# 816 Polar moment of inertia and radius of gyration at one corner of rectangle

**Problem 816**

A rectangle is 3 in. by 6 in. Determine the polar moment of inertia and the radius of gyration with respect to a polar axis through one corner.

**Solution 816**

Moment of inertia about the x-axis

$I_x = \dfrac{bh^3}{3} = \dfrac{6(3^3)}{3}$

$I_x = \dfrac{bh^3}{3} = \dfrac{6(3^3)}{3}$

$I_x = 54 \, \text{ in.}^4$

Moment of inertia about the y-axis

$I_y = \dfrac{hb^3}{3} = \dfrac{3(6^3)}{3}$

$I_y = 216 \, \text{ in.}^4$

Polar moment of inertia

$J = I_x + I_y = 54 + 216$

$J = 270 \, \text{ in.}^4$ *answer*

Radius of gyration about the corner

$k_z = \sqrt{\dfrac{J}{A}} = \sqrt{\dfrac{270}{3(6)}}$

$k_z = 3.873 \, \text{ in.}$ *answer*

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