Calculus: Center of Mass and Centroids

Vocabulary practice flashcards covering master formulas, point masses, 1-D wires, 2-D plates, centroids, 3-D solids, and common exam traps for Section 3.8: Center of Mass.

Master Center-of-Mass Formula

The foundational principle behind all center-of-mass problems: $\text{coordinate} = \frac{\text{moment}}{\text{mass}} = \frac{\int (\text{position of slice})\,dm}{\int dm}$. The general procedure is to find $dm$, determine the position of the slice, integrate $\text{position} \times dm$, and divide by $\int dm$.

For example, masses of $2\,\text{kg}$ at $(0,0)$, $3\,\text{kg}$ at $(4,0)$, and $5\,\text{kg}$ at $(1,3)$ give total mass $M = 10$, with coordinates $\bar{x} = 1.7$ and $\bar{y} = 1.5$.

Center of Mass for Discrete Point Masses: Calculated via a weighted average without integrals: $\bar{x} = \frac{\sum m_i x_i}{\sum m_i}$ and $\bar{y} = \frac{\sum m_i y_i}{\sum m_i}$.

For example, $8\,\text{kg}$ at $(0,0)$ and $4\,\text{kg}$ at $(t,t)$ yield $\bar{x} = \frac{4t}{12} = \frac{t}{3}$ and $\bar{y} = \frac{t}{3}$.

Time-Dependent Point Masses: Point mass systems where coordinates are functions of time, such as $\vec{p}_i(t)$. The standard discrete weighted average formula is applied, producing center-of-mass coordinates that are functions of $t$.

For a thin wire positioned along the x-axis from $a$ to $b$ with length-density $\delta_\ell(x)$ ,

total mass is $M = \int_a^b \delta_\ell(x)\,dx$, the moment is $M\bar{x} = \int_a^b x\delta_\ell(x)\,dx$, and the center of mass is $\bar{x} = \frac{M\bar{x}}{M}$.

Constant-Density Wire

A one-dimensional wire along the x-axis from $a$ to $b$ with uniform density. Its center of mass is located directly at the geometric midpoint $\bar{x} = \frac{a+b}{2}$ without requiring any integration.

Integration-by-Parts Moment Integrals

Wire center-of-mass problems where the density function $\delta(x)$ involves expressions such as $e^x$, $\ln(x)$, $x\ln(x)$, or $\cos(x)$. In these cases, computing the moment integral $M\bar{x} = \int x\delta(x)\,dx$ yields integrands like $xe^x$ or $x^2 \ln(x)$ that require integration by parts.

Mass of a Two-Dimensional Plate ($M$)

The total mass of a flat lamina bounded between top curve $f(x)$ and bottom curve $g(x)$ from $a$ to $b$ with density $\delta(x)$, obtained using vertical strips $dm = \delta(x)(f(x) - g(x))\,dx$ to give $M = \int_a^b \delta(x)(f(x) - g(x))\,dx$.

Plate Moment about the y-Axis ($M\bar{x}$)

The moment of a planar plate bounded by top curve $f(x)$ and bottom curve $g(x)$ with density $\delta(x)$ about the y-axis, defined by $M\bar{x} = \int_a^b x\delta(x)(f(x) - g(x))\,dx$.

Plate Moment about the x-Axis ($M\bar{y}$)

The moment of a planar plate bounded by top curve $f(x)$ and bottom curve $g(x)$ with density $\delta(x)$ about the x-axis, defined by $M\bar{y} = \int_a^b \delta(x) \frac{f(x)^2 - g(x)^2}{2}\,dx$. This arises because each vertical strip balances at its midpoint $\frac{f+g}{2}$, and $\frac{f+g}{2}(f - g) = \frac{f^2 - g^2}{2}$.

Crossing Curves Limits Method

The procedure used when finding the region between two curves without given interval limits: set $f(x) = g(x)$ to solve for the boundary points, then test a value strictly between those points to determine which curve is on top ($f$) and which is on the bottom ($g$).

