Planar lamina

<h2 id="definition">Definition</h2>
A planar lamina is defined as a figure (a <a href="/facts/Closed_set/upYnRTUj">closed set</a>) D of a finite area in a plane, with some mass m.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>
This is useful in calculating <a href="/facts/Moments_of_inertia/TpLLAVl7">moments of inertia</a> or <a href="/facts/Center_of_mass/n4Lr9GAz">center of mass</a> for a constant density, because the mass of a lamina is proportional to its area. In a case of a variable density, given by some (non-negative) <a href="/facts/Surface_density/kFSauAs4">surface density</a> function 
 
 
 
 ρ
 (
 x
 ,
 y
 )
 ,
 
 
 {\displaystyle \rho (x,y),}
 
 the mass 
 
 
 
 m
 
 
 {\displaystyle m}
 
 of the planar lamina D is a planar integral of ρ over the figure:<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

m
        =
        
          ∬
          
            D
          
        
        ρ
        (
        x
        ,
        y
        )
        
        d
        x
        
        d
        y
      
    
    {\displaystyle m=\iint _{D}\rho (x,y)\,dx\,dy}

<h2 id="properties">Properties</h2>
The center of mass of the lamina is at the point

(
          
            
              
                
                  M
                  
                    y
                  
                
                m
              
            
            ,
            
              
                
                  M
                  
                    x
                  
                
                m
              
            
          
          )
        
      
    
    {\displaystyle \left({\frac {M_{y}}{m}},{\frac {M_{x}}{m}}\right)}

where 
 
 
 
 
 M
 
 y
 
 
 
 
 {\displaystyle M_{y}}
 
 is the moment of the entire lamina about the y-axis and 
 
 
 
 
 M
 
 x
 
 
 
 
 {\displaystyle M_{x}}
 
 is the moment of the entire lamina about the x-axis:

M
          
            y
          
        
        =
        
          lim
          
            m
            ,
            n
            →
            ∞
          
        
        
        
          ∑
          
            i
            =
            1
          
          
            m
          
        
        
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
        x
        
          
            
              
              
                i
                j
              
            
          
          
            ∗
          
        
        
        ρ
         
        (
        x
        
          
            
              
              
                i
                j
              
            
          
          
            ∗
          
        
        ,
        y
        
          
            
              
              
                i
                j
              
            
          
          
            ∗
          
        
        )
        
        Δ
        D
        =
        
          ∬
          
            D
          
        
        x
        
        ρ
         
        (
        x
        ,
        y
        )
        
        d
        x
        
        d
        y
      
    
    {\displaystyle M_{y}=\lim _{m,n\to \infty }\,\sum _{i=1}^{m}\,\sum _{j=1}^{n}\,x{_{ij}}^{*}\,\rho \ (x{_{ij}}^{*},y{_{ij}}^{*})\,\Delta D=\iint _{D}x\,\rho \ (x,y)\,dx\,dy}

M
          
            x
          
        
        =
        
          lim
          
            m
            ,
            n
            →
            ∞
          
        
        
        
          ∑
          
            i
            =
            1
          
          
            m
          
        
        
        
          ∑
          
            j
            =
            1
          
          
            n
          
        
        
        y
        
          
            
              
              
                i
                j
              
            
          
          
            ∗
          
        
        
        ρ
         
        (
        x
        
          
            
              
              
                i
                j
              
            
          
          
            ∗
          
        
        ,
        y
        
          
            
              
              
                i
                j
              
            
          
          
            ∗
          
        
        )
        
        Δ
        D
        =
        
          ∬
          
            D
          
        
        y
        
        ρ
         
        (
        x
        ,
        y
        )
        
        d
        x
        
        d
        y
      
    
    {\displaystyle M_{x}=\lim _{m,n\to \infty }\,\sum _{i=1}^{m}\,\sum _{j=1}^{n}\,y{_{ij}}^{*}\,\rho \ (x{_{ij}}^{*},y{_{ij}}^{*})\,\Delta D=\iint _{D}y\,\rho \ (x,y)\,dx\,dy}

with summation and integration taken over a planar domain 
 
 
 
 D
 
 
 {\displaystyle D}
 
.

