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 I y  I y = 3* *(-2) 2 + 3*(-4) 2 = 76 Find I x  I x = 3* *(-4) 2 + 3*5 2 = 166 Find r 0  r 0- =

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, r y with respect to the y-axis is f(x) g(x) x = a x = b

8 Try It Out Given the region bounded by y 3 = x 2, y = 4 and the y-axis. Density = 4g/cm 2  Moment of Inertia about x-axis

9 Try It Out Given the region bounded by y 3 = x 2, y = 4 and the y-axis. Density = 4g/cm 2  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 = x 2, 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 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