The big idea: Velocity ratio VR = distance the effort moves ÷ distance the load moves.
For a rotating drive it is the same as input speed ÷ output speed, and it is fixed entirely by the geometry — the tooth counts or the diameters — with no reference to any force at all.
Velocity ratio worked on gears, pulleys and levers, beside mechanical advantage on the same drawings.
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| System | VR = | Worked example |
|---|---|---|
| Gear pair | Teeth on driven ÷ teeth on driver | 12 into 48 gives VR = 4, so 1,200 rpm in gives 300 rpm out |
| Belt or pulley drive | Driven diameter ÷ driver diameter | 40 mm into 160 mm gives VR = 4, so 2,800 rpm in gives 700 rpm out |
| Lever | Effort distance ÷ load distance, which equals effort arm ÷ load arm | Effort 750 mm out, load 50 mm out: VR = 15, and the effort end moves fifteen times as far |
| Compound train | The ratios of the stages multiplied together | 3:1 then 5:1 gives VR = 15, so 1,500 rpm in gives 100 rpm out |
Output speed = input speed ÷ VR: That is the rearrangement an exam asks for most often.
And the sense check is simple: a VR above 1 means the output is slower than the input. If your answer came out faster, the ratio went in upside down.
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Working the speed at every point
Take one mesh at a time
Work out the speed after the first pair, then use that as the input to the next. Doing the whole train in one step is where errors hide.
Two gears on one shaft turn together
In a compound train, the gear that was driven and the small gear beside it share a shaft, so they run at the same speed — that is what links the stages.
An idler changes nothing
Its tooth count cancels out of the overall ratio. It reverses the direction and bridges a gap, and the speed is set by the first and last gears only.
Check the direction at each mesh
Every external mesh reverses the rotation, so an even number of meshes gives the same direction as the input and an odd number reverses it.
A worked train: A 15-tooth driver meshes with a 45-tooth gear, which shares a shaft with a 12-tooth gear driving a 60-tooth gear. Input 1,200 rpm.
Stage 1: 45 ÷ 15 = 3, so the middle shaft runs at 1,200 ÷ 3 = 400 rpm. Stage 2: 60 ÷ 12 = 5, so the output runs at 400 ÷ 5 = 80 rpm. Overall VR = 3 × 5 = 15, and 1,200 ÷ 15 = 80 rpm — which is the check.
How this is tested — calculating velocity ratios for gear-, pulley- and belt-driven systems. It comes up two ways:
Paper 1 — multiple choice
- Calculate a velocity ratio from tooth counts or diameters.
- Find an output speed from an input speed and a ratio.
Paper 2 — analysing a product
- Calculate the speed at several points in a gear train.
- Explain what a velocity ratio means for a named product.
The trap: Confusing VR with MA. VR is a ratio of distances or speeds and comes purely from the geometry; MA is a ratio of forces and is reduced by friction.
A cordless drill motor runs at 12,000 rpm. Its gearbox has a 10-tooth driver into a 40-tooth gear, which shares a shaft with a 12-tooth gear driving a 48-tooth gear. Apply velocity ratio to find the chuck speed and comment on it.
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