In a gear train with driver 24 teeth and driven 12 teeth, what is the speed of the driven gear relative to the driver?

• A. Driven spins at the same speed as the driver.
• B. 2:1 (driven spins twice as fast).
• C. 1:2 (driven spins half as fast).
• D. 1:1 (equal speed)

Answer: (B)

When two gears mesh, the speed of the driven gear depends on the ratio of their teeth (or radii). The contact point on both gears moves at the same linear speed, so v = ωr must be the same for both gears.

With the driver having 24 teeth and the driven gear having 12 teeth, the driven gear has half the radius. Since the linear speed at the pitch line is the same for both gears, the driven gear must rotate faster by the same factor that its radius is smaller. That means ω_driven = ω_driver × (r_driver / r_driven) = ω_driver × (24 / 12) = 2 × ω_driver.

So the driven gear spins twice as fast as the driver. (Note: the directions are opposite, but the question asks only about speed.)