After logging out of my email, I saw an ad for a video that caught my eye. The video was for This bicycle generator can bring electricity to millions of people. Now I will tell you, that bike is cool! I am always looking for a way to keep in shape (sometimes more successfully than other times). I think this idea would be good for people with a New Year’s resolution to exercise more, especially if they cancel their power company.

But this post isn’t about the video directly. After watching the video, I did a search for “bicycle generator,” which brought me to the instructibles.com website. One of the members, notoriouslev, made the bike and posted a comment to help people try to determine the v-belt length needed for this project.

I think it is obvious that he is not a mathematician, but also obvious the he paid attention in school in math class. His explanation, I think, is perfect for the lay person. It gives some basic informaiton about why it works without getting too technical to scare people, and then gives the steps someone else would need to take to replicate his idea for their project. In his short reply, he references both the circumference of a circle and the Pythagorean Theorem.

Now if a student says, “when am I ever going to learn this,” I can point them to notoriouslev and say, “you might not, but he did!”

He starts by measuring the diameter of his rim to get 26 inches, and converts it to 13 inches as the radius.

Then he says, “The equation to get the circumference of the rim is 2 x pie x radius = 2 x 3.14 x 13 = 81.64″ around.” Although it is pi, not pie, I’ll give him a pass on this step. Alternately, he could have used πd and not had to convert the diameter to a radius.

I don’t know how these belts are measured or bought, but if he is able to get an accuracy of hundredths of an inch, he may want to use a more accurate number for pi. I got 81.68 inches.

Once he has this number, he says to, “Subtract 1/4 of the circumference: 81.64 * 75% = 61.23” I think this is classic! He says to subtract, but then he multiplies instead! Mathematically, it is correct, though. If we take the circumference and subtract 1/4 of the circumference, we can rewrite that as:

C – ¼C = C(1 – ¼) = C•¾ = C•0.75 = C•75%

In his last step, he said to, “Add the diameter: 61.23 + 26 = 87.23″ Long.” At first, I had to look at the diagram to see why he added the diameter. Once I looked, I saw that he was really adding two radaii, which is the same value as the diameter. Essentially, he created a circle with a square drawn on top. The square has a vertex at the center of the circle, and the square’s side length is the same as the radius of the circle. Two of the sides of the square are tangent to the circle.

He does have an optional step 4 listed which isn’t entirely accurate. He says to:

Optional Step 4: You can add extra length to the v belt in case you don’t want the generator sitting just below the rear wheel’s exterior. For instance to place the generator 1 foot behind the rear wheel, you add 12″ for the extra distance behind the wheel, and then you use the

Pythagorean theoremto get the additional v belt required to reach back to the tire: (12 x 12) + (13 x 13) …then get the square root = 17.7″. So to place the generator 1 ft behind the rear wheel of this bike, we add 29.7 inches to the 87.23″ we already had and get 116.93″

This model would assume (or imply) that there is some sort of pully at point D causing the belt to stay on the wheel rim all the way to point D before moving to the generator at point E. According to his description the side labeled d_{1} is 12 inches, the foot behind the read wheel. The side labeled e_{1} is 13 inches, the same as the radius of the wheel.

In actuallity, though, the belt will come off the wheel forming a tangent from the wheel to the generator at Point D.

For his purposes, though, using the hypotenuse might be close enough to get a belt length. In using the tangent line, the belt would leave the rim at point F, so even though the belt length from E to F is longer than the hypotenuse he calculated, the arclength from F to D is no longer traveled, so we lose some length from the circumference (less than 75% used now).

Using Geogebra, I found the measurement of the belt is 53.31 + 13 + 12 +25 = 103.31 inches.

Of note, since the segment BE is also a tangent, it will have the same length as EF (a much easier calculation than using Pythagorean). (Can you show why?)

If we bought a belt using his 116.93 inches calculation, I believe that the generator will be closer to 2 feet behind the back wheel. (About 20.19 inches behind the back wheel.)

What started as a signout video from my email turned into a neat little mathematical tangent. Thanks to notoriouslev for the comments that started this little journey.