Question

Determine the  moment of inertia Θ_a of a homogeneous torus with a circular cross section and mass m.

Answer 1

The mass moment of inertia Θ_a is determined from

We can determine the geometrical relations

r = R + c sinϕ , 

dm = ρ 2πr2c cosϕ dr 

from the figure. With 

d_r = c cosϕ dϕ

we obtain

Note that the integrals of the odd functions over the even interval are zero. Using m = 2π²ρc²R we finally get

This result reduces to Θ_a = mR² in the case of a thin ring (c ≪ R).