solid sphere help?

Question

anyone can proof this probem?

*** solid sphere
ICM=2/5 MR2 note:(2 superscript)

Answer 1

Have you learned calculus? You may understand my proof if you do. Let us take z-axis as the rotational axis. Considering a small mass element ρ*r^2sin(θ)drdθdφ located at (r, θ, φ) in the spherical coordinates, where ρ is the density, such that ρ*(4πR^3/3) = M, the distance to the rotational axis z is r sin(θ) and thus ICM of this element is r^2 [sin(θ)]^2*ρ*r^2sin(θ)drdθdφ = ρ*r^4 [sin(θ)]^3drdθdφ.
Integrate this element over the whole solid sphere, we have the result:
[integral sign][for r: 0 to R, for θ: 0 to π, and for φ: 0 to 2π]ρ*r^4 [sin(θ)]^3drdθdφ
= ρ*[integral sign][0 to R] r^4dr*[integral sign][0 to π][sin(θ)]^3dθ*[integral sign][0 to 2π]dφ
= 2πρ*[integral sign][0 to R] r^4dr*[integral sign][0 to π](1 - cos^2(θ))sin(θ)dθ
= 2πρ*(R^5/5)*[integral sign][0 to π](cos^2(θ) - 1)d(cos(θ))
= 2πρ*(R^5/5)*(4/3)
= (2/5)*ρ*(4πR^3/3)R^2
= (2/5)*MR^2