Additional Math Pages & Resources

Wednesday, June 8, 2011

Degrees, Gears and Ratios, Part III

This week I am investigating how GEARS allow us to visibly express ratios. They aren't just numbers written on paper in math class. We use them to do work (turn, lift, etc.). In the last few days we've looked at watches, but today I want to examine gears that are doing heavier work. Here's an example:


The gears on the left are the same size, and represent a ratio of 1:1. The gears on the right are different sizes, representing the ratio of 2:1. You have to turn the smaller gear twice to get the larger one to turn once - but in the process you are gaining "mechanical advantage" or strength.

You are probably familiar with the gears on a bicycle. They help us get up steep hills, and allow us to ride faster than we can walk. Most bike gears include the addition of a chain. A chain simply allows us to separate the gear sets from each other, placing them in convenient locations on the bike frame, while still connecting them (even though they don't physically touch). The front and rear shifters (derailleurs) move (derail) the chain from one gear to another.

In a car's transmission, the gears touch all the time. Transmissions are very complicated mechanisms. The active pairs of gears are brought into use by a shift lever that locks each set of gears to the shafts upon which they rotate.

You could think of pulleys as gears connected by a rope. The size and number of the pulleys determine the ratios of the system. The more pulleys, the more mechanical advantage you get, but the more length of rope you have to pull to lift something.

When I was the age that our Excel Math students are now, ropes and pulleys were a handy way to lift a train set up into the rafters of my garage. The number and size of the pulleys was determined by how many we had on hand, not by any math analysis.

If you have read this blog for a long time, you might remember the posting on non-circular gears.

As I said at the beginning, a set of gears is a way to represent a math ratio. If you can read this linked article without falling asleep, you might want to become an engineer!