Moments of Inertia

 

 

Example:  Calculate the moment of inertia of area for the trapezoidal area, BCDE, shown

below about the z-axis,           Izz  =  ʃ  y2 dA       

 

 

                              

 

 

Strategy:   Determine the limits of integration.  Then use a double integral to evaluate Izz .

 

 

                                                z = 1      y = z + 1

                                      Izz  =     ʃ            ʃ              y2 dy dz

                                                z = 0       y = 0

 

 

                                                z = 1                       z + 1

                                      Izz  =     ʃ           (1/3) y3  |     dz

                                                z = 0                       0

 

 

                                                z = 1                      

                                      Izz  =     ʃ           (1/3) [ z3 + 3z2 + 3z + 1 ]    dz

                                                z = 0                      

 

                                                                                                  1

                                      Izz =     (1/3) [ z4/4 + z3 + 3z/2 + z ]  |   =  5/4           (result)

                                                                                                  0

 

 
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