Name: balance point of a variable-density plate
Uploaded: 2026-06-08T09:38:40.125Z
Duration: 1 min 23 s
Channel: JoVE

一块平板金属在某个特定点（称为质心）处达到平衡。这种平衡取决于材料的密度，而密度可能在物体内部的不同位置有所变化。从概念上讲，质心是物体质量在 x 和 y 坐标上分布的加权平均值。此处，分子是相对于某一给定轴的力矩。为了求出质心，将该区域划分为许多微小单元。每个单元具有微小的面积和由局部密度决定的微小质量。通过在材料域上对这些质量微元进行二重积分，可得到平板的总质量。为了确定质心，需要计算相对于坐标轴的力矩。对 x 轴的矩等于质量与垂直距离的乘积，而对 y 轴的矩等于质量与水平距离的乘积。通过对整个区域内的各个单独效应进行求和，总力矩由二重积分给出。最后，将坐标轴上的力矩除以总质量，即可得到质心的坐标。

力矩与质心

Moments and Centers of Mass

一块平板金属或薄板在其质量中心点处达到平衡。该点不仅取决于薄板的形状，还取决于其内部质量的分布情况。如果材料的密度不均匀，某些区域对平衡点的贡献会大于其他区域。从数学上讲，质量中心被视为薄板内所有点的加权平均值，其中权重由密度决定。假设薄板占据 xy 平面上的一个区域 D，且其在每一点的密度由 ρ(x,y) 给出。为了分析质量分布，将该区域划分为许多微小的面积元素。每个微小元素的面积为 dA，其近似质量可通过将其面积乘以局部密度得到。将整个区域内的所有这些微小质量贡献相加，即可得到总质量：\begin{equation*}m = \iint_{D} \rho(x, y)\, dA\end{equation*}这个二重积分表示板上质量的连续总和。为了确定平衡点，需计算关于坐标轴的力矩。力矩用于衡量质量产生绕轴旋转趋势的程度。关于 x 轴的力矩取决于垂直距离，因此每个质量元素以其 y 坐标作为权重：\begin{equation*}M_x ≈ Σ y ρ(x,y) ΔA\end{equation*}对 y 轴的力矩取决于水平距离，因此每个质量元素都以其 x 坐标作为权重\begin{equation*}M_y \approx \Sigma x \,\rho(x,y)\,\Delta A\end{equation*}质心的坐标通过将每个力矩除以总质量得到：\begin{equation*}\bar{x} = \jfrac{M_y}{m}\end{equation*}\begin{equation*}\bar{y} = \jfrac{M_x}{m}\end{equation*}这些公式表明，质心是平板的质量在两个坐标方向上均达到平衡的点。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Multiple Integrals and Applications

多重积分及其应用

KEY TERMS AND CONCEPTS

关键术语与概念

矩与质心

balance point of a variable-density plate

A flat metal plate, or lamina, balances at a point called its center of mass. That balance point depends on the plate’s shape and on how mass is spread through the material. If the density changes from place to place, some regions affect the balance more than others.

The center of mass can be treated as a weighted average of every point in the plate. For a plate in the xy-plane, the density at each point is written as ρ(x, y). To study the mass distribution, the region D is divided into many small area elements. Each element has area dA, and its approximate mass is found by multiplying area by the local density.

Adding those small mass pieces across the whole region gives the total mass. This is written as a double integral, m = ∬D ρ(x, y) dA. The integral represents a continuous sum of mass over the plate.

To find the balance point, moments about the coordinate axes are computed. A moment measures the tendency of mass to cause rotation about an axis. The moment about the x-axis depends on vertical distance, so each mass element is weighted by its y-coordinate. The moment about the y-axis depends on horizontal distance, so each mass element is weighted by its x-coordinate.

The center of mass coordinates are found by dividing each moment by the total mass. The x-coordinate is x̄ = My/m, and the y-coordinate is ȳ = Mx/m. These formulas show the point where the distributed mass of the plate balances in both coordinate directions.

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Learn how a lamina’s center of mass depends on shape and density, and how double integrals and moments are used to find the balance point.

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The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Balance Point of a Variable-Density Plate

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