Moments and Centers of Mass: Fresh Take

  • Find the balance point (center of mass) of straight objects and flat surfaces
  • Utilize a shape’s symmetry to find the centroid, or geometric center, of flat objects
  • Use Pappus’s theorem to calculate the volume of an object

Center of Mass and Moments

The Main Idea 

  • Center of Mass:
    • The balancing point of an object or system
    • Point where the total mass can be concentrated without changing the object’s behavior
  • Moments:
      • Measure of the rotational force applied to an object
      • First moment with respect to origin: [latex]M = \sum_{i=1}^n m_i x_i[/latex]
  • One-Dimensional Systems:
    • Center of mass: [latex]\overline{x} = \frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i}[/latex]
  • Two-Dimensional Systems:
    • Moments: [latex]M_x = \sum_{i=1}^n m_i y_i[/latex], [latex]M_y = \sum_{i=1}^n m_i x_i[/latex]
    • Center of mass: [latex]\overline{x} = \frac{M_y}{m}[/latex], [latex]\overline{y} = \frac{M_x}{m}[/latex]

Problem-Solving Strategy

  1. Identify the system (point masses or continuous object)
  2. Choose an appropriate coordinate system
  3. Calculate the total mass of the system
  4. Compute moments with respect to axes
  5. Use formulas to find the center of mass coordinates

Find the center of mass for three point masses:

  • [latex]m_1 = 2\text{ kg}[/latex] at [latex](-1, 3)[/latex]
  • [latex]m_2 = 6\text{ kg}[/latex] at [latex](1, 1)[/latex]
  • [latex]m_3 = 4\text{ kg}[/latex] at [latex](2, -2)[/latex]

Suppose four point masses are placed on a number line as follows:

[latex]\begin{array}{cccc}{m}_{1}=12\text{kg,}\text{ placed at }{x}_{1}=-4\text{m}\hfill & & & {m}_{2}=12\text{kg,}\text{ placed at }{x}_{2}=4\text{m}\hfill \\ {m}_{3}=30\text{kg,}\text{ placed at }{x}_{3}=2\text{m}\hfill & & & {m}_{4}=6\text{kg,}\text{ placed at }{x}_{4}=-6\text{m}.\hfill \end{array}[/latex]

Find the moment of the system with respect to the origin and find the center of mass of the system.

Suppose three point masses are placed on a number line as follows (assume coordinates are given in meters):

[latex]\begin{array}{c}{m}_{1}=5\text{kg, placed at}(-2,-3),\hfill \\ {m}_{2}=3\text{kg, placed at}(2,3),\hfill \\ {m}_{3}=2\text{kg, placed at}(-3,-2).\hfill \end{array}[/latex]

Find the center of mass of the system.

Center of Mass of Thin Plates

The Main Idea 

  • Lamina:
    • Thin sheet with uniform density
    • Mass distributed continuously across a 2D region
  • Centroid:
    • Geometric center of a 2D shape
    • Coincides with center of mass for uniform density
  • Key Formulas:
    • Mass: [latex]m = \rho \int_a^b f(x) dx[/latex]
    • [latex]x[/latex]-axis Moment: [latex]M_x = \rho \int_a^b \frac{[f(x)]^2}{2} dx[/latex]
    • [latex]y[/latex]-axis Moment: [latex]M_y = \rho \int_a^b x f(x) dx[/latex]
    • Center of Mass: [latex]\overline{x} = \frac{M_y}{m}, \overline{y} = \frac{M_x}{m}[/latex]
  • Symmetry Principle:
    • If a region is symmetric about a line, its centroid lies on that line

Problem-Solving Strategy

  1. Identify the region [latex]R[/latex] and its bounding function [latex]f(x)[/latex]
  2. Set up integrals for mass and moments ([latex]M_x[/latex] and [latex]M_y[/latex])
  3. Evaluate the integrals
  4. Calculate center of mass coordinates using [latex]M_x[/latex], [latex]M_y[/latex], and [latex]m[/latex]
  5. Simplify and interpret the results

Let [latex]R[/latex] be the region bounded above by the graph of the function [latex]f(x)={x}^{2}[/latex] and below by the [latex]x[/latex]-axis over the interval [latex]\left[0,2\right].[/latex] Find the centroid of the region.

Let R be the region bounded above by the graph of the function [latex]f(x)=6-{x}^{2}[/latex] and below by the graph of the function [latex]g(x)=3-2x.[/latex] Find the centroid of the region.

Find the centroid of the region bounded by [latex]y = \sqrt{x}[/latex] and the [latex]x[/latex]-axis from [latex]x = 0[/latex] to [latex]x = 4[/latex].

Center of Mass of a Region Bounded by Two Functions

The Main Idea 

  • Region Definition:
    • Bounded above by function [latex]f(x)[/latex]
    • Bounded below by function [latex]g(x)[/latex]
    • Bounded on sides by [latex]x = a[/latex] and [latex]x = b[/latex]
  • Key Formulas:
    • Mass: [latex]m = \rho \int_a^b [f(x) - g(x)] dx[/latex]
    • [latex]x[/latex]-axis Moment: [latex]M_x = \rho \int_a^b \frac{1}{2}([f(x)]^2 - [g(x)]^2) dx[/latex]
    • [latex]y[/latex]-axis Moment: [latex]M_y = \rho \int_a^b x[f(x) - g(x)] dx[/latex]
    • Center of Mass: [latex]\overline{x} = \frac{M_y}{m}, \overline{y} = \frac{M_x}{m}[/latex]
  • Centroid of Rectangle:
    • Located at [latex](x_i^, \frac{f(x_i^) + g(x_i^*)}{2})[/latex]

Problem-Solving Strategy

  1. Identify bounding functions [latex]f(x)[/latex] and [latex]g(x)[/latex] and integration limits
  2. Set up integrals for mass and moments ([latex]M_x[/latex] and [latex]M_y[/latex])
  3. Evaluate the integrals, often requiring integration by parts or substitution
  4. Calculate center of mass coordinates using [latex]M_x[/latex], [latex]M_y[/latex], and [latex]m[/latex]
  5. Simplify and interpret the results

The Symmetry Principle

The Main Idea 

  • Symmetry Principle:
    • If a region is symmetric about a line, its centroid lies on that line
    • Simplifies calculations for symmetric shapes
  • Theorem of Pappus for Volume:
    • Volume = Area of region × Distance traveled by centroid
    • Applies to solids of revolution

Problem-Solving Strategy

  1. Identify symmetry in the given region
  2. Use symmetry to simplify centroid calculations
  3. For solids of revolution:
    1. Calculate the area of the region
    2. Determine the path of the centroid
    3. Apply the Theorem of Pappus

Let R be the region bounded above by the graph of the function [latex]f(x)=1-{x}^{2}[/latex] and below by [latex]x[/latex]-axis. Find the centroid of the region.

Let [latex]R[/latex] be a circle of radius [latex]1[/latex] centered at [latex](3,0).[/latex] Use the theorem of Pappus for volume to find the volume of the torus generated by revolving [latex]R[/latex] around the [latex]y[/latex]-axis.

Find the volume of a washer-shaped solid formed by revolving the region between concentric circles of radii [latex]3[/latex] and [latex]5[/latex] centered at [latex](4,0)[/latex] around the [latex]y[/latex]-axis.