Center of Mass & Pappus's Theorem Reference Cheat Sheet

Key Facts

• For point masses on a line, the center of mass is $\bar{x} = \frac{\sum m_i x_i}{\sum m_i}$.
• For point masses in the plane, the center of mass is $\bar{x} = \frac{\sum m_i x_i}{M}$ and $\bar{y} = \frac{\sum m_i y_i}{M}$, where $M = \sum m_i$.
• For a thin rod on $a \le x \le b$ with density $\rho(x)$, mass is $M = \int_a^b \rho(x)\,dx$ and center of mass is $\bar{x} = \frac{1}{M}\int_a^b x\rho(x)\,dx$.
• For a plane region $R$ with constant density, the centroid is $\bar{x} = \frac{M_y}{A}$ and $\bar{y} = \frac{M_x}{A}$.
• For a region between $y = f(x)$ and $y = g(x)$ on $a \le x \le b$, area is $A = \int_a^b [f(x)-g(x)]\,dx$.
• For a region between $y = f(x)$ and $y = g(x)$, the moments are $M_y = \int_a^b x[f(x)-g(x)]\,dx$ and $M_x = \frac{1}{2}\int_a^b [f(x)^2-g(x)^2]du$.
• Pappus's centroid theorem for volume states that $V = A(2\pi d)$, where $d$ is the distance from the centroid of the plane region to the axis of rotation.
• Pappus's centroid theorem for surface area states that $S = L(2\pi d)$, where $L$ is arc length and $d$ is the distance from the curve's centroid to the axis of rotation.

Vocabulary

Center of mass

The balance point of a system, found by dividing total moment by total mass.

Centroid

The geometric center of a region or curve when density is constant.

Moment

A measure of rotational tendency, usually computed as distance times mass, area, or density.

Density function

A function such as $\rho(x)$ that describes how mass is distributed along an object.

Pappus's theorem

A theorem that finds volume or surface area by multiplying a centroid's circular path length by area or arc length.

Axis of rotation

The line around which a region or curve is rotated to form a solid or surface.