Question

The moment of inertia of a rectangle with base b and height h about an axis through its centroid parallel to the base is

• A. \( bh^3 / 3 \)
• B. \( bh^3 / 12 \)
• C. \( bh^3 / 6 \)
• D. \( bh^3 / 9 \)

Hint:

Rectangle Moment of Inertia. About centroidal axis parallel to base b: \(I_{cx = bh^3 / 12\). About centroidal axis parallel to height h: \(I_{cy = hb^3 / 12\). About base b: \(I_{base = bh^3 / 3\).

Answer 1

The Correct Option is B

The area moment of inertia (\(I\)) measures the resistance of a cross-section to bending.
For a rectangular cross-section with base \(b\) and height \(h\), the moment of inertia about an axis passing through its centroid (geometric center) and parallel to the base (let's call this the x-axis, \(I_{cx}\)) is given by the standard formula: $$ I_{cx} = \frac{bh^3}{12} $$ The moment of inertia about an axis through the centroid parallel to the height (y-axis) would be \(I_{cy} = \frac{hb^3}{12}\).
The moment of inertia about the base itself (\(I_{base}\)) can be found using the parallel axis theorem: \(I_{base} = I_{cx} + A d^2 = \frac{bh^3}{12} + (bh)(h/2)^2 = \frac{bh^3}{12} + \frac{bh^3}{4} = \frac{bh^3}{3}\).
Option (1) is the MOI about the base, while option (2) is the MOI about the centroidal axis parallel to the base.