Area of ring = Circumference*dr

[teroenza]

Homework Statement

My professor put on the board today, that the area of a ring (used in discussion of moments of inertia) was = the circumference of the ring *dr = 2*pi*r*dr.

This may sound trivial, but I cannot seem to work out in my head how this related to the formula I know for the area of a ring = pi*(r^2_outer-r^2_inner). I see the units are ok in either, but the factor of 2 is confusing me.

Thanks

 

[phyzguy]

Call the inner radius r and the outer radius r+dr. Then the area is, as you said:
A = pi((r+dr)^2 - r^2) = pi(r^2 + 2r dr +dr^2 - r^2) = pi (2r dr +dr^2)

Now as dr gets very small, the dr^2 term becomes vanishingly small compared to the 2 r dr term, so we can ignore it, giving:

A = 2 pi r dr

which is what your professor said.

 

[teroenza]

Thank you very much.