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Moments  One Dimension
Two children on a weightless seesaw will balance perfectly if the product of their masses and their distances from the fulcrum are equal, m_{1}d_{1} = m_{2}d_{2} (Figure 6.6.5). Figure 6.6.5 For example, a 60 lb child 6 feet from the fulcrum will balance a 40 lb child 9 feet from the fulcrum, 60 · 6 = 40 · 9. If the fulcrum is at the origin x = 0, the masses m_{1 }and m_{2} have coordinates x_{l} = d_{l} and x_{2} = d_{2}. The equation for balancing becomes m_{1 }x_{l} + m_{2 }x_{2} = 0. Similarly, finitely many masses m_{1}, ..., m_{k} at the points x_{1},... ,x_{k} will balance about the point x = 0 if m_{1 }x_{l} + ... + m_{k }x_{k} = 0. Given a mass m at the point x, the quantity mx is called the moment about the origin. The moment of a finite collection of point masses m_{1} ..., m_{k} at x_{1}, ..., x_{k }about the origin is defined as the sum M = m_{1}x_{l} + ... + m_{k}x_{k}. Suppose the point masses are rigidly connected to a rod of mass zero. If the moment M is equal to zero, the masses will balance at the origin. In general they will balance at a point x called the center of gravity (Figure 6.6.6). x is equal to the moment divided by the total mass m, Since the mass m is positive, the moment M has the same sign as the center of gravity x. Figure 6.6.6 Now consider a length of wire between x = a and x = b whose density at x is p(x). The moment of the wire about the origin is defined as the integral This formula is justified by considering a piece of the wire of infinitesimal length Δx. On the piece from x to x + Δx the density remains infinitely close to ρ(x). Thus if ΔM is the moment of the piece, ΔM ≈ x Δm ≈ xp(x) Δx (compared to Δx). The moment of an object is equal to the sum of the moments of its parts. Hence by the Infinite Sum Theorem, If the wire has moment M about the origin and mass m, the center of mass of the wire is defined as the point x = M/m. A point of mass m located at x has the same moment about the origin as the whole wire, M = xm. Physically, the wire will balance on a fulcrum placed at the center of mass.


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