Gears, Belts, and Rotational Motion

Gear questions are among the most common on the OAR MCT. The key is understanding direction, speed, and torque relationships.

Gear Basics

Direction of Rotation

Gear Train Direction

For meshing gears in a train:

• Odd number of gears (1, 3, 5...): First and last rotate in the same direction
• Even number of gears (2, 4, 6...): First and last rotate in opposite directions

Example: Gear A meshes with B, B meshes with C, C meshes with D.

• A and B: opposite
• B and C: opposite (so A and C: same)
• C and D: opposite (so A and D: opposite)

Four gears → first and last rotate in opposite directions.

Gear Ratios

Speed Ratio

Gear Ratio = Teeth on driven gear / Teeth on driving gear

Or equivalently: Speed₁ × Teeth₁ = Speed₂ × Teeth₂

Example: Gear A (driving) has 20 teeth. Gear B (driven) has 60 teeth.

• Gear ratio = 60/20 = 3:1
• Gear B rotates at 1/3 the speed of Gear A
• If A spins at 300 RPM, B spins at 100 RPM

Torque Ratio

Torque is the rotational equivalent of force. Gear systems trade speed for torque:

Speed × Torque = constant (in an ideal system)

Example: An engine gear (12 teeth) drives a wheel gear (48 teeth).

• Speed ratio: Wheel turns at 12/48 = 1/4 engine speed
• Torque multiplied by 4 (48/12 = 4)

This is exactly how a car transmission works: lower gears provide more torque (for acceleration), higher gears provide more speed (for cruising).

Compound Gear Trains

When multiple gear pairs are connected on shared shafts, multiply the individual ratios:

Overall Ratio = Ratio₁ × Ratio₂ × Ratio₃ × ...

Example:

• Stage 1: 10-tooth gear drives 40-tooth gear (ratio = 4:1)
• Stage 2: 10-tooth gear (on same shaft as the 40-tooth) drives 30-tooth gear (ratio = 3:1)
• Overall ratio = 4 × 3 = 12:1

The output shaft rotates at 1/12 the input speed, but with 12 times the torque.

Belt and Chain Drives

Speed Relationships

Speed₁ × Diameter₁ = Speed₂ × Diameter₂

(For chains: replace diameter with number of sprocket teeth)

Example: A motor pulley (diameter 10 cm) drives a machine pulley (diameter 30 cm) via a belt.

• 10 × Speed_motor = 30 × Speed_machine
• Speed_machine = 10/30 × Speed_motor = 1/3 of motor speed

Belt Types

Torque and Moment

Torque (τ) = Force × Distance from pivot (perpendicular distance)

τ = F × r

Example: A wrench handle is 0.3 m long. If you apply 100 N at the end, what torque do you generate? τ = 100 × 0.3 = 30 N·m

Longer Handle = More Torque

This is why a longer wrench makes loosening a bolt easier — you are increasing r, which increases torque with the same applied force.

Angular Velocity

ω (omega) = 2π × RPM / 60 (in radians per second)

Or simply use RPM (revolutions per minute) for comparison problems on the OAR.

Relationship: v = ωr (linear velocity = angular velocity × radius)

Gears of different sizes that mesh together have the same linear velocity at their contact point, but different angular velocities.