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
= 3* *(-2)2 + 3*(-4)2 = 76
Find Ix
Ix = 3*32 + 4*(-4)2 + 3*52 = 166
Find r0
r0 =

6. 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 = a
x = b
Density of region

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

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 gyration about 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 of inertia 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 day 7, 13, 19, 21, 23