Moment of thin spherical shell about axis, Physics

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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:

768_Moment of Thin Spherical Shell About Axis.png

2014_Moment of Thin Spherical Shell About Axis1.png

It is prosperous to integrate with respect to φ first.

2095_Moment of Thin Spherical Shell About Axis2.png

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

549_Moment of Thin Spherical Shell About Axis3.png

We are able to write

89_Moment of Thin Spherical Shell About Axis4.png

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

We now have

2194_Moment of Thin Spherical Shell About Axis5.png

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

I =2/3mR2


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