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Moments of Inertia

Moments of Inertia

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Moments of Inertia

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  1. Moments of Inertia Lesson 7.6

  2. Review • Recall from previous lesson the first moment about y-axis • The moment of inertia (or second moment) is the measure of the tendency of an object to resist change in motion

  3. Moment of Inertia • For a system of n masses • If the masses were at the same distance r from the axis of rotation we have

  4. Radius of Gyration • Radius of gyration about x-axis(where m is total mass of system) • Radius of gyration about y-axis • Radius of gyration about the origin

  5. Example • Suppose we have 3g at (2,3), 4g at (-2,-4), and 3g at (-4,5) • Find Iy • Iy = 3*2 2 + 4*(-2)2 + 3*(-4)2 = 76 • Find Ix • Ix = 3*32 + 4*(-4)2 + 3*52 = 166 • Find r0 • r0- = 4.9193

  6. Density of region Moment of Inertia for a Region • Given a region bounded by curves of two functions and lines x = a, x = b • The moment of inertia about the y-axis f(x) g(x) x = b x = a

  7. Radius of Gyration f(x) • Given the same region • Radius of gyration, ry with respect to the y-axis is g(x) x = b x = a

  8. Try It Out • Given the region bounded by y3 = x2 , y = 4 and the y-axis. Density = 4g/cm2 • Moment of Inertia about x-axis

  9. Try It Out • Given the region bounded by y3 = x2 , y = 4 and the y-axis. Density = 4g/cm2 • Radius of gyrationabout x-axis

  10. For a Solid of Revolution • Moment of inertia of a solid of revolution formed by generating a region around the y-axis • The radius of gyration is

  11. Example • Consider region bounded by y = x2 , the y-axis, and y = 2 rotated • What is the moment ofinertia about the x-axis

  12. Interesting Application • Sweet spot for a baseball bat • What happens when you hit the ball … • At point A? • At point B or C? Note the interesting description of this lab assignment from Colorado State

  13. Assignment • Lesson 7.6 • Page 299 • Exercises 1, 3, 5, 9, 11, 15, 17 • Second day7, 13, 19, 21, 23