#8. A driver gear with 8 teeth is meshed with a driven gear with 12 teeth. If the driven gear is rotating at 1,200 RPM, what is the speed the driver gear? a. \( 1,800 \mathrm{RPM} \) b. 800 RPM c. \( 1,200 \mathrm{RPM} \) d. 2,400 RPM

To find the speed of the driver gear, you can use the gear ratio formula where the ratio of the number of teeth on the driven gear to the driver gear is inversely proportional to their speeds. The formula is: \[ \text{Speed of Driver Gear (RPM)} = \text{Speed of Driven Gear (RPM)} \times \left( \frac{\text{Teeth on Driven Gear}}{\text{Teeth on Driver Gear}} \right) \] Substituting the known values: \[ \text{Speed of Driver Gear} = 1200 \, \text{RPM} \times \left( \frac{12}{8} \right) = 1200 \times 1.5 = 1800 \, \text{RPM} \] So the answer is \(1,800 \text{ RPM}\) (option a). When gears mesh, they're like dance partners: the smaller one spins faster while the bigger one takes its time! Understanding gear ratios not only helps in machinery concepts but also applies to everyday items, like bicycles. The smaller front gear (chainring) makes it easier to pedal while providing speed, whereas the larger rear gear (cog) allows for easier climbing on steep hills. This is a real-world manifestation of those gear relationships! For smoother, efficient gear systems, avoid common mistakes like mismatched gear sizes or incorrect alignments. Ensuring that gears are well-lubricated can also prevent slippage and unnecessary wear. Always double-check the number of teeth before making assumptions about speed and power transfer—those tiny details can shift your whole understanding!