Chapter 6 | Applications of Integration
705
m 1 = 30kg,placed at x 1 =−2m m 3 = 10kg,placed at x 3 =6m
m 2 = 5kg,placed at x 2 =3m
m 4 = 15kg,placed at x 4 =−3m. Find the moment of the system with respect to the origin and find the center of mass of the system.
Solution First, we need to calculate the moment of the system: M = ∑ i =1 4 m i x i
=−60+15+60−45=−30. Now, to find the center of mass, we need the total mass of the system: m = ∑ i =1 4 m i =30+5+10+15=60kg. Then we have x – = M m = −30 60 = − 1 2 . The center of mass is located 1/2 m to the left of the origin.
6.29
Suppose four point masses are placed on a number line as follows: m 1 = 12kg,placed at x 1 =−4m
m 2 = 12kg,placed at x 2 =4m
m 3 = 30kg,placed at x 3 =2m m 4 = 6kg,placed at x 4 =−6m. Find the moment of the system with respect to the origin and find the center of mass of the system.
We can generalize this concept to find the center of mass of a system of point masses in a plane. Let m 1 be a point mass located at point ( x 1 , y 1 ) in the plane. Then the moment M x of the mass with respect to the x -axis is given by M x = m 1 y 1 . Similarly, the moment M y with respect to the y -axis is given by M y = m 1 x 1 . Notice that the x -coordinate of the point is used to calculate the moment with respect to the y -axis, and vice versa. The reason is that the x -coordinate gives the distance from the point mass to the y -axis, and the y -coordinate gives the distance to the x -axis (see the following figure).
Figure 6.64 Point mass m 1 is located at point ( x 1 , y 1 ) in the plane.
If we have several point masses in the xy -plane, we can use the moments with respect to the x - and y -axes to calculate the
Made with FlippingBook - professional solution for displaying marketing and sales documents online