{"id":472,"date":"2025-02-13T19:45:17","date_gmt":"2025-02-13T19:45:17","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/moments-and-centers-of-mass-learn-it-2\/"},"modified":"2025-02-13T19:45:17","modified_gmt":"2025-02-13T19:45:17","slug":"moments-and-centers-of-mass-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/moments-and-centers-of-mass-learn-it-2\/","title":{"raw":"Moments and Centers of Mass: Learn It 2","rendered":"Moments and Centers of Mass: Learn It 2"},"content":{"raw":"\n

Center of Mass of Thin Plates<\/h2>\n

We've examined point masses on a line and in a plane. Now, we look at systems where mass is distributed continuously across a thin sheet, called a lamina. We assume the lamina's density is constant.<\/p>\n

Laminas are often two-dimensional regions in a plane, with the geometric center called its centroid. The center of mass of a lamina depends only on the shape, not the density. For a rectangular lamina, the center of mass is where the diagonals intersect, which follows the symmetry principle.<\/p>\n

\n

the symmetry principle<\/h3>\n

If a region [latex]R[\/latex] is symmetric about a line [latex]l[\/latex], then the centroid of [latex]R[\/latex] lies on [latex]l[\/latex].<\/p>\n<\/section>\n

Let's examine general laminas. Suppose we have a lamina bounded above by the graph of a continuous function [latex]f(x),[\/latex] below by the [latex]x[\/latex]-axis, and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex].<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"261\"]\"This Figure 4. A region in the plane representing a lamina.[\/caption]\n\n\n

To find the center of mass, we need the total mass of the lamina. We divide the lamina into thin vertical strips, approximating each strip's mass using the density [latex]\u03c1[\/latex]. The mass of the strip is given by [latex]\\rho f({x}_{i}^{*})\\text{\u0394}x.[\/latex]<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"261\"]\"This Figure 5. A representative rectangle of the lamina.[\/caption]\n\n\n

To get the approximate mass of the lamina, we add the masses of all the rectangles to get<\/p>\n

[latex]m\\approx \\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho f({x}_{i}^{*})\\text{\u0394}x[\/latex]<\/div>\n

This is a Riemann sum. Taking the limit as [latex]n\\to \\infty [\/latex] gives the exact mass of the lamina:<\/p>\n

[latex]m=\\underset{n\\to \\infty }{\\text{lim}}\\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho f({x}_{i}^{*})\\text{\u0394}x=\\rho {\\displaystyle\\int }_{a}^{b}f(x)dx[\/latex]<\/div>\n

Next, we calculate the moment of the lamina with respect to the x-axis. For each rectangle, the center of mass is at [latex]x_{i}^{*}[\/latex]. The moment with respect to the [latex]x[\/latex]-axis is given by:<\/p>\n

[latex]{M}_{x}=\\underset{n\\to \\infty }{\\text{lim}}\\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho \\frac{{\\left[f({x}_{i}^{*})\\right]}^{2}}{2}\\text{\u0394}x=\\rho {\\displaystyle\\int }_{a}^{b}\\frac{{\\left[f(x)\\right]}^{2}}{2}dx.[\/latex]<\/div>\n

Similarly, the moment with respect to the [latex]y[\/latex]-axis is:<\/p>\n

[latex]{M}_{y}=\\underset{n\\to \\infty }{\\text{lim}}\\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho {x}_{i}^{*}f({x}_{i}^{*})\\text{\u0394}x=\\rho {\\displaystyle\\int }_{a}^{b}xf(x)dx[\/latex]<\/div>\n

The coordinates of the center of mass are:<\/p>\n

[latex]\\overline{x}=\\frac{{M}_{y}}{m} \\text{ and }\\overline{y}=\\frac{{M}_{x}}{m}[\/latex]<\/p>\n

\n

If we look closely at the expressions for [latex]{M}_{x},{M}_{y},\\text{ and }m,[\/latex] we notice that the constant [latex]\\rho [\/latex] cancels out when [latex]\\overline{x}[\/latex] and [latex]\\overline{y}[\/latex] are calculated.<\/p>\n<\/section>\n

We summarize these findings in the following theorem.<\/p>\n

\n

center of mass of a thin plate in the [latex]xy[\/latex]-plane<\/h3>\n

Let [latex]R[\/latex] denote a region bounded above by the graph of a continuous function [latex]f(x),[\/latex] below by the [latex]x[\/latex]-axis, and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex] respectively. Let [latex]\\rho [\/latex] denote the density of the associated lamina. Then we can make the following statements:<\/p>\n

