Question

Calculate numericaly what is the moment of inertia of a solid sphere of density ¶ and radius R about its diameter

Answer 1

Contrary to the opinions of others, I believe the rationale behind the derivation of the I=2/5MR2 formula is far more important than the formula itself.  The radius of the sphere is R. Let is conceive of a disk of radius r located at a distance x from the center of the sphere. For purposes of calculation herein, let us state its thickness to be dx.  r=R2−x2−−−−−−−√  The volume of this disk is thus:  dV=πr2dx=π(R2−x2)dx  The mass of this disk is:  dm=ρdV=πρ(R2−x2)dx  We know that, over a uniform body:  I=∫r2dm  Therefore,  I=∫r2(πρ(R2−x2)dx  =πρ∫R0(R2−x2)2dx  Carrying out the integration, we obtain:  I=8πρ15R5  And since M=ρV=4πρR33  Substituting the given value for ρ  I=2/5MR2