratio	V.R.	no unit	Ratio of two distances or speeds
Efficiency	η	no unit or %	0 < η ≤ 1, or 0-100%
Number of teeth	T or N	no unit	A count

Do not confuse N m with J: both have the same base units, but torque is a vector-like turning effect while work is scalar energy transfer. Write torque as N m, not J.

3. Complete Formula Sheet

General machine

M.A. = L/E
V.R. = distance by effort / distance by load
η = output work / input work
η = M.A./V.R.
Percentage efficiency = (M.A./V.R.) × 100

Resultant parallel forces

Like: R = A + B
Unlike: R = |A - B|
Position from moments:
A × distance from A = B × distance from B

Lever

Effort × effort arm = Load × load arm
M.A. = L/E = effort arm / load arm (ideal)

Inclined plane

M.A. = length/height = l/h = 1/sin θ (ideal)
E = Lh/l = L sin θ

Torque and couple

τ = Fd = Fr sin θ
d = perpendicular distance
Moment of couple = F × perpendicular separation

Equilibrium

ΣF = 0
Στ = 0
Clockwise moments = anticlockwise moments

Wheel and axle

M.A. = L/E = R/r (ideal)
One turn: effort moves 2πR; load moves 2πr

Screw jack

One turn: effort distance = 2πl
Load distance = pitch p
M.A. = 2πl/p (ideal)

Two meshing gears

Speed ratio: ω_driver/ω_driven = T_driven/T_driver
Adjacent external gears rotate in opposite directions

4. The Three Classes of Levers

Class	Middle part	Pattern	M.A.	Examples
Class I	Fulcrum	E-F-L	May be <1, =1 or >1	Seesaw, scissors, crowbar, pliers
Class II	Load	F-L-E	Always >1 ideally	Wheelbarrow, nutcracker, bottle opener
Class III	Effort	F-E-L	Always <1 ideally	Tweezers, fishing rod, human forearm, fire tongs

Memory trick: FRE 123 - class 1 has Fulcrum in the middle, class 2 has Resistance/load in the middle, and class 3 has Effort in the middle.

5. Key Derivations

1. General relation η = M.A./V.R.

Efficiency η = output work/input work = (L × distance moved by load)/(E × distance moved by effort). Rearrange: η = (L/E) ÷ (distance by effort/distance by load). Therefore η = M.A./V.R. For an ideal machine η = 1, so M.A. = V.R.

2. Law of the lever

In rotational equilibrium, clockwise moment equals anticlockwise moment. If effort and load act on opposite sides of the fulcrum, E × effort arm = L × load arm. Dividing by E × load arm gives L/E = effort arm/load arm. Thus a longer effort arm produces a larger ideal M.A.

3. Mechanical advantage of an inclined plane

Let slope length be l and vertical height be h. For a smooth ideal plane, input work = E × l and useful output work = L × h. Therefore El = Lh, so M.A. = L/E = l/h. Since sin θ = h/l, M.A. = 1/sin θ. A gentler, longer slope needs less effort but requires a longer travel distance.

4. Torque formula

Only the component of force perpendicular to the radius produces rotation. That component is F sin θ, so torque = radius × perpendicular component = τ = rF sin θ. Equivalently, if d is the perpendicular distance from the pivot to the line of action, τ = Fd. Torque is greatest when θ = 90°.

5. Position of a resultant of two like parallel forces

If A and B act at M and N and their resultant R acts at O between them, take moments about O. The resultant passes through O and has zero moment there. Hence A × MO = B × NO, while R = A + B. The stronger force lies closer to the resultant.

6. Wheel and axle

In one complete revolution, the effort at wheel radius R moves 2πR while the load at axle radius r moves 2πr. For an ideal machine, E(2πR) = L(2πr). Cancelling 2π gives L/E = R/r. Thus M.A. = R/r.

7. Screw jack

For one turn, effort at a handle of length l moves around a circle of circumference 2πl. The load rises by one pitch p. Ideally, E(2πl) = Lp. Therefore M.A. = L/E = 2πl/p. Increasing handle length or decreasing pitch increases mechanical advantage.

8. Gear speed relation

At the contact point, both gears have the same tangential speed. Since circumference and tooth count are proportional, revolutions are inversely proportional to tooth count. Thus ω_driver/ω_driven = T_driven/T_driver. A small driver turning a large driven gear reduces speed and increases torque; reversing their roles increases speed.

