Gears & Gear Trains
Trace rotation, compare gear speeds, use tooth ratios, and solve multi-gear systems without getting lost in the diagram.
Gears & Gear Trains
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Read the contact before you calculate
Gear questions usually test two separate ideas: direction and speed. Solve those separately. A large gear is not automatically the driver, and a small gear is not automatically faster unless the gears are actually connected in a way that transmits motion.
1. Meshed gears reverse direction
Two external gears touching tooth-to-tooth rotate in opposite directions.
2. Trace one contact at a time
A→B reverses once. B→C reverses again. So A and C rotate in the same direction.
3. Smaller gear = greater angular speed
If a 20-tooth gear meshes with a 40-tooth gear, the 20-tooth gear makes twice as many revolutions in the same time.
4. Tooth count and rpm trade off
teeth₁ × rpm₁ = teeth₂ × rpm₂For a simple pair of meshed gears, doubling the tooth count halves the rotational speed.
5. An idler does not change the final ratio
The middle gear changes the direction relationship and spacing. The overall speed ratio still depends on the first and last gears.
6. Same shaft means same rpm
If two gears are rigidly attached to the same axle, they rotate together at the same angular speed and in the same direction, even if their sizes differ.
Use TRACE → COMPARE → RATIO
TRACE: Mark the direction at every gear contact. Each external-gear mesh reverses direction.
COMPARE: Decide whether the question only asks which gear is faster/slower. Smaller meshed gear → faster rotation.
RATIO: Only calculate when numbers are needed. Use N₁n₁ = N₂n₂, where N is tooth count and n is rpm.
Two fast checks
Direction check: an even number of external-gear contacts gives the same final direction; an odd number gives the opposite direction.
Magnitude check: if the driven gear has more teeth than the driver, your calculated driven rpm should be smaller.
Five forms you should recognize
Problem: A turns clockwise and meshes directly with B.
Reasoning: One contact = one reversal.
Answer: B turns counterclockwise.
Problem: A–B–C–D–E form a line. A turns clockwise.
Trace: A ↻ → B ↺ → C ↻ → D ↺ → E ↻.
Answer: B and D are the counterclockwise gears.
Problem: A 15-tooth gear meshes with a 45-tooth gear.
Reasoning: The smaller gear has one-third as many teeth, so it turns three times as fast.
Answer: The 15-tooth gear has three times the rpm of the 45-tooth gear.
Problem: A 24-tooth driver turns at 150 rpm and drives a 60-tooth gear.
24(150) = 60n → n = 60 rpmCheck: The driven gear is larger, so 60 rpm being less than 150 rpm makes sense.
Problem: A 20-tooth gear drives a 30-tooth idler, which drives a 50-tooth output. A turns clockwise at 200 rpm.
Direction: Two contacts → output turns clockwise.
20(200) = 50n → n = 80 rpmThe idler's 30 teeth do not appear in the final speed ratio.
Check before you commit
- Assuming all gears in a train alternate speed by the same amount
- Using gear size to decide direction
- Forgetting that every external-gear contact reverses direction
- Multiplying rpm by tooth count in the wrong direction and making a larger driven gear faster
- Including a simple idler in the final driver-to-output ratio
- Assuming two gears on the same shaft have different rpm because they have different sizes
- Using the simple idler shortcut on a compound train where two gears share a shaft
- Answering a direction-only question with unnecessary calculations
Do you need the lesson—or just practice?
One original question in each form checks direction, gear trains, relative speed, ratios, and transfer. It recommends your next step but does not yet verify mastery.
Work at the level you need.
Foundations
Build direction, reversal counting, size-speed intuition, and idler basics.
Core Practice
Mix direction with tooth counts, rpm, and longer gear trains.
DOST-Style Transfer
Apply the rules to racks, shared shafts, compound trains, and unfamiliar setups.
Ready to verify this competency?
A score of 5/5 verifies mastery. An unsuccessful attempt loads a different five-question bank.
Gears & Gear Trains FAQ
Do bigger gears always turn more slowly?
Only compare speeds after identifying how the gears are connected. Two gears rigidly fixed to the same shaft have the same rpm. Two gears meshing at their teeth have speeds inversely related to tooth count.
Does an idler gear change the gear ratio?
In a simple train, no. It changes direction transmission and allows spacing between the driver and output. The overall speed ratio depends on the first driver and final driven gear.
How can I tell the final direction quickly?
Count external-gear contacts from the starting gear to the target. Even number of contacts → same direction. Odd number → opposite direction.
What if two gears are on the same shaft?
Treat them as one rotating unit for rpm and direction. Their different tooth counts matter only when each meshes with another gear.
Do I need to memorize many formulas?
No. For the gear questions in this reviewer, the main numerical relationship is N₁n₁ = N₂n₂. Most errors come from direction tracing or from misunderstanding the connection, not from algebra.
Continue your Mechanical Reasoning review.
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