if the radius of the gear is 2 feet and the radius of the cog is 8 inches, what is the angular velocity of the cog (in radians per second)?

Question

this is a questions about a lighthouse.  the lighthouse is 175 feet tall, the beam of light shines up to 25 miles off the coast, and it rotates through one revolution per 15 seconds. it has a motor, that turns a cog, that drives a gear, that rests upon a set of chariot wheels.

 

there are three parts to this question, the next two are:

 

2.  if the radius of any given chariot wheel is 3 inches, how long does it take for it to complete one full revolution?

 

3.  suppose the motor turns the cog (2π)/3 radians.  assuming a clear sky (i.e., the beam of light will go the farthest possible: 25 miles), how many square miles are swept out by the resulting shift in the beams direction?

Answer 1

I'll work through all three parts of this lighthouse problem systematically.

Part 1: Angular velocity of the cog

Given information:

1. Gear radius: 2 feet = 24 inches
2. Cog radius: 8 inches
3. The lighthouse rotates through one revolution per 15 seconds

Since the gear rests on the chariot wheels and drives the lighthouse rotation, the gear completes one revolution in 15 seconds.

The gear's angular velocity: ωgear = 2π/15 radians per second

For meshing gears, the linear velocity at the point of contact is equal:

1. vgear = ωgear × rgear
2. vcog = ωcog × rcog

Since vgear = vcog: ωgear × rgear = ωcog × rcog

(2π/15) × 24 = ωcog × 8

ωcog = (2π × 24)/(15 × 8) = 48π/120 = 2π/5 radians per second

Part 2: Time for one chariot wheel revolution

The chariot wheels are in contact with the gear (radius 24 inches), and the gear rotates at 2π/15 radians per second.

Linear velocity of gear's edge: v = ωgear × rgear = (2π/15) × 24 = 48π/15 = 16π/5 inches per second

The chariot wheel (radius 3 inches) has the same linear velocity at contact: v = ωwheel × rwheel

16π/5 = ωwheel × 3

ωwheel = 16π/15 radians per second

Time for one revolution: T = 2π/ωwheel = 2π/(16π/15) = 2π × 15/(16π) = 15/8 seconds = 1.875 seconds

Part 3: Area swept by the beam

When the motor turns the cog (2π)/3 radians, we need to find how much the gear (and lighthouse) rotates.

Using the gear ratio from Part 1: θgear/θcog = rcog/rgear = 8/24 = 1/3

So: θgear = (1/3) × (2π/3) = 2π/9 radians

The beam sweeps out a circular sector with:

1. Radius: 25 miles
2. Central angle: 2π/9 radians

Area = (1/2)r²θ = (1/2)(25²)(2π/9) = (1/2)(625)(2π/9) = 625π/9 square miles

≈ 218.2 square miles