12.1 Calculation of mass, moments, and centers of mass

Double integrals are powerful tools for calculating mass, moments, and centers of mass in planar regions. They help us understand how mass is distributed across objects, which is crucial for engineering and physics applications.

By integrating density functions over regions, we can find total mass, moments, and centers of mass. These calculations are essential for analyzing object behavior, designing structures, and solving real-world problems involving mass distribution and rotation.

Mass and Density

Mass Density Function and Total Mass

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• Mass density function $\rho(x,y)$ represents the mass per unit area at a point $(x,y)$ in a planar region
• Total mass $M$ of a planar region $R$ can be calculated using a double integral of the mass density function over the region: $M = \iint_R \rho(x,y) \, dA$
• If the mass density is constant, the total mass simplifies to the product of the density and the area of the region: $M = \rho A$
• Example: A rectangular plate with dimensions $2 \, \text{m} \times 3 \, \text{m}$ has a constant mass density of $5 \, \text{kg/m}^2$. The total mass is $M = 5 \, \text{kg/m}^2 \times (2 \, \text{m} \times 3 \, \text{m}) = 30 \, \text{kg}$

Planar Region and Double Integrals

• A planar region is a two-dimensional area in the $xy$-plane over which a double integral is evaluated
• Double integrals are used to calculate quantities such as mass, moments, and center of mass for planar regions
• The limits of integration for a double integral are determined by the boundaries of the planar region
• Example: For a circular region with radius $R$ centered at the origin, the double integral in polar coordinates is: $\iint_R f(r,\theta) \, dA = \int_0^{2\pi} \int_0^R f(r,\theta) \, r \, dr \, d\theta$

Moments

First and Second Moments

• The first moment of a planar region about the $x$-axis is given by: $M_x = \iint_R y \, dA$
• The first moment about the $y$-axis is: $M_y = \iint_R x \, dA$
• The second moment (moment of inertia) of a planar region about the $x$-axis is: $I_x = \iint_R y^2 \, dA$
• The second moment about the $y$-axis is: $I_y = \iint_R x^2 \, dA$
• These moments are used to calculate the center of mass and describe the distribution of mass in a planar region

Moment of Inertia and Applications

• The moment of inertia measures an object's resistance to rotational acceleration about a given axis
• For a planar region with mass density $\rho(x,y)$, the moment of inertia about the $z$-axis (perpendicular to the $xy$-plane) is: $I_z = \iint_R (x^2 + y^2) \rho(x,y) \, dA$
• Moments of inertia are important in engineering applications involving rotating objects, such as flywheels, gears, and propellers
• Example: A thin circular plate with radius $R$ and constant mass density $\rho$ has a moment of inertia about its center given by: $I_z = \frac{1}{2} \rho \pi R^4$

Center of Mass

Center of Mass and Centroid

• The center of mass $(\bar{x}, \bar{y})$ of a planar region with mass density $\rho(x,y)$ is given by: $\bar{x} = \frac{M_y}{M}, \quad \bar{y} = \frac{M_x}{M}$ where $M_x$, $M_y$, and $M$ are the first moments and total mass, respectively
• For a planar region with constant density, the center of mass coincides with the centroid, which is the geometric center of the region
• The centroid $(\bar{x}, \bar{y})$ of a planar region $R$ can be calculated using: $\bar{x} = \frac{1}{A} \iint_R x \, dA, \quad \bar{y} = \frac{1}{A} \iint_R y \, dA$ where $A$ is the area of the region

Calculating Center of Mass with Double Integrals

• To find the center of mass of a planar region with variable mass density, use double integrals to calculate the first moments and total mass: $M_x = \iint_R y \rho(x,y) \, dA, \quad M_y = \iint_R x \rho(x,y) \, dA, \quad M = \iint_R \rho(x,y) \, dA$
• Substitute these values into the center of mass formulas: $\bar{x} = \frac{M_y}{M}, \quad \bar{y} = \frac{M_x}{M}$
• Example: For a semicircular region with radius $R$ and mass density $\rho(x,y) = xy$, the center of mass is located at $(\frac{4R}{3\pi}, \frac{4R}{3\pi})$