Transforming Cartesian to Polar Coordinates

[shanepitts]

Homework Statement

I am currently trying to calculate the moment and products of inertia of a ring rotating about the x-axis at the moment the ring lies in the xy plane. The problem is that the notations I have from textbook are denoted for Cartesian coordinates. i.e. I_xx=∑_i m_i(y_i²+z_i²), and I_xy=∑_im_ix_iy_i. How can I convert these to polar coordinates?

Homework Equations

x=rcosθ, y=rsinθ, z=?

The Attempt at a Solution

I'd assume that the moment of inertia would become I_xx=∑mr²dθ.

but since the problem is for a ring with linear mass density, I am also wondering must I exclude certain coordinates?

thank you in advance.

 

[DrClaude]

You can find descriptions of how to calculate such moments of inertia at http://hyperphysics.phy-astr.gsu.edu/hbase/ihoop.html
Be careful that for continuous objects, not point masses, you have to use density instead of masses and integrals instead of sums.

For symmetry reasons, you have that I_xy = 0, I_xz = 0, etc., and also I_xx = I_yy.