## anonymous 5 years ago A circular wire hoop of constant density delta lies along the circle and x^2 + y^2 = a^2 in the xy-plane. Find the hoop's moment inertia and radius of gyration about the z axis

1. anonymous

If you consider a differential element of your ring, by the definition of moment of inertia,$dI=a^2dm$Now, since the hoop has constant density, delta, you can relate the mass to length as$\delta = \frac{m}{L}\iff m=L {\delta} \rightarrow dm=\delta dL$Hence$dI=a^2 \delta dL \rightarrow I = a^2 \delta \int\limits_{L}{}dL=a^2 \delta 2\pi a=2 \pi \delta a^3$

2. anonymous

The radius of gyration is$r_g=\sqrt{\frac{I}{m}}=\sqrt{\frac{2\pi a^3(m/2\pi a)}{m}}=a$

3. anonymous

thank you!!

4. anonymous

welcome :)