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Moments of Inertia. Lesson 7.6. 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. Moment of Inertia. For a system of n masses

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Moments of inertia

Moments of Inertia

Lesson 7.6


Review
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


Moment of inertia
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


Radius of gyration
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


Example
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


Moment of inertia for a region

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


Radius of gyration1
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


Try it out
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


Try it out1
Try It Out

  • Given the region bounded by y3 = x2 , y = 4 and the y-axis. Density = 4g/cm2

    • Radius of gyrationabout x-axis


For a solid of revolution
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


Example1
Example

  • Consider region bounded by y = x2 , the y-axis, and y = 2 rotated

    • What is the moment ofinertia about the x-axis


Interesting application
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


Assignment
Assignment

  • Lesson 7.6

  • Page 299

  • Exercises 1, 3, 5, 9, 11, 15, 17

  • Second day7, 13, 19, 21, 23


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