How is the moment of inertia of a rectangle calculated about its base?

The moment of inertia of a rectangle about its base is calculated using the formula ( I = \frac{b h^3}{3} ). In this formula, ( b ) represents the base width of the rectangle, and ( h ) represents its height. The factor of ( \frac{1}{3} ) accounts for the distribution of the area with respect to the base, which is where the axis of rotation is located.

This specific calculation is derived from the principles of mechanics and is important for understanding how materials resist bending and flexural forces. When the rectangle bends or deflects under load, the moment of inertia determines how rigidly it can withstand such deformation. The further away the area is from the base, the more significant its contribution to the moment of inertia. The ( h^3 ) term emphasizes that the height of the rectangle plays a crucial role; as the height increases, the moment of inertia increases at a cubic rate.

In contrast, options that propose ( \frac{b h^3}{4} ), ( \frac{b h^3}{12} ), and ( \frac{b h^3}{24} ) do not correspond to the standard calculation for