Center Of Mass

The center of mass of the object is the point with coordinates

JUSTIFICATION

The piece of the object on an element of area ΔD has mass

Δm ≈ ρ(x, y) ΔA (compared to ΔA).

A point mass m at (x, y) has moments

M_x = ym, My = xm.

Therefore the piece of the object at ΔD has moments

ΔM, ≈ y Δm ≈ yp(x, y) ΔA (compared to ΔA),

ΔM_y ≈ x Δm ≈ xp(x, y) ΔA (compared to ΔA).

The double integrals for M_x and M_y now follow from the Infinite Sum Theorem.

An object will balance on a pin at its center of mass (Figure 12.4.3). The center of mass is useful in finding the work done against gravity when moving the object. The work is the same as if the mass were all concentrated at the center of mass, and is given by

W = mgs

where s is the distance the center of mass is raised and g is constant.

Figure 12.4.3

Example 2: Center of Mass

Example 3: Work