12.1 Calculation of mass, moments, and centers of mass
3 min read•august 6, 2024
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 ρ(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 of the mass density function over the region: M=∬Rρ(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=ρA
Example: A rectangular plate with dimensions 2m×3m has a constant mass density of 5kg/m2. The total mass is M=5kg/m2×(2m×3m)=30kg
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 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 is: ∬Rf(r,θ)dA=∫02π∫0Rf(r,θ)rdrdθ
Moments
First and Second Moments
The first moment of a planar region about the x-axis is given by: Mx=∬RydA
The first moment about the y-axis is: My=∬RxdA
The second moment () of a planar region about the x-axis is: Ix=∬Ry2dA
The second moment about the y-axis is: Iy=∬Rx2dA
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 ρ(x,y), the moment of inertia about the z-axis (perpendicular to the xy-plane) is: Iz=∬R(x2+y2)ρ(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 ρ has a moment of inertia about its center given by: Iz=21ρπR4
Center of Mass
Center of Mass and Centroid
The center of mass (xˉ,yˉ) of a planar region with mass density ρ(x,y) is given by: xˉ=MMy,yˉ=MMx where Mx, My, and M are the first moments and total mass, respectively
For a planar region with , the center of mass coincides with the , which is the geometric center of the region
The centroid (xˉ,yˉ) of a planar region R can be calculated using: xˉ=A1∬RxdA,yˉ=A1∬RydA 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: Mx=∬Ryρ(x,y)dA,My=∬Rxρ(x,y)dA,M=∬Rρ(x,y)dA
Substitute these values into the center of mass formulas: xˉ=MMy,yˉ=MMx
Example: For a semicircular region with radius R and mass density ρ(x,y)=xy, the center of mass is located at (3π4R,3π4R)