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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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Presentation Transcript
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