(Note: Boldface words are basic mathematical terms that you will need to know in order to understand this reading. They are explained in the Mathematical Glossary. Your teacher will discuss these and other important math terms as part of the preparation for this lesson.)
Gears are wheels with teeth. They are used to make work easier, either by changing the direction of power, or exchanging speed for power (or power for more speed).
If we know the diameters of two gears that are touching and the speed and direction of the first gear, we can calculate the speed and direction of the second gear by using an algebraic formula. The second gear always turns in the opposite direction to the first gear. Working with the gear ratio of the two gears will provide us with the ability to calculate the speed of the second gear.
The gear ratio is the diameter of the primary gear (or first gear) divided by the diameter of the next touching gear, or secondary gear.
We define the following variables, or unknowns:
ds = diameter of secondary gear
dp = diameter of primary gear
Ns = speed of secondary gear
Np = speed of primary gear
The basic equation for the speeds and diameters of two gears that are meshed together is
Ns x ds = Np x dp
This means that when two meshed gears are turning, the product of the speed and diameter of one is equal to the product of the speed and diameter of the other. As you will see in this lesson, if you know the values (that is, actual numbers) for any three of the variables in this equation, you can always find the value of the fourth variable.
Suppose you know that
Ns=100 rpm Np=50 rpm ds=2"
Use the gear equation to find dp:
Ns x ds = Np x dp
Divide both sides of the equation by Np:
Ns/Np x ds=dp
The unknown dp is now one side of the equation. Then
100 rpm/50 rpm x 2" = dp
(by substituting known values for variables), and thus
2 x 2" = dp
4" = diameter of primary gear
Further note: The direction of the gears has changes. If the primary gear is going clockwise, the secondary gear is moving counterclockwise.
The same rules also apply to three or four gears.
1. If the diameter of gear A in Figure 1 is 6" and the gear rotates clockwise at a speed of 12 rpm and the diameter of gear B is 8", then gear B rotates (in what direction) _________ and its speed is ________rpm.
Then, gear C, with a diameter if 3", rotates (in what direction) __________ and with a speed of _________ rpm.
Remember: Ns x ds = Np x dp
2. By using what you have learned about gears, explain why putting a car in reverse makes it go backwards.
3. The same rules of gear ratios apply when using a chain, as in a bicycle (see Figure 2). Let's calculate the speed in rpm of the small sprocket (attached to the wheel) if its diameter is 2" and that of the big sprocket (driven by pedals) is 4". The rider can pedal at 20 RPM. Note that the sprockets do not turn in opposite directions when a chain connects them as in Figure 2.
Now calculate the speed of the sprocket attached to the wheel if it has a diameter of 4" and the sprocket attached to the pedals has a diameter of 2". Again, lets suppose the rider pedals at 20 rpm. Why is it helpful for a bicyclist to be able to choose different gear ratios?
4. The same rules of ratios that apply to gears also apply to a fan belt on a car (see Figure 3). From the diameters of the pulleys in Figure 3, let's calculate the speed of each of the engine components. The crankshaft pulley is the pulley that's driven directly by the engine; it turns at 1000 rpm. If the crankshaft pulley is turning clockwise, what is the direction of the other three pulleys?