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

1

1 1 2 2 ...

n

y i ii

n n

M m x

m x m x m x

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

2

1

2 2 21 1 2 2 ...

n

y i ii

n n

I m x

m x m x m x

2

1

n

y ii

I m r

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

yy

Ir

m

xx

Ir

m

2 20 x yr r r

5

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

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

2 ( ) ( )b

y

a

I x f x g x dx Density of

region

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

yy

Ir

m

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

2 ( ) ( )b

x

a

I y f y g y dy 4

2 3/ 2

0

4 0yI y y dy

9

Try It Out

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

Radius of gyrationabout x-axis

xx

Ir

m

82/3

0

4m x dx

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

3 32 2 ( ) ( )b b

y

a a

I x y dx x f x g x dx

yy

Ir

m

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

32 ( )b

x

a

I y f y dy 2

3 1/ 2

0

2xI y y dy 2

3/ 2

0

2m y dy

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 day7, 13, 19, 21, 23