    \n\t
  1. The mass of the lamina is
    \n
    [latex]m=\\rho {\\displaystyle\\int }_{a}^{b}f(x)dx.[\/latex]<\/div>\n<\/li>\n\t
  2. The moments [latex]{M}_{x}[\/latex] and [latex]{M}_{y}[\/latex] of the lamina with respect to the [latex]x[\/latex]- and [latex]y[\/latex]-axes, respectively, are
    \n
    [latex]{M}_{x}=\\rho {\\displaystyle\\int }_{a}^{b}\\frac{{\\left[f(x)\\right]}^{2}}{2}dx\\text{ and }{M}_{y}=\\rho {\\displaystyle\\int }_{a}^{b}xf(x)dx.[\/latex]<\/div>\n<\/li>\n\t
  3. The coordinates of the center of mass [latex](\\overline{x},\\overline{y})[\/latex] are
    \n
    [latex]\\overline{x}=\\frac{{M}_{y}}{m}\\text{ and }\\overline{y}=\\frac{{M}_{x}}{m}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n

    In the next example, we use this theorem to find the center of mass of a lamina.<\/p>\n

    \n

    Let R<\/em> be the region bounded above by the graph of the function [latex]f(x)=\\sqrt{x}[\/latex] and below by the [latex]x[\/latex]-axis over the interval [latex]\\left[0,4\\right].[\/latex] Find the centroid of the region.<\/p>\n\n\n[reveal-answer q=\"fs-id1167793960791\"]Show Solution[\/reveal-answer] [hidden-answer a=\"fs-id1167793960791\"]\n\n\n

    The region is depicted in the following figure.<\/p>\n\n\n[caption id=\"\" align=\"aligncenter\" width=\"266\"]\"This Figure 6. Finding the center of mass of a lamina.[\/caption]\n\n\n

    Since we are only asked for the centroid of the region, rather than the mass or moments of the associated lamina, we know the density constant [latex]\\rho [\/latex] cancels out of the calculations eventually. Therefore, for the sake of convenience, let\u2019s assume [latex]\\rho =1.[\/latex]<\/p>\n

    First, we need to calculate the total mass:<\/p>\n

    [latex]\\begin{array}{cc}\\hfill m& =\\rho {\\displaystyle\\int }_{a}^{b}f(x)dx={\\displaystyle\\int }_{0}^{4}\\sqrt{x}dx\\hfill \\\\ & ={\\frac{2}{3}{x}^{3\\text{\/}2}|}_{0}^{4}=\\frac{2}{3}\\left[8-0\\right]=\\frac{16}{3}.\\hfill \\end{array}[\/latex]<\/div>\n

    Next, we compute the moments:<\/p>\n

    [latex]\\begin{array}{cc}\\hfill {M}_{x}& =\\rho {\\displaystyle\\int }_{a}^{b}\\frac{{\\left[f(x)\\right]}^{2}}{2}dx\\hfill \\\\ & ={\\displaystyle\\int }_{0}^{4}\\frac{x}{2}dx={\\frac{1}{4}{x}^{2}|}_{0}^{4}=4\\hfill \\end{array}[\/latex]<\/div>\n

    and<\/p>\n

    [latex]\\begin{array}{cc}\\hfill {M}_{y}& =\\rho {\\displaystyle\\int }_{a}^{b}xf(x)dx\\hfill \\\\ & ={\\displaystyle\\int }_{0}^{4}x\\sqrt{x}dx={\\displaystyle\\int }_{0}^{4}{x}^{3\\text{\/}2}dx\\hfill \\\\ & ={\\frac{2}{5}{x}^{5\\text{\/}2}|}_{0}^{4}=\\frac{2}{5}\\left[32-0\\right]=\\frac{64}{5}.\\hfill \\end{array}[\/latex]<\/div>\n

    Thus, we have<\/p>\n

    [latex]\\overline{x}=\\frac{{M}_{y}}{m}=\\frac{64\\text{\/}5}{16\\text{\/}3}=\\frac{64}{5}\u00b7\\frac{3}{16}=\\frac{12}{5}\\text{ and }\\overline{y}=\\frac{{M}_{x}}{y}=\\frac{4}{16\\text{\/}3}=4\u00b7\\frac{3}{16}=\\frac{3}{4}.[\/latex]<\/div>\n

    The centroid of the region is [latex](12\\text{\/}5,3\\text{\/}4).[\/latex]<\/p>\n

    [\/hidden-answer]<\/p>\n<\/section>\n

    \n

    [ohm_question]5664[\/ohm_question]<\/p>\n<\/section>\n","rendered":"

    Center of Mass of Thin Plates<\/h2>\n

    We’ve examined point masses on a line and in a plane. Now, we look at systems where mass is distributed continuously across a thin sheet, called a lamina. We assume the lamina’s density is constant.<\/p>\n

    Laminas are often two-dimensional regions in a plane, with the geometric center called its centroid. The center of mass of a lamina depends only on the shape, not the density. For a rectangular lamina, the center of mass is where the diagonals intersect, which follows the symmetry principle.<\/p>\n

    \n

    the symmetry principle<\/h3>\n

    If a region [latex]R[\/latex] is symmetric about a line [latex]l[\/latex], then the centroid of [latex]R[\/latex] lies on [latex]l[\/latex].<\/p>\n<\/section>\n

