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 6.6.5

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 m₁ and m₂ have coordinates x_l = -d_l and x₂ = d₂. The equation for balancing becomes

m₁ x_l + m₂ x₂ = 0.

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

m₁ 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 m₁ ..., m_k at x₁, ..., x_k about the origin is defined as the sum

M = m₁x_l + ... + m_kx_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.

Example 3