moments of inertia lesson 7.6. 2 review recall from previous lesson the first moment about y-axis...
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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
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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
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Assignment
• Lesson 7.6
• Page 299
• Exercises 1, 3, 5, 9, 11, 15, 17
• Second day7, 13, 19, 21, 23