How is the number of revolutions (r) calculated using diameter and teeth?

The calculation of the number of revolutions (r) using diameter and teeth is rooted in the relationship between the circumference of the gear and the number of teeth it has. The correct formula takes into account the diameter of the gear (D), the total number of teeth on the gear (R), and a given parameter (d), which in many cases can refer to another diameter or a gear's radius.

The formula for the number of revolutions can generally be understood as relating the distance traveled around the diameter of the gear to the distance covered per revolution, which is determined by the number of teeth. Essentially, when you multiply the diameter by the number of teeth, you can establish a proportion that helps compute the total revolutions. This instructional relationship emphasizes how larger diameters and higher tooth counts can together influence the overall motion.

By utilizing the correct formula, one effectively accounts for these variables and arrives at the calculated number of revolutions. The coherence of this formula with physical laws governing rotating systems makes it the appropriate choice for deriving r.