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## PowerPoint Slideshow about ' Moments of Inertia' - stuart-smith

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### 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 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

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 = b

x = a

Radius of Gyration

f(x)

- Given the same region
- Radius of gyration, ry with respect to the y-axis is

g(x)

x = b

x = a

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 gyrationabout 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 ofinertia 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 day7, 13, 19, 21, 23

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