Question

Two gears are connected and are rotating simultaneously. The smaller gear has a radius of 2 inches, and the larger gear has a radius of 8 inches.
What it looks like: two circles touching at one point. Larger circle has radius of 8 inches. Smaller circle has radius of 2 inches.

Part 1: What is the angle measure, in degrees and rounded to the nearest tenth, through which the larger gear has rotated when the smaller gear has made one complete rotation? Part 2: How many rotations will the smaller gear make during one complete rotation of the larger gear? Show all work.

Answer 1

The two gears are shown n the diagram below.

ω₁ and ω₂ are the angular velocities of the larger and smaller gears respectively.

Part 1.
When the smaller gear makes one revolution, it turns through an angle of 2π radians or 360°.
Because the gears do not slip, the larger gear turns through an angle of θ, so that
 (θ radians)*(8 in) = (2π radians)*(2 in)
or
8θ = 4π
θ = π/2 radians = 90°

Answer: 90.0°

Part 2.
When the larger gear makes one revolution, it turns through an angle of 2π radians.
Because the gears do not slip, the smaller gear turns through an angle φ, such that
(2 in)*(φ radians) = (8 in)*(2π radians)
or
2φ = 16π
φ = 8π radians
    = (8π radians)*(1/2π rotations/radian)
    = 4 rotations

Answer:  4 rotations