Moment of Thin Spherical Shell About Axis

This problem can be set up in spherical coordinates so that conversions from Cartesian coordinates are not required. The mass element for this case is supposed to exist on the surface of a sphere. The element of the surface area of a sphere δS is related to an element of mass by δm = σδS where σ is the mass per unit area ρ = R sin θ is the length of the moment arm for the mass element. Therefore δI = ρ² δm. The area of the surface element is Rδθ × Rsin θδφ or R² sin θδθδφ.

Expanding, we have

δI = (Rsin θ)²δm

δI = R²sin²θσ δS

δI = R²sin²θσR²sin θ δθ δφ

δI = R⁴σ sin³θ δθ δφ

Summing the increments and taking limits, we may write the following integral:

It is prosperous to integrate with respect to φ first.

To solve this integral evoke that sin θ dθ = d(- cos θ) Making the substitution mindful that changing the variable of integration requires changing the integration limits as follows

We are able to write

It perhaps convenient to replace cos θ in 8.3 with a easier variable say x

We now have

Noting that the total mass m is σ4πR2 we are able to reduce to

I =2/3mR²