Given a thin rectangular plate of width $b$ and height $h$, with its base along the $x$-axis and one side along the $y$-axis, determine the moment of inertia of the plate about the $y$-axis using integration.

• A. $\frac{1}{3}m{h}^2$
• B. $\frac{1}{3}m{b}^2$
• C. $\frac{1}{12}m{h}^2$
• D. $\frac{1}{12}m{b}^2$

Answer: (B)

Verified step by step guidance

1. Identify the geometry and coordinate system: The plate is a thin rectangle with width \(b\) along the \(x\)-axis and height \(h\) along the \(y\)-axis. The base lies along the \(x\)-axis, so the plate extends from \(x=0\) to \(x=b\) and from \(y=0\) to \(y=h\).
2. Express the mass element \(dm\) in terms of the area element \(dA\): Since the plate is thin and uniform, its mass per unit area (surface density) is \(\sigma = \frac{m}{bh}\). An infinitesimal area element is \(dA = dx\,dy\), so \(dm = \sigma \, dA = \frac{m}{bh} dx dy\).
3. Write the expression for the moment of inertia about the \(y\)-axis: The moment of inertia \(I_y\) is defined as \(I_y = \int r^2 \, dm\), where \(r\) is the perpendicular distance from the \(y\)-axis. Here, \(r = x\), so \(I_y = \int x^2 \, dm\).
4. Set up the double integral over the plate area: Substitute \(dm\) and integrate over \(x\) and \(y\) limits: \( I_y = \int_0^h \int_0^b x^2 \left( \frac{m}{bh} \right) dx dy \)
5. Perform the integration step-by-step: - Integrate with respect to \(x\): \(\int_0^b x^2 dx = \frac{b^3}{3}\) - Integrate with respect to \(y\): \(\int_0^h dy = h\) Combine these results to express \(I_y\) in terms of \(m\), \(b\), and \(h\).