The total mass of an object is determined by integrating its density function over its volume, and this process is significantly influenced by the chosen coordinate system (e.g., Cartesian for a cube versus spherical for a sphere) and the shape's symmetry. The animation effectively highlights this by showing how the same distance-dependent density function yields distinct total mass values when applied to different geometries like a cube and a sphere. This emphasizes the interplay between geometry and density distribution in determining an object's overall mass.
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The total mass of an object with varying density is calculated by integrating the density function over its volume.
Choosing the appropriate coordinate system (Cartesian for a cube, spherical for a sphere) simplifies the calculation of the volume integral.
For shapes with symmetry, like the cube in this example, recognizing that parts of the integral are identical allows for solving only one part and multiplying the result, streamlining the computation.
The given density function $\rho(x)=\frac{\rho_0}{L^2} r^2$ indicates that the material density increases with the square of the distance from the origin.
For a cube of side length $L$ centered at the origin (or with edges aligned along the axes from 0 to $L$ ), the total mass is $M_{\text {cube }}=\rho_0 L^3$.
For a sphere of radius $L$ centered at the origin, the total mass is $M_{\text {splexe }}=\frac{4 \pi \rho_0 L^3}{5}$.
Even for the same characteristic length $L$ and density factor $\rho_0$, the total mass enclosed within a cube is significantly different from the mass within a sphere due to the different geometries and the density function's dependence on distance.
The animation cycles between two different volumes: a cube and a sphere. This feature directly addresses the two parts of the problem and allows for a direct comparison of the results. You can see how the same density function, when integrated over different shapes, yields two distinct total mass values. The on-screen text updates to display the calculated mass for each volume, reinforcing the final answers derived in the analysis.
how to calculate mass in a non-uniform density field by using volume integration
how to calculate mass in a non-uniform density field by using volume integration