Finding the angular momentum using the inertia tensor/matrix

[shanepitts]

Homework Statement

A thin ring of radius r is constrained to rotate with constant angular velocity ω as shown in attached picture. Let the linear mass density of the ring be ρ(θ)=ρ₀(2+sin2θ) where ρ₀ is constant.

a) Find the angular momentum L of the ring about O, at the instant the ring is in the xy plane as shown. Answer this part twice: (i) by using the moments and products of inertia I_ij, and (ii) by directly integrating L=∫dmr x v.

Homework Equations

The moment of inertia tensor/ matrix.
L=Iω=n_transposeIn

The Attempt at a Solution

Not sure if I am starting this problem properly, attached below is my attempt.

Knowing that I must plug these moments and products of India inside the tensor matrix.

Please help

 

[SteamKing]

When you calculated the moment of inertia I_xx, you treated the density of the ring, ρ, as a constant. According to the OP, ρ(θ) = ρ_o(2+sin2θ), where ρ_o is a constant. If you are going to calculate the MOI matrix for the ring, you must take this arbitrary density function into account. This extends even to calculating the mass of the ring.

 

[shanepitts]

When you calculated the moment of inertia I_xx, you treated the density of the ring, ρ, as a constant. According to the OP, ρ(θ) = ρ_o(2+sin2θ), where ρ_o is a constant. If you are going to calculate the MOI matrix for the ring, you must take this arbitrary density function into account. This extends even to calculating the mass of the ring.

Thank you.

But are my integral limits correct considering it is a ring? Moreover, shall I integrate with respect to y, x, and/or θ, or just one variable?

 

[SteamKing]

Thank you.

But are my integral limits correct considering it is a ring? Moreover, shall I integrate with respect to y, x, and/or θ, or just one variable?

I don't think so.

You seem to have integrands which use cartesian coordinates while the limits appear to be expressed in polar coordinates (what does x = 2πr mean? Isn't that the circumference of the ring?)

 

[shanepitts]

I don't think so.

You seem to have integrands which use cartesian coordinates while the limits appear to be expressed in polar coordinates (what does x = 2πr mean? Isn't that the circumference of the ring?)

Thanks.