Calculate the moment of inertia of a thin plate, in the shape of a rectangle, ab

Question

Calculate the moment of inertia of a thin plate, in the shape of a rectangle, about an axis that passes
through one of the shorter ends and is parallel to the opposite side. Let M represent the mass of the plate
and L the length of the base perpendicular to the axis of rotation. Let h represent the height of the plate
and w the thickness of the plate, much smaller than L or h. The moment of inertia when rotating through
the center of mass is: Iz = M (L2 + h2) / 12 and Ix = M ( h2) / 12.

Calculate the moment of inertia of a thin plate, in the shape of a rectangle, about an axis that passes through one of the shorter ends and is parallel to the opposite side. Let M represent the mass of the plate and L the length of the base perpendicular to the axis of rotation. Let h represent the height of the plate and w the thickness of the plate, much smaller than L or h. The moment of inertia when rotating through the center of mass is: Iz = M (L^2 + h^2) / 12 and Ix = M ( h^2) / 12.

Explanation / Answer

Let us consider an small element "dx" along the length, which is situated at a linear distance "x" from

the axis.

(i) Elemental mass :

Linear density    ?, is the appropriate density type in this case.

                                       ?   = M/L

here "M" and "L" are the mass and length of the rod respectively. Elemental mass (m) is, thus, given

as :                                              m =   ? dx = M/L dx

           Moment of inertia for elemental mass :

                                     I =   r^2 m

                                        =   X^2 (M/L) dx

iv) Moment of inertia when rotating through x    I? = ? r^2m dx

                                                       

                                                       = ?x^2(M/L)dx

While setting limits we should cover the total length of the rod. The appropriate limits of integral in

this case are -L/2 and L/2. Hence

                                             +L/2

                                                ? -L/2 x^2(M/L)dx

                                                       

                                    =    (M/L) [X^3/3] +L2-L/2        = ML^2/12-----(1)

Similarly, we can also calculate MI of the rectangular plate about a line parallel to its length and through

the center (limits of integral in in this case are -h/2 and h/2.) Hence

                                                  Iy=     mh^2/12      -------(2)

Therefore MI of the rectangular plate about the center of mass is Iz =    eq1 + eq 2

               

                                               M(L^2+h^2)/12

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