Training Engineering Math Gear Ratios and Mechanical Advantage
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Gear Ratios and Mechanical Advantage

24 min Engineering Math

Gear ratios and mechanical advantage are fundamental concepts that describe how simple machines trade force for distance or torque for speed. A gear train transmits rotational motion between shafts, and the gear ratio — determined by the number of teeth on each gear — dictates how much the output speed and torque change relative to the input. These principles apply to everything from bicycle drivetrains and automotive transmissions to robotic actuators and industrial conveyor systems. Understanding mechanical advantage also extends to levers, pulleys, and inclined planes.

Gear Ratios

Every transmission, clock, and machine tool uses gear ratios — which are nothing more than proportions.

Gear Ratio

$$GR = \frac{N_{\text{driven}}}{N_{\text{driver}}} = \frac{\omega_{\text{in}}}{\omega_{\text{out}}}$$

More teeth on driven gear → slower output but higher torque.

Mechanical Advantage

$$MA = \frac{F_{\text{out}}}{F_{\text{in}}} = \frac{d_{\text{in}}}{d_{\text{out}}}$$

For levers, pulleys, and inclined planes.

Example 1

A driver gear has 20 teeth, driven has 60 teeth. Input RPM = 300. Find output RPM.

  1. $GR = 60/20 = 3$.
  2. Output RPM $= 300/3 = 100$ RPM.
Example 2

A lever has effort arm 2 m and load arm 0.5 m. Find mechanical advantage and force needed to lift 200 N.

  1. $MA = 2/0.5 = 4$.
  2. $F_{\text{in}} = 200/4 = 50$ N.
Example 3

A gear train: gear A (15T) drives B (45T), B is on the same shaft as C (10T), C drives D (40T). Overall ratio?

  1. Stage 1: $45/15 = 3$.
  2. Stage 2:
  3. $40/10 = 4$.
  4. Overall: $3 \times 4 = 12:1$.

Practice Problems

1. Driver: 30T, driven: 90T. Find gear ratio.
2. Input 600 RPM through a 4:1 gear ratio. Output RPM?
3. Input torque 10 N·m through 5:1 ratio. Output torque?
4. A lever: effort arm 3 m, load arm 1 m. MA?
5. An inclined plane is 5 m long, 1 m high. MA?
6. Two-stage gear train: 2:1 then 3:1. Overall ratio?
Show Answer Key

1. 3:1

2. 150 RPM

3. 50 N·m

4. 3

5. 5

6. 6:1

⚙ Gear Ratio & Torque Calculator
Gear ratio
Output speed
Output torque
Direction
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