Calculus II - 11
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This document discusses finding the center of mass of objects using calculus. It provides examples of finding the center of mass for discrete objects, continuous objects defined by a function, and regions bounded by curves. The key steps are to calculate the moments about the x- and y-axes and then use these to determine the x- and y-coordinates of the center of mass. Examples include finding the center of a semicircular plate and the centroid of a region bounded by a line and parabola.
![Continuous case 2:
The moment of the system
= [ ( ) ( )]
= [ ( ) ( ) ]
The center of mass is (¯, ¯ ) = ,
= [ ( ) ( )] .
y=f(x)
y=g(x)](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-10-320.jpg)
![Ex: find the centroid of the region bounded
by the line = and the parabola = .
( )= , ( )=
( )− ( ) = .
[ ( )− ( )] [ ( ) − ( ) ]
¯= /
¯= /](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-12-320.jpg)
![Ex: find the centroid of the region bounded
by the line = and the parabola = .
( )= , ( )=
( )− ( ) = .
[ ( )− ( )] [ ( ) − ( ) ]
¯= /
¯= /
= −](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-13-320.jpg)
![Ex: find the centroid of the region bounded
by the line = and the parabola = .
( )= , ( )=
( )− ( ) = .
[ ( )− ( )] [ ( ) − ( ) ]
¯= /
¯= /
= −
= .](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-14-320.jpg)
![Ex: find the centroid of the region bounded
by the line = and the parabola = .
( )= , ( )=
( )− ( ) = .
[ ( )− ( )] [ ( ) − ( ) ]
¯= /
¯= /
= − = ( − )
= .](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-15-320.jpg)
![Ex: find the centroid of the region bounded
by the line = and the parabola = .
( )= , ( )=
( )− ( ) = .
[ ( )− ( )] [ ( ) − ( ) ]
¯= /
¯= /
= − = ( − )
= . = .](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-16-320.jpg)
![Ex: find the centroid of the region bounded
by the line = and the parabola = .
( )= , ( )=
( )− ( ) = .
[ ( )− ( )] [ ( ) − ( ) ]
¯= /
¯= /
= − = ( − )
= . = .
The center of mass is at point , .](https://image.slidesharecdn.com/math12722011-09-30-111203212029-phpapp02/85/Calculus-II-11-17-320.jpg)
Calculus II - 11
- 1. 8.3 Applications to Physics and Engineering Moments and Center of Mass Question: Given a thin plate with arbitrary shape, where is the center of mass?
- 2. Discrete case: m1 m2 d1 d2 =
- 3. Discrete case: , , ( , ), ( , ), ( , ). The moment of the system = + + m1 = + + The center of mass is o (¯, ¯ ) = , m2 m3
- 4. Continuous case 1: The moment of the system = ( ) ( ) = The center of mass is (¯, ¯ ) = , = ( ) . y=f(x)
- 5. Ex: find the center of mass of a semicircular plate of radius .
- 6. Ex: find the center of mass of a semicircular plate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − −
- 7. Ex: find the center of mass of a semicircular plate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − − − ( − ) =
- 8. Ex: find the center of mass of a semicircular plate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − − − ( − ) = =
- 9. Ex: find the center of mass of a semicircular plate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − − − ( − ) = = The center of mass is at point , .
- 10. Continuous case 2: The moment of the system = [ ( ) ( )] = [ ( ) ( ) ] The center of mass is (¯, ¯ ) = , = [ ( ) ( )] . y=f(x) y=g(x)
- 11. Ex: find the centroid of the region bounded by the line = and the parabola = .
- 12. Ex: find the centroid of the region bounded by the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= /
- 13. Ex: find the centroid of the region bounded by the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = −
- 14. Ex: find the centroid of the region bounded by the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = .
- 15. Ex: find the centroid of the region bounded by the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = ( − ) = .
- 16. Ex: find the centroid of the region bounded by the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = ( − ) = . = .
- 17. Ex: find the centroid of the region bounded by the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = ( − ) = . = . The center of mass is at point , .