Moments of Inertia
Lesson 7.6
Review
• Recall from previous lesson the first
moment about y-axis
n
M y   mi xi
i 1
 m1 x1  m2 x2  ...  mn xn
• The moment of inertia (or second moment)
is the measure of the tendency of an
object to resist change in motion
2
Moment of Inertia
• For a system of n masses
n
I y   mi xi 2
i 1
 m1 x12  m2 x2 2  ...  mn xn 2
• If the masses were at the same distance r
from the axis of rotation we have
n
I y   mi r 2
i 1
3
Radius of Gyration
Ix
• Radius of gyration about x-axis
rx 
(where m is total mass of system)
m
• Radius of gyration about y-axis
ry 
Iy
m
• Radius of gyration about the origin
r0  rx  ry
2
2
4
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
5
Moment of Inertia for a Region
• Given a region bounded by curves of two
functions and lines x = a, x = b
f(x)
g(x)
x=a
x=b
• The moment of inertia about the y-axis
Density of
region
b
I y    x 2  f ( x)  g ( x)  dx
a
6
Radius of Gyration
• Given the same region
f(x)
g(x)
x=a
x=b
• Radius of gyration, ry with respect to the
y-axis is
ry 
Iy
m
7
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
b
I x    y 2  f ( y )  g ( y )  dy
a
4
I y  4 y 2  y 3/ 2  0  dy
0
8
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
rx 
Ix
m
8
m  4  x 2 / 3 dx
0
9
For a Solid of Revolution
• Moment of inertia of a
solid of revolution
formed by generating
a region around the y-axis
b
b
a
a
I y  2  x3 y dx  2  x 3  f ( x)  g ( x)  dx
• The radius of gyration is
ry 
Iy
m
10
Example
• Consider region bounded by y = x2 , the yaxis, and y = 2 rotated

What is the moment of
inertia about the x-axis
b
I x  2  y 3 f ( y ) dy
a
2
I x  2  y 3  y1/ 2 dy
0
2
m  2  y 3/ 2 dy
0
11
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
12
Assignment
•
•
•
•
Lesson 7.6
Page 299
Exercises 1, 3, 5, 9, 11, 15, 17
Second day
7, 13, 19, 21, 23
13