Moments of Inertia Examples

A rectangle is $b=100\ \text{mm}$ wide and $h=200\ \text{mm}$ deep. Determine $I_x$ about its horizontal centroidal axis.

For the same $100\times200\ \text{mm}$ rectangle, determine $I_y$ about the vertical centroidal axis and compare it with $I_x$.

A rectangular area has $A=6000\ \text{mm}^2$ and centroidal $I_c=8.00\times10^6\ \text{mm}^4$. Find its inertia about a parallel axis $d=50.0\ \text{mm}$ away.

A rectangular tube has outer dimensions $200\times300\ \text{mm}$ and a centered rectangular void $160\times260\ \text{mm}$. Determine $I_x$ about the common horizontal centroidal axis.

Two identical plates each have $A=2000\ \text{mm}^2$ and centroidal $I_c=1.50\times10^6\ \text{mm}^4$. Their centroids are each $60.0\ \text{mm}$ from the composite centroidal x-axis. Determine total $I_x$.

At point $O$, an area has $I_x=4.00\times10^6\ \text{mm}^4$ and $I_y=7.00\times10^6\ \text{mm}^4$. Determine $J_O$.

A section has area $A=5000\ \text{mm}^2$ and $I_x=2.00\times10^7\ \text{mm}^4$. Determine $k_x$.

A section-property table reports $I_x=8.50\times10^8\ \text{mm}^4$, while a rotating mechanical component is reported to have $I=12.0\ \text{kg}\cdot\text{m}^2$. Are these the same physical property?

A T-section consists of a $100\times20\ \text{mm}$ flange above a centered $20\times80\ \text{mm}$ web. Its centroid is at $\bar{y}=67.78\ \text{mm}$ above the bottom. Determine $I_x$ about the composite centroidal axis.

A $200\times120\ \text{mm}$ rectangle contains a $60\times40\ \text{mm}$ rectangular cutout centered at $(150,60)\ \text{mm}$. Determine the centroidal $I_y$ of the remaining area. The outer rectangle centroid is at $(100,60)\ \text{mm}$.

Two sections have the same area, $A=4000\ \text{mm}^2$. Section A has $I_x=8.00\times10^6\ \text{mm}^4$ and Section B has $I_x=2.00\times10^7\ \text{mm}^4$. Compare their radii of gyration about the $x$-axis.

A doubly symmetric I-shaped area has horizontal and vertical centroidal symmetry axes. What can be stated about its product of inertia and principal axes?