6. Visual Explanations

Lever and moments

Balance occurs when E × effort arm = L × load arm. Moving the effort farther from the fulcrum reduces the effort needed.

Inclined plane

For the same height h, a longer slope l gives larger M.A. = l/h and smaller ideal effort. You trade force for distance.

Couple

Net force is zero, but the turning effects add. Moment of couple = Fd. Examples: steering wheel, tap and key.

Meshing gears

External gears rotate in opposite directions. A smaller driver turning a larger driven gear provides speed reduction and torque multiplication.

7. Worked Examples

Example 1: Mechanical advantage

A machine of M.A. 5 raises a 25 N load. M.A. = L/E, so E = L/M.A. = 25/5 = 5 N.

Example 2: Lever

A 10 N effort acts 50 cm from a fulcrum; the load is 10 cm away. M.A. = 50/10 = 5. Load = M.A. × effort = 5 × 10 = 50 N.

Example 3: Nutcracker

A class-II nutcracker has effort arm 25 cm, load arm 8 cm and effort 32 N. From E × effort arm = L × load arm: L = 32 × 25/8 = 100 N.

Example 4: Inclined plane

A smooth plank is 4 m long and 1 m high. It raises a 48 N load. M.A. = l/h = 4/1 = 4. Effort E = L/M.A. = 48/4 = 12 N.

Example 5: Torque

An 8 N force acts perpendicularly 20 cm from a pivot. Convert 20 cm = 0.20 m. τ = Fd = 8 × 0.20 = 1.6 N m.

Example 6: Longer spanner

A 120 N force on a 0.40 m spanner gives τ = 48 N m. To get the same torque with 40 N, r = τ/F = 48/40 = 1.2 m. Tripling the arm reduces the required force to one-third.

Example 7: Moment of a couple

Two opposite 4 N forces act on a steering wheel 30 cm apart. d = 0.30 m. Moment of couple = Fd = 4 × 0.30 = 1.2 N m. Do not double it; Fd already equals the sum of the two moments.

Example 8: Wheel and axle

Wheel radius R = 35 cm and axle radius r = 7 cm. M.A. = R/r = 35/7 = 5. With effort 10 N, load L = 5 × 10 = 50 N.

Example 9: Screw jack

Handle length l = 0.35 m and pitch p = 5 mm = 0.005 m. Ideal M.A. = 2πl/p = 2π(0.35)/0.005 ≈ 440. Ideally, a 20 N effort could balance an 8800 N load; a real jack lifts less because η < 100%.

Example 10: Gears

An 11-tooth driver rotates at 120 rpm and drives a 22-tooth gear. n_driven = 120 × 11/22 = 60 rpm, in the opposite direction. Speed halves while ideal torque doubles.

Example 11: Efficiency

A machine has M.A. 5 and efficiency 80% = 0.80. Since η = M.A./V.R., V.R. = 5/0.80 = 6.25. If effort moves 20 cm, load distance = 20/6.25 = 3.2 cm.

Example 12: Resultant of like parallel forces

15 N and 5 N act upward 4 m apart. R = 20 N upward. If R is x from the 15 N force, take moments about that end: 20x = 5 × 4, so x = 1 m from the 15 N force. The resultant is closer to the larger force.

8. Exam Checklist and Common Mistakes

Mark F, L and E before deciding the class of a lever. Do not classify it by appearance.
Use the perpendicular distance to the line of action when calculating a moment.
Convert centimetres or millimetres to metres before giving torque in N m.
Clockwise and anticlockwise moments have opposite signs. In equilibrium their algebraic sum is zero.
A couple has zero resultant force but a non-zero turning effect.
M.A., V.R. and efficiency have no units. Convert percentage efficiency to a decimal in calculations.
For an ideal machine, input work = output work and M.A. = V.R. For a real machine, friction makes M.A. < V.R.
A machine never reduces total ideal work: a smaller effort must move through a greater distance.
On an inclined plane, l is the length along the slope; h is the vertical height.
For a screw jack, use pitch in metres and handle length from the screw axis to the point where effort acts.
Meshing external gears rotate in opposite directions. An idler gear changes direction but not the overall speed ratio.
For like parallel forces the resultant lies between them; for unlike forces it lies outside, on the side of the larger force.

Five-step numerical method: draw and label the machine → list known values with units → choose the formula → substitute after unit conversion → write the answer with unit and physical meaning.

Physics