Question

The moment of inertia of a door of mass $m$, length $2l$, and width $l$ about its longer side is

• A. $\frac{{11ml}^2}{24}$
• B. $\frac{{5ml}^2}{24}$
• C. $\frac{{ml}^2}{3}$
• D. None of the above

Answer 1

The correct option is C

$\frac{{ml}^2}{3}$

Step 1: Given data

Mass of the door$\to m$

Length of the door is $2l$ and the width of the door is $l$

Step 2: Diagram

Step 3: Calculation

We know by definition that the moment of inertia along the central axis $\mathrm{Y}$ is given as :${\mathrm{I}}_{\mathrm{Y}}=$$\frac{m{l}^2}{12}$

With the help of the parallel axis theorem, we can conclude that,

$$\begin{gathered} \Rightarrow {\mathrm{I}}_{\mathrm{AD}}={\mathrm{I}}_{\mathrm{Y}}+\mathrm{m}{\left(\frac{\mathrm{l}}{2}\right)}^2 \\ \Rightarrow {\mathrm{I}}_{\mathrm{AD}}=\frac{m{l}^2}{12}+\frac{m{l}^2}{4} \\ \Rightarrow {\mathrm{I}}_{\mathrm{AD}}=\frac{m{l}^2+3m{l}^2}{12} \\ \Rightarrow {\mathrm{I}}_{\mathrm{AD}}=\frac{4m{l}^2}{12} \\ \Rightarrow {\mathrm{I}}_{\mathrm{AD}}=\frac{m{l}^2}{3}. \end{gathered}$$

Hence, option (C) is correct.