Plate Defined by Vertices

A planar region defined by vertex coordinates (such as a triangle with vertices at $(0,0)$, $(1,1)$, and $(1,-1)$). Solving requires writing out the linear equation of each bounding edge to identify top curve $f(x)$ and bottom curve $g(x)$ over the relevant interval.

Plate with Constant Horizontal Boundaries

A rectangular plate or region bounded by lines such as $y = 1$ and $y = 5$, represented by constant functions $f(x) = 5$ and $g(x) = 1$. For a rectangle $[1,6] \times [1,5]$ where density $\delta = e^x$ does not depend on $y$, $\bar{y} = 3$ by inspection.

Centroid of a Two-Dimensional Region

The geometric center of a planar region where density is constant and cancels out. Given area $A = \int_a^b (f - g)\,dx$, the coordinates are $\bar{x} = \frac{1}{A} \int_a^b x(f - g)\,dx$ and $\bar{y} = \frac{1}{2A} \int_a^b (f^2 - g^2)\,dx$.

Centroid Symmetry Principle

A rule stating that if both the shape of a region and its density are symmetric across a line, the center of mass must lie along that line of symmetry, allowing one to skip evaluating that coordinate's integral (e.g., symmetry across the y-axis guarantees $\bar{x} = 0$).

Centroid of a Half-Disk

For the upper half-disk $y = \sqrt{r^2 - x^2}$, symmetry yields $\bar{x} = 0$, area is $A = \frac{\pi r^2}{2}$, and the vertical centroid is $\bar{y} = \frac{1}{2A} \int_{-r}^r (r^2 - x^2)\,dx = \frac{4r}{3\pi}$. Squaring $f$ eliminates the square root.

Centroid of a Right Triangle

Located $\frac{1}{3}$ of the way inward from the right-angled corner along each leg. For example, for the region beneath $y = kx$ on the interval $[0,b]$, the centroid is given by $\left(\frac{2b}{3}, \frac{kb}{3}\right)$.

Center of Mass of a 3-D Solid with Cross-Sectional Area

For a three-dimensional solid with horizontal slab cross-sectional area $A(z)$ and uniform density, the slab mass element is $dm = \delta A(z)\,dz$ and the vertical centroid is $\bar{z} = \frac{\int zA(z)\,dz}{\int A(z)\,dz}$.

Centroid Height of a Cone

A standard geometric result stating that the centroid of any cone, regardless of base shape, is located $\frac{1}{4}$ of the total height up from the base (or $\frac{3}{4}$ of the height down from the apex).

Density Cancellation Fallacy

The mistaken idea that a variable density function $\delta(x)$ can be cancelled from the ratio $\frac{\int x\delta\,dx}{\int \delta\,dx}$. Density can only factor out and cancel when $\delta$ is constant across the entire region.

Mid-Mass Point vs. Center of Mass

The mid-mass point divides the total mass into two equal halves ($50\%$ on each side), whereas the center of mass additionally weights mass by its distance from the origin. For $\delta = e^{-x}$ on $[0,4]$, the mid-mass point is approximately $0.675$, whereas $\bar{x} \approx 0.925$.

Validity of a Density Function

The requirement that a real-world density function must satisfy $\delta \ge 0$ everywhere on the specified interval, because negative mass density is physically meaningless. For example, $\delta(x) = x^2 - 3$ is valid on $[2,5]$ but invalid on $[1,4]$ where $x < \sqrt{3}$.

“Set Up Only, Do Not Evaluate” Protocol

A problem-solving instruction requiring the student to write the definite integrals for $M$, $M\bar{x}$, and $M\bar{y}$ with explicit integration limits, boundary curves $f$ and $g$, and density $\delta$, state $\bar{x} = \frac{M\bar{x}}{M}$ and $\bar{y} = \frac{M\bar{y}}{M}$, and stop without computing antiderivatives.

Center-of-Mass Sanity Check

A verification step confirming that the computed center of mass shifts toward the heavier or wider portions of the object and, for a single region with $f \ge g$, lies strictly within the bounding box of the geometric shape.