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Chapter 6 | Applications of Integration
Figure 6.63 The center of mass x – is the balance point of the system.
Thus, we have
– | = m
– | – ⎞ ⎠
m 1 | x 1 − x
2 | x 2 − x ⎛ ⎝ x 2 − x
⎛ ⎝ x – − x
⎞ ⎠ = m 2
m 1
1
– − m
1 x 1 = m 2 x 2 − m 2 x –
m 1 x
x – ( m
1 + m 2 ) = m 1 x 1 + m 2 x 2 x – = m 1 x 1 + m 2 x 2 m 1 + m 2 .
The expression in the numerator, m 1 x 1 + m 2 x 2 , is called the first moment of the system with respect to the origin. If the context is clear, we often drop the word first and just refer to this expression as the moment of the system. The expression in the denominator, m 1 + m 2 , is the total mass of the system. Thus, the center of mass of the system is the point at which the total mass of the system could be concentrated without changing the moment. This idea is not limited just to two point masses. In general, if n masses, m 1 , m 2 ,…, m n , are placed on a number line at points x 1 , x 2 ,…, x n , respectively, then the center of mass of the system is given by
∑ i =1 n
m i x i
x – =
.
∑ i =1 n
m i
Theorem 6.9: Center of Mass of Objects on a Line Let m 1 , m 2 ,…, m n be point masses placed on a number line at points x 1 , x 2 ,…, x n , respectively, and let m = ∑ i =1 n m i denote the total mass of the system. Then, the moment of the system with respect to the origin is given by (6.14) M = ∑ i =1 n m i x i and the center of mass of the system is given by (6.15) x – = M m .
We apply this theorem in the following example.
Example 6.29 Finding the Center of Mass of Objects along a Line
Suppose four point masses are placed on a number line as follows:
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