Question

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

Answer 1

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\[dI=a^2 \delta dL \rightarrow I = a^2 \delta \int\limits_{L}{}dL=a^2 \delta 2\pi a=2 \pi \delta a^3\]

Answer 2

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

Answer 3

thank you!!

Answer 4

welcome :)