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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, m1d1 = m2d2 (Figure 6.6.5).

Figure 6.6.506_applications_of_the_integral-339.gif

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 m1 and m2 have coordinates xl = -dl and x2 = d2. The equation for balancing becomes

m1 xl + m2 x2 = 0.

Similarly, finitely many masses m1, ..., mk at the points x1,... ,xk will balance about the point x = 0 if

m1 xl + ... + mk xk = 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 m1 ..., mk at x1, ..., xk about the origin is defined as the sum

M = m1xl + ... + mkxk.

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.

Example 3

Last Update: 2010-11-25