In other circumstances however this is not accepteble. Mathematically, the moment of inertia can be expressed in terms of its individual masses as the sum of the product of each individual mass and the squared. It is rather acceptable to ignore the centroidal term for the flange of an I/H section for example, because d is big and flange thickness (the h in the above formulas) is quite small. Usually in enginnereing cross sections the parallel axis term $Ad^2$ is much bigger than the centroidal term $I_o$. The Equation for Moment of Inertia for Circular Cross Section: The moment of inertia for a circular cross-section is given by I d 4 /64 where dDiameter of the circle. and the moment of inertia of a thin spherical shell is. the moment of inertia of a solid sphere is. By extending our previous example, we can find the moment of inertia of an arbitrary collection of particles of masses m and distances to the rotation axis r (where runs over all particles), and write: (5.4.3) I m r 2. $$ I = 13333333.3 \,mm^4 = 1333.33 cm^4 $$ The moment of inertia of a sphere about its central axis and a thin spherical shell are shown. Equation 5.4.2 is the rotational analog of Newton’s second law of motion. The method is demonstrated in the following examples. $$ I = 2\left(1666666.7 5000000 \right) \,mm^4 $$ Moments of inertia are always calculated relative to a specific axis, so the moments of inertia of all the sub shapes must be calculated with respect to this same axis, which will usually involve applying the parallel axis theorem. I_x
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