<h2 id="example">Example</h2>

Find the center of mass of a lamina with edges given by the lines 
 
 
 
 x
 =
 0
 ,
 
 
 {\displaystyle x=0,}
 
 
 
 
 
 y
 =
 x
 
 
 {\displaystyle y=x}
 
 and 
 
 
 
 y
 =
 4
 −
 x
 
 
 {\displaystyle y=4-x}
 
 where the density is given as 
 
 
 
 ρ
  
 (
 x
 ,
 y
 )
 
 =
 2
 x
 +
 3
 y
 +
 2
 
 
 {\displaystyle \rho \ (x,y)\,=2x+3y+2}
 
.
For this the mass 
 
 
 
 m
 
 
 {\displaystyle m}
 
 must be found as well as the moments 
 
 
 
 
 M
 
 y
 
 
 
 
 {\displaystyle M_{y}}
 
 and 
 
 
 
 
 M
 
 x
 
 
 
 
 {\displaystyle M_{x}}
 
.
Mass is 
 
 
 
 m
 =
 
 ∬
 
 D
 
 
 ρ
 (
 x
 ,
 y
 )
 
 d
 x
 
 d
 y
 
 
 {\displaystyle m=\iint _{D}\rho (x,y)\,dx\,dy}
 
 which can be equivalently expressed as an <a href="/facts/Iterated_integral/R4D7751o">iterated integral</a>:

m
        =
        
          ∫
          
            x
            =
            0
          
          
            2
          
        
        
          ∫
          
            y
            =
            x
          
          
            4
            −
            x
          
        
        
        (
        2
        x
        +
        3
        y
        +
        2
        )
        
        d
        y
        
        d
        x
      
    
    {\displaystyle m=\int _{x=0}^{2}\int _{y=x}^{4-x}\,(2x+3y+2)\,dy\,dx}

The inner integral is:

∫
          
            y
            =
            x
          
          
            4
            −
            x
          
        
        
        (
        2
        x
        +
        3
        y
        +
        2
        )
        
        d
        y
      
    
    {\displaystyle \int _{y=x}^{4-x}\,(2x+3y+2)\,dy}

=
        
          
            
            
              (
              
                2
                x
                y
                +
                
                  
                    
                      3
                      
                        y
                        
                          2
                        
                      
                    
                    2
                  
                
                +
                2
                y
              
              )
            
            |
          
          
            y
            =
            x
          
          
            4
            −
            x
          
        
      
    
    {\displaystyle \qquad =\left.\left(2xy+{\frac {3y^{2}}{2}}+2y\right)\right|_{y=x}^{4-x}}

=
        
          [
          
            2
            x
            (
            4
            −
            x
            )
            +
            
              
                
                  3
                  (
                  4
                  −
                  x
                  
                    )
                    
                      2
                    
                  
                
                2
              
            
            +
            2
            (
            4
            −
            x
            )
          
          ]
        
        −
        
          [
          
            2
            x
            (
            x
            )
            +
            
              
                
                  3
                  (
                  x
                  
                    )
                    
                      2
                    
                  
                
                2
              
            
            +
            2
            (
            x
            )
          
          ]
        
      
    
    {\displaystyle \qquad =\left[2x(4-x)+{\frac {3(4-x)^{2}}{2}}+2(4-x)\right]-\left[2x(x)+{\frac {3(x)^{2}}{2}}+2(x)\right]}

=
        −
        4
        
          x
          
            2
          
        
        −
        8
        x
        +
        32
      
    
    {\displaystyle \qquad =-4x^{2}-8x+32}

Plugging this into the outer integral results in:

m
              
              
                
                =
                
                  ∫
                  
                    x
                    =
                    0
                  
                  
                    2
                  
                
                
                  (
                  
                    −
                    4
                    
                      x
                      
                        2
                      
                    
                    −
                    8
                    x
                    +
                    32
                  
                  )
                
                
                d
                x
              
            
            
              
              
                
                =
                
                  
                    
