706
Chapter 6 | Applications of Integration
x - and y -coordinates of the center of mass of the system.
Theorem 6.10: Center of Mass of Objects in a Plane Let m 1 , m 2 ,…, m n be point masses located in the xy -plane at points ( x 1 , y 1 ), ( x 2 , y 2 ),…, ( x n , y n ), respectively, and let m = ∑ i =1 n m i denote the total mass of the system. Then the moments M x and M y of the system with respect to the x - and y -axes, respectively, are given by (6.16) M x = ∑ i =1 n m i y i and M y = ∑ i =1 n m i x i . Also, the coordinates of the center of mass ⎛ ⎝ x – , y – ⎞ ⎠ of the system are (6.17) x – = M y m and y – = M x m .
The next example demonstrates how to apply this theorem.
Example 6.30 Finding the Center of Mass of Objects in a Plane
Suppose three point masses are placed in the xy -plane as follows (assume coordinates are given in meters): m 1 = 2kg, placed at(−1, 3), m 2 = 6kg, placed at(1, 1), m 3 = 4kg, placed at(2, −2). Find the center of mass of the system.
Solution First we calculate the total mass of the system: m = ∑ i =1 3
m i =2+6+4=12kg.
Next we find the moments with respect to the x - and y -axes: M y = ∑ i =1 3
m i x i =−2+6+8=12,
M x = ∑ i =1 3
m i y i =6+6−8=4.
Then we have
M y m = 12 12 =1and
M x m = 4 12 =
x – =
y – =
1 3 .
The center of mass of the system is (1, 1/3), in meters.
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