This lesson plan will help engineering/technology or physics teachers who are teaching either basic mechanics or energy fundamentals teach some concepts using a bicycle.The main idea is that machines require an energy input, and that energy is used to trade off force and movement to accomplish some goal. It is designed to be done in one or two standard class periods. It requires a basic foundation in geometry and algebra, though some of these concepts are explicitly reviewed.An interactive CAD exercise lets students use Autodesk Inventor to change the gears and the effect on the rotations of the wheel.
Bike gears can be easier or harder to turn. They have different uses, such as for cruising as a commuter, or for racing as a competitor. This lesson will explore some of the properties of bike gears and energy input/output.
Energy is one of the most important concepts in science. It is NOT what makes things go – think of it instead as a constraint on what can be done. A bike rider with a lot of energy can bike hard and fast before running low on energy, but the same rider starting with the same amount of energy could just as easily sit at home doing nothing.
Machines can be thought of as “force amplifiers,” where bigger (or smaller) output forces can be generated with smaller (or bigger) movements. The lever is a classic simple machine that gets at this force/movement tradeoff.In physics, mechanical energy is often called “work” and is set equal to the force times distance, W = F x d. From this, the tradeoff between output force and movement (distance) can be seen more mathematically.NOTE: This is for mechanical machines. We might refer to a computer or electrical device as a “machine,” but that’s a more modern usage.
Students who bike should realize that fast biking is often easy (perhaps on a flat straightaway), and slow biking is often hard (perhaps on a hill). They might both make you equally tired, though students will probably say hard, slow pedaling is more tiring and thus requires more energy.When going uphill it is harder to pedal, but changing to a lower gear makes it easier. Students might remember that the easier gear for going uphill requires more pedaling to go the same speed. They might also remember that when going downhill when it’s easy to pedal, they sometimes lose the ability to pedal, and changing to a higher gear makes it “harder” but possible.The important point here is that the hard/easy, fast/slow distinctions are related. The lesson is meant to help formalize the relationships.
The lesson is something of an exercise in geometry and algebra, but the goal is to make it relevant to real bikes and to build some intuition about energy, force, and movement.
The “gear ratio” compares the number of teeth on the gears, front (with the pedals) to rear (with the wheel).MATH NOTE: It might be useful to point out that the mathematical idea of a “ratio,” which compares two quantities, is more general than “gear ratio.” An example is the ratio of boys to girls in the class (say 15 boys / 15 girls = 1:1), or boys to number of students in the class (here 15 boys / 30 students = 1:2). See https://en.wikipedia.org/wiki/Ratio
By converting fractions to difference between 44/11 = 4, and 11/44 = 0.25, should convey how in “high gear” the rear gear turns a lot more times than the front gear, and in “low gear” the front gear turns a lot more times than the rear gear. Students who have ridden multi-speed bikes should be able to remember having to pedal more or less when changing gears.
This is a review of the geometric properties of circles, specifically circumference, diameter, and the number π (which we’ll approximate at 3.14), using an example of a 27” tire.At the bottom there is a conversion from millimeters to meters (1000mm = 1m), and a conversion from inches to feet (12” = 1 foot).
These are ways to make the circumference more intuitive. Since this lesson involves linear distances of motions of gears, wheels, and pedals, this is an useful point for students to get some comfort with.
“Meters of development” is basically just a combination of gear ratio and wheel circumference.
The “gain ratio” provides a direct relationship between the pedal motion and the wheel. It was first suggested here: http://sheldonbrown.com/gain.htmlSo if the pedal, at the end of the crank arm, moves a certain amount, the wheel moves a certain amount – either more or less, depending on the “gain ratio.” The next two slides provide examples for high and low gear.
This is the relationship for high gear.
This is just the example for low gear.
Since the gain ratio is the distance the wheel moves compared to the distance the pedal moves, all the necessary info is on the diagram.
Mechanical advantage is the main idea for all machines – what is the output force for a given input force? This can be seen as “force amplification.” Since the mechanical advantage for the two gears above is less than 1 (0.44 and 0.22), the output force is LESS than the input force, so it’s more of a “force reducer.”If students are unfamiliar with forces, you can explain them as pushes and pulls (here forces are pushes). The arrow shows the direction and relative size of the force, and this “vector representation” is common in physics.One math idea included is the “inverse” of gain ratio. This is just 1 / gain ratio. For the lower gear, that means either 1/ 2.27, or since gain ratio is 34cm/15cm, it can also be seen as 15cm/34cm, or the ratio of the input distance to the output distance.Another math idea is that the equation MA = Fout / Fin is rearranged to be Fout = Fin x MA. This can be made explicit if necessary.
Point out that the front pedal force and distance are the same in both cases, but the output force and distance are different.
Energy is a crosscutting topic in science, and it is often difficult to understand. The goal here is only to introduce the idea that for mechanical systems, we get a measure of energy inputs by measuring the forces and distances involved in movement. The unit of energy is the Joule (J), which is equivalent to 1 Newton-meter (Nm).
Energy is a “conserved” physical quantity, which means it can neither be created nor destroyed. If energy is being transferred, it must be coming from somewhere and going somewhere. The idea in this example is that for the same force/distance input (i.e. energy input), different force/distance outputs are possible, though the will multiply together to equal the input energy.
For any machine, an energy source can be identified, along with how the energy is transferred to the output. The idea of “efficiency” of a machine is not really covered in this lesson, but students should realize that just as changing gears on a bike can make it easier or harder to perform certain tasks (such as climing hills), the same is true for cars. Manual transmission stick-shift cars make gear changing more obvious than automatic transmission cars.
Prepare students for the CAD activity by explaining what they will be doing. The math reviewed earlier in this presentation will be part of the required calculations on their worksheets. You can do the calculations as a class if necessary.
