Rotational inertia

Editor's Notes

• #2: This lesson plan should be useful to high school engineering/technology, physics, or math teachers. The lesson aims to teach students the basic concept of moment of inertia, and how it relates to the energy of a spinning body, all in the context of a bicycle wheel. It is designed to be completed in one to two standard class sessions.Within Inventor students will quickly alter a CAD model and take data that will be used in additional calculations about energy. A real-world application – a flywheel bicycle – is included to add authenticity.More challenging math calculations are included.
• #3: Here are two bike wheels made from two different materials. What might make one easier to spin? (Mass, and distribution of mass.)This lesson will explore the factors that affect their rotation.
• #4: Moment of inertia is probably a new concept for most students.
• #5: Students should recognize that pushing something heavy (more mass) is harder than pushing something light. They can think of the greater mass resisting the change in motion.For rotational systems, the moment of inertia plays that role.
• #6: Students might know the term “axis” from math class, as in the x-axis on a graph. Here, it refers to an imaginary line in space used as a reference around which the object rotates. IT MATTERS WHICH AXIS IS USED!Point out the same cylinder, but different axis, in the two images, and how the equations are different. You might also point out that the length L doesn’t show up in the moment of inertia equation for the central axis (left), but it does in the central diameter (right) equation.
• #7: Here are equations for numerous other shapes, taken from: https://webspace.utexas.edu/cokerwr/www/index.html/RI.htmThe main idea is that we can model the behavior of an object by using a close approximation to its shape, and that the moment of inertia is different for different shapes and axes.
• #8: The animation is on the next slide.Before showing it, ask students to guess what order they think the objects will finish in, assuming they are all the same mass and radius.
• #9: The animation should play automatically in presentation mode. If not, it can be viewed here (along with detailed math):https://en.wikipedia.org/wiki/File:Rolling_Racers_-_Moment_of_inertia.ogvThe results are shown frozen on next slide.
• #10: The order of finishing is:1) Orange solid sphere, 2) Blue disk, 3) Red hollow sphere, 4) Green ring.ANSWERS: Highest moment of inertia? Green ring (lower).Lowest moment of inertia? Orange solid sphere (winner)
• #11: These are the takeaways for moment of inertia.
• #12: This is a demo that you could do in class. See http://www.exploratorium.edu/snacks/momentum_machine/ for more info.
• #13: Here are two examples of where someone might want a lower moment of inertia.
• #14: Here are three examples of where higher moment of inertia is useful.The key idea is to get students to appreciate how resistance to change in motion can affect the purpose to which a design is put.
• #15: As energy storage devices, flywheels are massive objects that have high moments of inertia. The higher the moment, and the faster it spins, the more energy it stores.
• #16: Click the link to watch a 2:58 video of a college senior who built a bike with a flywheel for recovering energy from the back wheel.NOTES: Video can be downloaded from that site. And here is a version with Spanish subtitles: http://www.sciencefriday.com/video/08/18/2011/una-bicicleta-con-volante-de-inercia.html
• #17: This is the start of the more physics/math heavy section.
• #18: We assume a bike rider pedals about 60 times a minute, or once per second.MATH NOTE: Students might know “radians” from geometry class as a measure of angle. One full circle, 360°, is equivalent to 2π radians. Since 2π is around 6.28, one revolution = 6.28 radians. Radians are “dimensionless,” so it is just a number (e.g. no equivalent to “meters”). More info: https://en.wikipedia.org/wiki/RadianSECOND MATH NOTE: The word “per,” which is often taken to mean “divide,” can also be interpreted as “for every,” e.g. “25 radians every second.”
• #19: This is math is fairly straightforward, and sets up the equation for the worksheet.INVENTOR NOTE: The moments of inertia from Inventor are given in kg*mm2. To convert to m2, multiply by 0.000001 (10-6).MATH INTERPRETATION: One square millimeter equals one-millionth (0.000001) of a square meter. The image shows a square centimeter, and the relationship to square millimeters. Just picture a million instead of 100 to see the relationship.
• #20: Break for Inventor sessions.
• #21: This is another extension activity that can be done to practice some algebra calculations, and build some intuition about flywheel design.The flywheel in the video is 6.8 kg. The quote is from a Scientific American article, just to give an idea of how popular science articles discuss these kinds of things.Scientific American story: http://blogs.scientificamerican.com/observations/2011/06/24/a-bike-that-uses-its-brakes-for-a-speed-boost-and-other-student-engineer-inventions-video
• #22: We can estimate the moment of inertia for the flywheel by modeling it as a ring, and calculate it with the given mass (6.8 kg) and an estimate of its dimensions (10” diameter, which is about 1/3 of the diameter of a 27-inch bike wheel, based on a guess from the picture).
• #23: The 32.5 J estimate is based on the Full Bicycle Wheel setup analysis. Here we are solving for angular velocity, to get an estimate of how fast the flywheel spins as the wheel stops spinning, assuming all of the energy goes to the flywheel.
• #24: The main ideas here are that energy can be stored and transferred, and in some cases “reused.” And in the case of something spinning, its amount of stored energy depends on its rotational speed, mass, and shape/mass distribution.