Derive the expression for the moment of inertia of a rectangle of base $b$ and height $h$ about an axis passing through its centroid and parallel to its base ($I_{xc}$).

A wooden joist has a rectangular cross-section with width $b = 50\text{ mm}$ and height $h = 200\text{ mm}$. Calculate the moment of inertia about its base (the bottom edge).

Calculate the moment of inertia $I_x$ about the centroidal horizontal axis for a symmetrical steel I-beam. The top and bottom flanges are $200\text{ mm}$ wide and $20\text{ mm}$ thick. The web is $300\text{ mm}$ tall and $10\text{ mm}$ thick.

Calculate the polar radius of gyration ($k_O$) for a solid circular cross-section with diameter $D = 100\text{ mm}$.

Calculate the moment of inertia of a right triangle with base $b = 150\text{ mm}$ and height $h = 300\text{ mm}$ about its centroidal horizontal axis parallel to the base.

Determine the polar moment of inertia ($J_O$) for a hollow steel pipe with an outer diameter $D = 200\text{ mm}$ and an inner diameter $d = 160\text{ mm}$.

Calculate the moment of inertia $I_x$ about the centroidal horizontal axis for a rectangular block $200\text{ mm}$ wide and $300\text{ mm}$ high, which has a circular hole of diameter $100\text{ mm}$ perfectly centered within it.

For a given unsymmetrical cross-section, the properties relative to a set of $x$ and $y$ centroidal axes are computed as: $I_x = 40\text{ in}^4$, $I_y = 20\text{ in}^4$, and the product of inertia $I_{xy} = -10\text{ in}^4$. Determine the maximum principal moment of inertia ($I_{max}$).

Why is the I-beam (or Wide Flange section) the ubiquitous shape for structural steel girders in skyscrapers and bridges, rather than solid rectangular or circular bars of the same weight? Explain using the concept of the Moment of Inertia.

Automotive drive shafts are almost always hollow tubes rather than solid steel rods, even though solid rods are nominally "stronger." Why is a hollow tube preferred for transmitting torque? Explain using the Polar Moment of Inertia ($J$).

When designing structural columns, engineers rarely rely solely on the total cross-sectional area or the Moment of Inertia alone to determine capacity. Instead, the "Radius of Gyration" ($k$) is often the primary geometric parameter used. Explain why $k$ is used to evaluate column buckling.

A heavy load is placed on an asymmetric Z-shaped purlin on a roof. Even though the load is applied perfectly vertically, the beam begins to twist and bend sideways. Explain this behavior using the concept of Principal Axes and the Product of Inertia.