Rotational inertia


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  • 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.
  • 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.
  • Moment of inertia is probably a new concept for most students.
  • 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.
  • 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.
  • Here are equations for numerous other shapes, taken from: 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.
  • 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.
  • The animation should play automatically in presentation mode. If not, it can be viewed here (along with detailed math): results are shown frozen on next slide.
  • 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)
  • These are the takeaways for moment of inertia.
  • This is a demo that you could do in class. See for more info.
  • Here are two examples of where someone might want a lower moment of inertia.
  • 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.
  • 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.
  • 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:
  • This is the start of the more physics/math heavy section.
  • 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: 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.”
  • 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.
  • Break for Inventor sessions.
  • 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:
  • 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).
  • 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.
  • 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.
  • Rotational inertia

    1. 1. Autodesk Sustainability Workshop Bike Wheels, Rotational Inertia, and Energy Adam Kenvarg, Joel Rosenberg, and James Regulinski © 2013 Autodesk
    2. 2. Which of These is Easier to Spin? Aluminum © 2013 Autodesk Stainless Steel
    3. 3. What is Rotational Inertia? • Rotational inertia, often called moment of inertia, is the resistance of an object to a change in angular velocity. • It can be thought of as the rotational version of the role that mass plays as the resistance of an object to a change in straight-line velocity. © 2013 Autodesk
    4. 4. What is Rotational Inertia? II • Said another way, the higher the moment of inertia of an object, the greater its resistance to a change in the rotational velocity. • Similarly, the higher the mass of an object, the greater its resistance to a change in the straightline velocity. © 2013 Autodesk
    5. 5. Axis of rotation  The moment of inertia is almost always different for each different axis of rotation. Solid cylinder (or disk) about central axis Solid cylinder (or disk) about central diameter © 2013 Autodesk
    6. 6. More Moment of Inertia Values © 2013 Autodesk
    7. 7. Which object will finish first? (All are equal mass and radius) Note: the red sphere is a hollow shell © 2013 Autodesk
    8. 8. Go! Note: the red sphere is a hollow shell © 2013 Autodesk
    9. 9. Results   Which object has the highest moment of inertia? Which has the lowest? Note: the red sphere is a hollow shell © 2013 Autodesk
    10. 10. Moment of Inertia  Remember, the higher the moment of inertia of an object, the greater its resistance to a change in the rotational velocity. That’s why the green ring has the HIGHEST moment of inertia (most resistance).  Moment of inertia is determined by the amount of mass and its distance from the axis of rotation. © 2013 Autodesk
    11. 11. Classic Moment of Inertia Demo  Figure skaters control their moment of inertia by moving the distribution of their mass – closer to the axis means faster rotation. A simple demo of this can be done by spinning someone in a chair while the person brings weights closer and farther from their body. © 2013 Autodesk
    12. 12. When Do You Want a Low Moment of Inertia?  Examples include all kinds of wheels – allows for faster acceleration and deceleration of wheels and therefore the vehicle pg  Baseball bats – lets the batter swing the bat around more quickly (this can be accomplished by “choking up” on the bat) © 2013 Autodesk
    13. 13. When Do You Want a High Moment of Inertia?  Flywheels – intended to smooth power variations in mechanical systems and store energy  Tightrope walker’s sticks – to slow the rate of rotation as they tip back and forth pg ng © 2013 Autodesk  Juggling clubs – prevents slight mistakes by the jugglers from greatly altering the spin of the clubs. NOTE: You can buy both “fast-spinning” and “slowspinning” clubs, i.e., ones with lower and higher moments of inertia.
