The center of mass of a solid is the balance point of a three-dimensional object. If the solid has uniform density, this point depends only on its shape, but if the density varies, heavier regions pull the center of mass toward them. Calculus gives a precise way to find this point by adding up tiny pieces of mass throughout the volume.
This idea is important in physics, engineering, robotics, astronomy, and any situation where the stability or motion of a solid object matters.
For a solid region E with density rho(x,y,z), the total mass is found with a triple integral over the volume. The coordinates of the center of mass are ratios of moment integrals to total mass, so each coordinate measures how mass is distributed relative to an axis or coordinate plane. Symmetry can greatly simplify the work because the center of mass must lie on any plane, axis, or point of symmetry.
A worked calculation usually begins by describing the region, choosing bounds, integrating the density for mass, then computing the three moments and dividing by mass.
Key Facts
- Mass of a solid: M = ∭_E rho(x,y,z) dV
- Center of mass coordinates: xbar = M_yz / M, ybar = M_xz / M, zbar = M_xy / M
- Moment about the yz-plane: M_yz = ∭_E x rho(x,y,z) dV
- Moment about the xz-plane: M_xz = ∭_E y rho(x,y,z) dV
- Moment about the xy-plane: M_xy = ∭_E z rho(x,y,z) dV
- For constant density rho = k, the density cancels in xbar, ybar, and zbar, so the center of mass equals the centroid.
Vocabulary
- Center of mass
- The point where the mass of a solid can be considered concentrated for analyzing balance and translational motion.
- Density function
- A function rho(x,y,z) that gives mass per unit volume at each point in a solid.
- Triple integral
- An integral over a three-dimensional region used to add quantities throughout a volume.
- Moment
- A weighted integral that measures how mass is distributed relative to a coordinate plane or axis.
- Centroid
- The geometric center of a region, equal to the center of mass when density is constant.
Common Mistakes to Avoid
- Using volume instead of mass for nonuniform density is wrong because density must be included in both the numerator moments and the denominator mass.
- Forgetting which moment matches which coordinate is wrong because xbar uses the moment M_yz, ybar uses M_xz, and zbar uses M_xy.
- Writing bounds that do not describe the entire solid is wrong because the triple integral will then add mass over the wrong region.
- Assuming the center of mass is always at the visual middle is wrong because varying density or an asymmetric shape can move it away from the geometric center.
Practice Questions
- 1 A rectangular box has 0 <= x <= 2, 0 <= y <= 4, 0 <= z <= 6 and constant density. Find its center of mass.
- 2 A solid occupies the unit cube 0 <= x <= 1, 0 <= y <= 1, 0 <= z <= 1 with density rho(x,y,z) = 2x. Find its mass M and x-coordinate xbar of the center of mass.
- 3 A solid has uniform density and is symmetric about the planes x = 0 and y = 0, but not about any horizontal plane. What can you conclude about xbar and ybar, and why can you not immediately determine zbar?