                    
                      (
                      
                        −
                        
                          
                            
                              4
                              
                                x
                                
                                  3
                                
                              
                            
                            3
                          
                        
                        −
                        4
                        
                          x
                          
                            2
                          
                        
                        +
                        32
                        x
                      
                      )
                    
                    |
                  
                  
                    x
                    =
                    0
                  
                  
                    2
                  
                
              
            
            
              
              
                
                =
                
                  
                    112
                    3
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}m&=\int _{x=0}^{2}\left(-4x^{2}-8x+32\right)\,dx\\&=\left.\left(-{\frac {4x^{3}}{3}}-4x^{2}+32x\right)\right|_{x=0}^{2}\\&={\frac {112}{3}}\end{aligned}}}

Similarly are calculated both moments:

M
          
            y
          
        
        =
        
          ∬
          
            D
          
        
        x
        
        ρ
        (
        x
        ,
        y
        )
        
        d
        x
        
        d
        y
        =
        
          ∫
          
            x
            =
            0
          
          
            2
          
        
        
          ∫
          
            y
            =
            x
          
          
            4
            −
            x
          
        
        x
        
        (
        2
        x
        +
        3
        y
        +
        2
        )
        
        d
        y
        
        d
        x
      
    
    {\displaystyle M_{y}=\iint _{D}x\,\rho (x,y)\,dx\,dy=\int _{x=0}^{2}\int _{y=x}^{4-x}x\,(2x+3y+2)\,dy\,dx}

with the inner integral:

∫
          
            y
            =
            x
          
          
            4
            −
            x
          
        
        x
        
        (
        2
        x
        +
        3
        y
        +
        2
        )
        
        d
        y
      
    
    {\displaystyle \int _{y=x}^{4-x}x\,(2x+3y+2)\,dy}

=
        
          
            
            
              (
              
                2
                
                  x
                  
                    2
                  
                
                y
                +
                
                  
                    
                      3
                      x
                      
                        y
                        
                          2
                        
                      
                    
                    2
                  
                
                +
                2
                x
                y
              
              )
            
            |
          
          
            y
            =
            x
          
          
            4
            −
            x
          
        
      
    
    {\displaystyle \qquad =\left.\left(2x^{2}y+{\frac {3xy^{2}}{2}}+2xy\right)\right|_{y=x}^{4-x}}

=
        −
        4
        
          x
          
            3
          
        
        −
        8
        
          x
          
            2
          
        
        +
        32
        x
      
    
    {\displaystyle \qquad =-4x^{3}-8x^{2}+32x}

which makes:

M
                  
                    y
                  
                
              
              
                
                =
                
                  ∫
                  
                    x
                    =
                    0
                  
                  
                    2
                  
                
                (
                −
                4
                
                  x
                  
                    3
                  
                
                −
                8
                
                  x
                  
                    2
                  
                
                +
                32
                x
                )
                
                d
                x
              
            
            
              
              
                
                =
                
                  
                    
                    
                      (
                      
                        −
                        
                          x
                          
                            4
                          
                        
                        −
                        
                          
                            
                              8
                              
                                x
                                
                                  3
                                
                              
                            
                            3
                          
                        
                        +
                        16
                        
                          x
                          
                            2
                          
                        
                      
                      )
                    
                    |
                  
                  
                    x
                    =
                    0
                  
                  
                    2
                  
                
              
            
            
              
              
                
                =
                
                  
                    80
                    3
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}M_{y}&=\int _{x=0}^{2}(-4x^{3}-8x^{2}+32x)\,dx\\&=\left.\left(-x^{4}-{\frac {8x^{3}}{3}}+16x^{2}\right)\right|_{x=0}^{2}\\&={\frac {80}{3}}\end{aligned}}}

and

M
                  
                    x
                  
                
              
              
                
                =
                
                  ∬
                  
                    D
                  
                
                y
                
                ρ
                (
                x
                ,
                y
                )
                
                d
                x
                
                d
                y
                =
                
                  ∫
                  
                    x
                    =
                    0
                  
                  
                    2
                  
                
                
