Center of Mass

Introduction to Coordinate Systems in Integration

• Purpose: Exploring a different coordinate system for integrals.

Understanding the Setup

• Begin with the equation of a circular cylinder:

  • $x^2 + y^2 = 4$

  • The cylinder has a radius of 2.

  • Domain of x: $-2 \leq x \leq 2$

  • y values lead to both positive and negative square roots of y.

  • Area of integration is within the cylinder, represented in the xy-plane.

• Note: The footprint (area) defined by $x^2 + y^2 = 4$ is not part of the solid over which we are integrating.

Visual Representation

• Visualizing the cylinder:

  • It encompasses all points inside a circle of radius 2 on the xy-plane.

• Analysis of z values:

  • Rearranging gives:

  • $z = 2 \pm \sqrt{4 - x^2 - y^2}$

  • The equation represents a sphere centered at another point. Specifically, it translates to:

  • Relation: $x^2 + y^2 + (z - 2)^2 = 4$

  • This shape resembles a ball positioned above the origin, resting on the xy-plane.

Coordinate Transformation

• Spherical Coordinates:

  • Transitioning into spherical coordinates involves using:

  • $x = \rho \sin \phi \cos \theta$

  • $y = \rho \sin \phi \sin \theta$

  • $z = \rho \cos \phi$

• Upon squaring the coordinates, the simplification leads to:

  • $\rho^2 - 4z = 0$

• Transforming to spherical coordinates gives us the equation:

  • $\rho = 4 \cos \phi$

  • Which describes a sphere on the xy-plane.

Graphical Representation in Spherical Coordinates

• When analyzing the angle phi:

  • At $\phi = 0$, $\rho = 4$ which is at the top of the sphere.

  • As $\phi$ increases, $\rho$ shrinks, demonstrating the volume of integration.

• Limitations:

  • Avoid values where $\rho$ becomes negative; valid limits only exist if the sphere remains above the xy-plane.

Integration in Spherical Coordinates

• Importance of maintaining the volume element for integration:

  • $dV = \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta$

• Function for integration, since $x^2 + y^2 + z^2$ translates into spherical coordinates as:

  • $\rho^2$

• The integral involves calculating:

  • $\int \rho^3 \, dV$

  • Result: $\frac{4096}{21} \pi$ (simplified result of integration).

Triple and Double Integrals in Applications

• Starting point: Integrals can be applied to find mass when combined with density functions.

• Density Function (Coefficient of mass density):

  • Notation: $\delta$ for two-dimensional analysis.

  • Indicates mass density in area (grams per square centimeter).

• The double integral of a density function gives the total mass:

  • For mass calculation, consider tiny rectangle areas, $mass \approx density \times area$.

Mass Moments and Center of Mass

• First Mass Moment:

  • Notated as $M_y$, represents value weighted by distance from y-axis.

• Moment statistical description:

  • Involves density and distance tallies.

  • Approach: Integrate the density function weighted by the respective coordinate (x or y).

• To find the center of mass:

  • Use the relations $\bar{x} = \frac{M<em>y}{m}$ and $\bar{y} = \frac{M</em>x}{m}$ where m is the total mass.

Example Problem Demonstration

• Consider the curve $y = x$ and the area between $0\leq x \leq 1$.

• Density function is proportional to distance from the y-axis (thus $kx$).

• Integrals needed to find mass and moments:

  • $m = k \int_0^1 (x^2)dx$.

  • Final calculation indicates $\frac{k}{12}$ for mass.

• Calculation involves repeated integration for moments, leading to coordinates of the center of mass:

  • Result: Center of mass located at $\left(\frac{3}{5}, \frac{1}{2} \right)$

Moving into Three-Dimensional Problems

• Explore triple integrals focusing on volumes, specifically tetrahedrons:

  • Example defined by density function over a triangular base.

• Integrations must be set up correctly to account for volume:

  • $mass = \int \int \int \delta(x,y,z) \, dz \, dy \, dx$

  • For valid regions, arrange appropriate limits based on the geometric shape.

• Employ density functions to approximate mass accurately - yields real-life applications in physics and engineering contexts.

Conclusion

• Review forming integrals in various coordinate systems, recognizing changes in geometry.

• Maintain clarity in definitions and transformations for smooth calculations in multiple dimensions.