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 If the masses were at the same distance r from the axis of rotation we have

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 Suppose we have 3g at (2,3), 4g at (-2,-4), and 3g at (-4,5) Find = 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 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

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

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 Out Given the region bounded by y3 = x2 , y = 4 and the y-axis. Density = 4g/cm2 Radius of gyration about x-axis

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

Example Consider region bounded by y = x2 , the y-axis, and y = 2 rotated What is the moment of inertia about the x-axis

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 Lesson 7.6 Page 299 Exercises 1, 3, 5, 9, 11, 15, 17 Second day 7, 13, 19, 21, 23