                  ∫
                  
                    y
                    =
                    x
                  
                  
                    4
                    −
                    x
                  
                
                y
                
                (
                2
                x
                +
                3
                y
                +
                2
                )
                
                d
                y
                
                d
                x
              
            
            
              
              
                
                =
                
                  ∫
                  
                    0
                  
                  
                    2
                  
                
                (
                x
                
                  y
                  
                    2
                  
                
                +
                
                  y
                  
                    3
                  
                
                +
                
                  y
                  
                    2
                  
                
                )
                
                  
                    
                      |
                    
                  
                  
                    y
                    =
                    x
                  
                  
                    4
                    −
                    x
                  
                
                
                d
                x
              
            
            
              
              
                
                =
                
                  ∫
                  
                    0
                  
                  
                    2
                  
                
                (
                −
                2
                
                  x
                  
                    3
                  
                
                +
                4
                
                  x
                  
                    2
                  
                
                −
                40
                x
                +
                80
                )
                
                d
                x
              
            
            
              
              
                
                =
                
                  
                    
                    
                      (
                      
                        −
                        
                          
                            
                              x
                              
                                4
                              
                            
                            2
                          
                        
                        +
                        
                          
                            
                              4
                              
                                x
                                
                                  3
                                
                              
                            
                            3
                          
                        
                        −
                        20
                        
                          x
                          
                            2
                          
                        
                        +
                        80
                        x
                      
                      )
                    
                    |
                  
                  
                    x
                    =
                    0
                  
                  
                    2
                  
                
              
            
            
              
              
                
                =
                
                  
                    248
                    3
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}M_{x}&=\iint _{D}y\,\rho (x,y)\,dx\,dy=\int _{x=0}^{2}\int _{y=x}^{4-x}y\,(2x+3y+2)\,dy\,dx\\&=\int _{0}^{2}(xy^{2}+y^{3}+y^{2}){\Big |}_{y=x}^{4-x}\,dx\\&=\int _{0}^{2}(-2x^{3}+4x^{2}-40x+80)\,dx\\&=\left.\left(-{\frac {x^{4}}{2}}+{\frac {4x^{3}}{3}}-20x^{2}+80x\right)\right|_{x=0}^{2}\\&={\frac {248}{3}}\end{aligned}}}

Finally, the center of mass is

(
          
            
              
                
                  M
                  
                    y
                  
                
                m
              
            
            ,
            
              
                
                  M
                  
                    x
                  
                
                m
              
            
          
          )
        
        =
        
          (
          
            
              
                
                  80
                  3
                
                
                  112
                  3
                
              
            
            ,
            
              
                
                  248
                  3
                
                
                  112
                  3
                
              
            
          
          )
        
        =
        
          (
          
            
              
                5
                7
              
            
            ,
            
              
                31
                14
              
            
          
          )
        
      
    
    {\displaystyle \left({\frac {M_{y}}{m}},{\frac {M_{x}}{m}}\right)=\left({\frac {\frac {80}{3}}{\frac {112}{3}}},{\frac {\frac {248}{3}}{\frac {112}{3}}}\right)=\left({\frac {5}{7}},{\frac {31}{14}}\right)}

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Atkins, Tony; Escudier, Marcel (2013), "Plane lamina", A Dictionary of Mechanical Engineering (1 ed.), Oxford University Press, doi:10.1093/acref/9780199587438.001.0001, ISBN 9780199587438, retrieved 2021-06-08 <a href="9780199587438" target="_blank">9780199587438</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">"Planar Laminae", WolframAlpha, retrieved 2021-03-09 <a href="https://www.wolframalpha.com/examples/mathematics/geometry/plane-geometry/planar-laminae/" target="_blank">https://www.wolframalpha.com/examples/mathematics/geometry/plane-geometry/planar-laminae/</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">"Lamina". MathWorld. Retrieved 2021-03-09. <a href="https://mathworld.wolfram.com/Lamina.html" target="_blank">https://mathworld.wolfram.com/Lamina.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
</ol>

Planar lamina open-in-new

Planar lamina