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# Calculus II - 11

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Stewart Calculus Section 8.3

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• ### Calculus II - 11

1. 1. 8.3 Applications toPhysics and Engineering Moments and Center of Mass Question: Given a thin plate with arbitrary shape, where is the center of mass?
2. 2. Discrete case: m1 m2 d1 d2 =
3. 3. Discrete case: , , ( , ), ( , ), ( , ).The moment of the system = + + m1 = + +The center of mass is o (¯, ¯ ) = , m2 m3
4. 4. Continuous case 1:The moment of the system = ( ) ( ) =The center of mass is (¯, ¯ ) = , = ( ) . y=f(x)
5. 5. Ex: find the center of mass of a semicircularplate of radius .
6. 6. Ex: find the center of mass of a semicircularplate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − −
7. 7. Ex: find the center of mass of a semicircularplate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − − − ( − ) =
8. 8. Ex: find the center of mass of a semicircularplate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − − − ( − ) = =
9. 9. Ex: find the center of mass of a semicircularplate of radius . √ ( )= − ( ) ( ) − ¯= − ( ) ¯= ( ) − − − ( − ) = =The center of mass is at point , .
10. 10. Continuous case 2:The moment of the system = [ ( ) ( )] = [ ( ) ( ) ]The center of mass is (¯, ¯ ) = , = [ ( ) ( )] . y=f(x) y=g(x)
11. 11. Ex: find the centroid of the region boundedby the line = and the parabola = .
12. 12. Ex: find the centroid of the region boundedby the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= /
13. 13. Ex: find the centroid of the region boundedby the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = −
14. 14. Ex: find the centroid of the region boundedby the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = .
15. 15. Ex: find the centroid of the region boundedby the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = ( − ) = .
16. 16. Ex: find the centroid of the region boundedby the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = ( − ) = . = .
17. 17. Ex: find the centroid of the region boundedby the line = and the parabola = . ( )= , ( )= ( )− ( ) = . [ ( )− ( )] [ ( ) − ( ) ] ¯= / ¯= / = − = ( − ) = . = . The center of mass is at point , .