Centre of mass

Definition

Let $E \subseteq \mathbb{R}^n$ be a bounded Jordan measurable set, and let $\delta \colon E \to [0, \infty)$ be a continuous density with total mass $M = \int_E \delta(x) \, dx > 0$. The centre of mass of $E$ with density $\delta$ is the point $\bar{x} = (\bar{x}_1, \dots, \bar{x}_n)$ with $$\bar{x}_i = \frac{1}{M} \int_E x_i \, \delta(x) \, dx, \qquad i = 1, \dots, n.$$ If $\delta$ is constant, $\bar{x}$ is the centroid of $E$, and it does not depend on the value of the constant.

Remarks

For $n = 3$ the integrals are triple integrals, and for a plane lamina, $n = 2$, they are double integrals. For point masses $m_1, \dots, m_k$ at $p_1, \dots, p_k$, the centre of mass is $\sum_j m_j p_j / \sum_j m_j$. The definition above replaces the sum by an integral. Example: the half disc $\{x^2 + y^2 \le a^2, \ y \ge 0\}$ with constant density has centroid $(0, 4a / (3\pi))$. If $E$ and $\delta$ are symmetric under a reflection, then $\bar{x}$ lies in the mirror of that reflection.

Explanations

L1 human

The centre of mass is the balance point of a body. If the body is supported at that point, gravity does not turn it in any direction.