    Let’s examine general laminas. Suppose we have a lamina bounded above by the graph of a continuous function [latex]f(x),[\/latex] below by the [latex]x[\/latex]-axis, and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex].<\/p>\n

    \"This
    Figure 4. A region in the plane representing a lamina.<\/figcaption><\/figure>\n

    To find the center of mass, we need the total mass of the lamina. We divide the lamina into thin vertical strips, approximating each strip’s mass using the density [latex]\u03c1[\/latex]. The mass of the strip is given by [latex]\\rho f({x}_{i}^{*})\\text{\u0394}x.[\/latex]<\/p>\n

    \"This
    Figure 5. A representative rectangle of the lamina.<\/figcaption><\/figure>\n

    To get the approximate mass of the lamina, we add the masses of all the rectangles to get<\/p>\n

    [latex]m\\approx \\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho f({x}_{i}^{*})\\text{\u0394}x[\/latex]<\/div>\n

    This is a Riemann sum. Taking the limit as [latex]n\\to \\infty[\/latex] gives the exact mass of the lamina:<\/p>\n

    [latex]m=\\underset{n\\to \\infty }{\\text{lim}}\\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho f({x}_{i}^{*})\\text{\u0394}x=\\rho {\\displaystyle\\int }_{a}^{b}f(x)dx[\/latex]<\/div>\n

    Next, we calculate the moment of the lamina with respect to the x-axis. For each rectangle, the center of mass is at [latex]x_{i}^{*}[\/latex]. The moment with respect to the [latex]x[\/latex]-axis is given by:<\/p>\n

    [latex]{M}_{x}=\\underset{n\\to \\infty }{\\text{lim}}\\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho \\frac{{\\left[f({x}_{i}^{*})\\right]}^{2}}{2}\\text{\u0394}x=\\rho {\\displaystyle\\int }_{a}^{b}\\frac{{\\left[f(x)\\right]}^{2}}{2}dx.[\/latex]<\/div>\n

    Similarly, the moment with respect to the [latex]y[\/latex]-axis is:<\/p>\n

    [latex]{M}_{y}=\\underset{n\\to \\infty }{\\text{lim}}\\underset{i=1}{\\overset{n}{\\text{\u2211}}}\\rho {x}_{i}^{*}f({x}_{i}^{*})\\text{\u0394}x=\\rho {\\displaystyle\\int }_{a}^{b}xf(x)dx[\/latex]<\/div>\n

    The coordinates of the center of mass are:<\/p>\n

    [latex]\\overline{x}=\\frac{{M}_{y}}{m} \\text{ and }\\overline{y}=\\frac{{M}_{x}}{m}[\/latex]<\/p>\n

    \n

    If we look closely at the expressions for [latex]{M}_{x},{M}_{y},\\text{ and }m,[\/latex] we notice that the constant [latex]\\rho[\/latex] cancels out when [latex]\\overline{x}[\/latex] and [latex]\\overline{y}[\/latex] are calculated.<\/p>\n<\/section>\n

    We summarize these findings in the following theorem.<\/p>\n

    \n

    center of mass of a thin plate in the [latex]xy[\/latex]-plane<\/h3>\n

    Let [latex]R[\/latex] denote a region bounded above by the graph of a continuous function [latex]f(x),[\/latex] below by the [latex]x[\/latex]-axis, and on the left and right by the lines [latex]x=a[\/latex] and [latex]x=b,[\/latex] respectively. Let [latex]\\rho[\/latex] denote the density of the associated lamina. Then we can make the following statements:<\/p>\n

      \n
    1. The mass of the lamina is\n
      [latex]m=\\rho {\\displaystyle\\int }_{a}^{b}f(x)dx.[\/latex]<\/div>\n<\/li>\n
    2. The moments [latex]{M}_{x}[\/latex] and [latex]{M}_{y}[\/latex] of the lamina with respect to the [latex]x[\/latex]– and [latex]y[\/latex]-axes, respectively, are\n
      [latex]{M}_{x}=\\rho {\\displaystyle\\int }_{a}^{b}\\frac{{\\left[f(x)\\right]}^{2}}{2}dx\\text{ and }{M}_{y}=\\rho {\\displaystyle\\int }_{a}^{b}xf(x)dx.[\/latex]<\/div>\n<\/li>\n
    3. The coordinates of the center of mass [latex](\\overline{x},\\overline{y})[\/latex] are\n
      [latex]\\overline{x}=\\frac{{M}_{y}}{m}\\text{ and }\\overline{y}=\\frac{{M}_{x}}{m}.[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/section>\n

      In the next example, we use this theorem to find the center of mass of a lamina.<\/p>\n

      \n

      Let R<\/em> be the region bounded above by the graph of the function [latex]f(x)=\\sqrt{x}[\/latex] and below by the [latex]x[\/latex]-axis over the interval [latex]\\left[0,4\\right].[\/latex] Find the centroid of the region.<\/p>\n