Bevel gears worksheet: predict speed and direction around a corner

Turn the handwheel at the bottom of a tower and the sails at the top spin. Count the teeth and you know how fast. But which *way* they spin isn't set by the gears at all: it's set by which side of the corner each gear sits on. Move one gear across, and the sails reverse.

Bevel gears, or corner gears, have slanted teeth, so two of them can meet at a right angle and send turning round a corner. This worksheet practises the two things to predict about every corner: speed (count teeth, just like flat gears) and direction (look at where the gears sit). It follows on from the gear ratio worksheet, which uses the same tooth rule for flat gears, and the Bevel Gears page.

What you need

• A Bevel-Gear Windmill (level 3, about 25 minutes), built or ready to build. Its build page lists the Gear Speed Lab as the build to do first.
• Optional: an Egg Beater (level 2, about 20 minutes) and a Chair-Swing Carousel (level 2, about 35 minutes).
• Paper for a test record, and a sticky dot or tape flag on each axle so you can see turns.

Part 1: speed round a corner

The rule is the same as for flat gears: output turns = input turns × driver teeth ÷ driven teeth.

1. The windmill's bottom pair is two 12-tooth corner gears. One handwheel turn gives 1 × 12 ÷ 12 = 1 turn of the tower axle. Its top pair is also 12 to 12, so the sails turn once too.
2. The Egg Beater's handle carries a 20-tooth corner gear that drives a 12-tooth one: 20 ÷ 12 ≈ 1.67 whisk turns per handle turn. Two 16-tooth gears side by side then make the whisks spin in opposite directions at the same speed.
3. The Chair-Swing Carousel uses two stages. A 40-tooth gear drives an 8-tooth one (× 5), then a 12-tooth corner gear drives a 20-tooth one (× 0.6). Multiply the stages: 5 × 0.6 = 3 ride turns per crank turn. Its build file animates the crank at 10 turns a minute and the ride at 30.

Test it: mark the handwheel and the sails, turn the handwheel slowly five times and count. Write down your prediction first.

Part 2: direction round a corner

Here's where people get stuck. A corner pair turns the motion through a right angle, so "same way" or "opposite way" only makes sense once you say where you're looking from. Use one rule for the whole sheet: look along each axle from the front, or from above for an upright axle.

Now the key idea: a corner gear's partner pushes it on one edge. Move the gear to the other side of its partner and the teeth push it on the opposite edge, so it spins the other way at the same speed. Move *both* gears of a pair across and the two flips cancel.

Try it in the lab before you touch the real windmill. It starts as built.

Part 3: test the real windmill

1. Looking from the front, turn the handwheel anticlockwise. Which way do the sails turn? (Predict, then look. On the model as built they turn the opposite way.)
2. Look down the tower. Which way does the tower axle turn?
3. Take the sail axle's corner gear and move it to the other side of the tower axle, if your build has room. Predict again, then test.

Answers

• Part 1: 1 turn; about 1.67 turns; 3 turns.
• Part 3: as built, the sails turn the opposite way to the handwheel from the front. Moving the sail gear makes them turn the same way. Moving both top gears puts them back to opposite.

Where this matters

Put a corner gear on the wrong side and a machine fights itself. The four-wheel drive guide shows this in the Driveshaft 4×4: its build page tells you to fit the corner gears on opposite sides at the two ends, so both axles turn the same way. A differential is built from corner gears too.

Check yourself