Gear Ratios & Velocity Ratio Calculations

This lesson focuses on the mathematical calculations associated with gears and pulleys. Being able to calculate gear ratios, output speeds and velocity ratios is essential for AQA GCSE Design and Technology (8552), Section 3.1.5, and these calculations frequently appear on Paper 1.

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Gear Ratio (GR)

The gear ratio tells you how the speed and torque change between the driver (input) gear and the driven (output) gear.

Formula

$\text{Gear Ratio} = \frac{\text{Number of teeth on driven gear}}{\text{Number of teeth on driver gear}}$Gear Ratio=Number of teeth on driver gearNumber of teeth on driven gear​

Interpreting the Gear Ratio

Gear Ratio	Meaning	Effect
GR > 1	The driven gear has MORE teeth than the driver	Speed decreases, torque increases (gearing down)
GR < 1	The driven gear has FEWER teeth than the driver	Speed increases, torque decreases (gearing up)
GR = 1	Both gears have the same number of teeth	Speed and torque are unchanged (direction changes)

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Calculating Output Speed

Once you know the gear ratio, you can calculate the output speed:

$\text{Output speed (RPM)} = \frac{\text{Input speed (RPM)}}{\text{Gear Ratio}}$Output speed (RPM)=Gear RatioInput speed (RPM)​

Worked Example 1: Simple Gear Train

A motor drives a gear with 20 teeth at 600 RPM. This meshes with a gear with 60 teeth. Calculate the output speed.

Step 1: Calculate gear ratio

$GR = \frac{60}{20} = 3$GR=2060​=3

Step 2: Calculate output speed

$\text{Output speed} = \frac{600}{3} = 200 \text{ RPM}$Output speed=3600​=200 RPM

Answer: The output gear rotates at 200 RPM. The system has geared down — slower speed but higher torque.

Worked Example 2: Gearing Up

A motor drives a gear with 40 teeth at 300 RPM. This meshes with a gear with 10 teeth. Calculate the output speed.

$GR = \frac{10}{40} = 0.25$GR=4010​=0.25

$\text{Output speed} = \frac{300}{0.25} = 1200 \text{ RPM}$Output speed=0.25300​=1200 RPM

Answer: The output gear rotates at 1200 RPM. The system has geared up — faster speed but lower torque.

AQA Exam Tip: Always show each step of your working clearly. Even if you get the final answer wrong, you can gain marks for correct method (gear ratio formula, output speed formula).

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Compound Gear Ratios

For a compound gear train, multiply the individual gear ratios together:

$\text{Overall GR} = GR_1 \times GR_2 \times GR_3 \times \ldots$Overall GR=GR1​×GR2​×GR3​×…

The diagram below shows the flow of motion through a compound gear train. Gears B and C share the same shaft, so the slow speed at B is carried directly into C and reduced again by the C-to-D mesh.

graph LR
    Motor["Motor / Input<br/>1000 RPM"] --> A["Gear A (Driver)<br/>10 teeth"]
    A -->|"meshes with"| B["Gear B (Driven)<br/>40 teeth<br/>GR1 = 40/10 = 4<br/>250 RPM"]
    B -.->|"same shaft"| C["Gear C (Driver)<br/>10 teeth<br/>250 RPM"]
    C -->|"meshes with"| D["Gear D (Driven, Output)<br/>50 teeth<br/>GR2 = 50/10 = 5<br/>50 RPM"]
    D --> Out["Output Shaft<br/>Overall GR = 4 x 5 = 20<br/>Torque x 20"]

Worked Example 3: Compound Gear Train

Gear	Teeth	Notes
A (driver)	10	Input at 1000 RPM
B	40	Meshes with A
C (same shaft as B)	10	Meshes with D
D (output)	50	Output

Step 1: Individual gear ratios

$GR_1 = \frac{40}{10} = 4$GR1​=1040​=4

$GR_2 = \frac{50}{10} = 5$GR2​=1050​=5

Step 2: Overall gear ratio

$\text{Overall GR} = 4 \times 5 = 20$Overall GR=4×5=20

Step 3: Output speed

$\text{Output speed} = \frac{1000}{20} = 50 \text{ RPM}$Output speed=201000​=50 RPM

Answer: The output speed is 50 RPM — a significant reduction from the 1000 RPM input, with a corresponding increase in torque.

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Velocity Ratio (VR)

The velocity ratio (VR) is another way to express the relationship between input and output speeds. For gears, VR is the same as gear ratio.

$VR = \frac{\text{Distance moved by effort}}{\text{Distance moved by load}} = \frac{\text{Teeth on driven}}{\text{Teeth on driver}}$VR=Distance moved by loadDistance moved by effort​=Teeth on driverTeeth on driven​

For other mechanisms (pulleys, levers), VR is calculated differently but the principle is the same.

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Calculating Torque

Torque is the turning force applied to a shaft. When gears change speed, they also change torque.

$\text{Torque (Nm)} = \text{Force (N)} \times \text{Distance from axis (m)}$Torque (Nm)=Force (N)×Distance from axis (m)

Relationship Between Speed and Torque

$\text{Input torque} \times \text{Input speed} = \text{Output torque} \times \text{Output speed}$Input torque×Input speed=Output torque×Output speed

(Assuming 100% efficiency — no friction losses)

Worked Example 4: Torque Calculation

A motor produces 2 Nm of torque at 600 RPM. A gear ratio of 3:1 reduces the speed. What is the output torque? (Assume 100% efficiency)

$\text{Output speed} = \frac{600}{3} = 200 \text{ RPM}$Output speed=3600​=200 RPM

$2 \times 600 = \text{Output torque} \times 200$2×600=Output torque×200

$\text{Output torque} = \frac{2 \times 600}{200} = 6 \text{ Nm}$Output torque=2002×600​=6 Nm

Answer: The output torque is 6 Nm — three times the input torque, because the gear ratio is 3:1.

AQA Exam Tip: The relationship is: if speed is halved, torque is doubled (and vice versa), assuming no friction. This is a direct consequence of energy conservation and is a very commonly tested concept.

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Worm Gear Ratio Calculation

For a worm and worm wheel:

$GR = \frac{\text{Number of teeth on worm wheel}}{\text{Number of starts on worm}}$GR=Number of starts on wormNumber of teeth on worm wheel​

Most worms are single-start (one helical thread), so the gear ratio equals the number of teeth on the worm wheel.

Worked Example 5

A single-start worm drives a worm wheel with 30 teeth. The worm rotates at 600 RPM.

$GR = \frac{30}{1} = 30$GR=130​=30

$\text{Output speed} = \frac{600}{30} = 20 \text{ RPM}$Output speed=30600​=20 RPM

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Practice Problems

Try these yourself before checking the answers:

Problem 1: A driver gear has 15 teeth and rotates at 900 RPM. The driven gear has 45 teeth. What is the output speed?

Problem 2: A compound gear train has: A (12T) → B (48T), C (10T, same shaft as B) → D (60T). The input is 3600 RPM. What is the output speed?

Problem 3: A worm gear system has a single-start worm and a worm wheel with 50 teeth. If the worm is turned at 500 RPM, what is the output speed?

Answers

Problem 1: GR = 45/15 = 3. Output = 900/3 = 300 RPM

Problem 2: GR₁ = 48/12 = 4. GR₂ = 60/10 = 6. Overall GR = 4 × 6 = 24. Output = 3600/24 = 150 RPM

Problem 3: GR = 50/1 = 50. Output = 500/50 = 10 RPM

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Summary