    14. 14. How Does Moment of Inertia Relate to Energy?  In F1 racing cars are now allowed to store energy from braking in flywheels in a “kinetic energy recovery system” (a.k.a. KERS). These flywheels allow the driver to later use the additional energy for a speed boost to overtake opponents. .jpg!slide=590081 © 2013 Autodesk  Some companies are developing large scale flywheels to help even out the power output from power plants. These flywheels obviate the need for additional power plants running as full-time backup (which is a huge waste of energy and a large contributor to CO2 emissions)
    15. 15. A Bike With a Flywheel (2:58 video) © 2013 Autodesk
    16. 16. Calculating Energy  The energy, E, stored in a rotating object is related to its moment of inertia, I, and its angular velocity, w, by this equation: E = 1/2 * I * w2  The angular velocity, w, is the rate at which something rotates, in radians per second. © 2013 Autodesk
    17. 17. Angular velocity of Wheel  An average bike rider can pedal around once every second (1 revolution per second).  1 revolution = 6.28 radians (= 2π)  Average rider pedal speed = 6.28 radians / second NOTE: Radians are “dimensionless,” so w ≈ 6.28 / s  EXAMPLE: For a “high gear” ratio of 44/11 = 4, every revolution of the pedal/front gear turns the rear gear/wheel four revolutions, so: w = 4 * (6.28 radians / second) ≈ 25 / s © 2013 Autodesk
    18. 18. Energy Stored in Bike Wheel  Plug in the moment of inertia and angular velocity to find the energy stored in a bike wheel. Here is the start of the calculation: E = 1/2 * I * w2 = 0.5 * I * (25 / s)2 = I * (312.5 / sec2)  INVENTOR NOTE: The units of moment of inertia are given in kg*mm2. To convert to kg*m2, multiply by 10-6 (= 0.000001): 1 m2 0.000001 m2 © 2013 Autodesk = 1,000,000 mm2 = 1 mm2
    19. 19. Let’s Explore This Using an Example  Open Bike_Rim_For_Rotation.ipt  Find the moment of inertia using iProperties for variations on the rim.  Follow the instruction on the handout © 2013 Autodesk
    20. 20. Back To the Flywheel  6.8 kg flywheel from a Porche  “The flywheel increases maximum acceleration and nets 10 percent pedal energy savings where speeds are between 20 and 24 kilometers per hour.” © 2013 Autodesk
    21. 21. Moment of Inertia of Flywheel  The moment of inertia for the flywheel, modeled as a ring, is: I = 1/2 * M * (R12 + R22)  If we assume R2= 5” = 0.127 m and R1= 4” = 0.102 m, with mass M = 6.8 kg, then for the flywheel: I © 2013 Autodesk = 0.5 * 6.8 kg * ((0.102m) 2 + (0.127m) 2 ) = 3.4kg * (0.0161 m2 + 0.0104 m2) = 0.0901 kg * m2
    22. 22. Energy and Angular Velocity of Flywheel  Let’s say that the wheel has 32.5 Joules of energy. If we assume that all of that energy is transferred to the flywheel, we can calculate its angular velocity, w: E = 32.5 J = 1/2 * I * w2 = 0.5 * 0.0901 kg*m2 * w2 32.5 J 0.0451 kg*m2 = w2 26.8 radians / s = w (4.25 revolutions / s) © 2013 Autodesk
    23. 23. Summary  Moment of inertia is determined by both the mass and shape of an object. Higher moment of inertia results from more mass further from the axis of rotation.  The amount of energy a rotating object stores is determined by its moment of inertia (resistance to motion) and angular velocity.  When a bike rider pedals, energy is transferred to the parts of the bike, including the front wheel.  Some of that energy can be stored in a flywheel, and returned from the flywheel later. © 2013 Autodesk
    24. 24. Autodesk is a registered trademark of Autodesk, Inc., and/or its subsidiaries and/or affiliates in the USA and/or other countries. All other brand names, product names, or trademarks belong to their respective holders. Autodesk reserves the right to alter product and services offerings, and specifications and pricing at any time without notice, and is not responsible for typographical or graphical errors that may appear in this document. © 2013 Autodesk, Inc. All rights reserved.