**Problem 817**

Determine the moment of inertia and radius of gyration with respect to a polar centroidal axis of the cross section of a hollow tube whose outside diameter is 6 in. and inside diameter is 4 in.

**Solution 817**

Polar moment of inertia

$\bar{J} = \frac{1}{2}\pi(R^4 - r^4)$

$\bar{J} = \frac{1}{2}\pi(R^4 - r^4)$

$\bar{J} = \frac{1}{2}\pi(3^4 - 2^4)$

$\bar{J} = 32.5\pi \, \text{ in.}^4 = 102.10 \, \text{ in.}^4$ *answer*

Area

$A = \pi(R^2 - r^2)$

$A = \pi(3^2 - 2^2)$

$A = 5\pi \, \text{ in.}^2$

Radius of gyration

$\bar{k}_z = \sqrt{\dfrac{\bar{J}}{A}}$

$\bar{k}_z = \sqrt{\dfrac{32.5\pi}{5\pi}}$

$\bar{k}_z = 0.7071 \, \text{ in.}$ *answer*