Stewart Calculus Section 8.3

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 , .