2 / 5

Gear Ratios and Mechanical Advantage

24 min Engineering Math

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.

$GR = 60/20 = 3$. 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.

$MA = 2/0.5 = 4$.

$F_{\text{in}} = 200/4 = 50$ 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?

Stage 1: $45/15 = 3$. Stage 2: $40/10 = 4$.

Overall: $3 \times 4 = 12:1$.

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