Question

In the rectangular lamina shown in the $\text{AB = }\dfrac{\text{BC}}{\text{2}}$. The moment of inertia of the lamina is minimum along the axis passing through:

A. AB
B. BC
C. EG
D. FH

Answer 1

Hint: We must know that Moment of inertia is defined as a quantity which expresses a body's tendency to resist angular acceleration. For each of the options given, we are required to find the distance of the axis. One with minimum distance to the axis will have the least moment of inertia.

Complete step by step answer:
Moment of inertia can be denoted as:
$\begin{align}
  & \sum\limits_{{}}^{{}}{{{\text{m}}_{\text{i}}}{{\text{r}}_{\text{i}}}^{2}} \\
 & \Rightarrow \text{ }\!\!\alpha\!\!\text{ }{{\text{r}}^{\text{2}}} \\
\end{align}$
Moment of inertia is dependent on the distance from the axis of the body. It is directly proportional to the distance from the axis of the body.
If we assume AD as l, we can derive that,
$\text{AB = }\dfrac{\text{l}}{\text{2}}$
From the following images we can clearly see that the distance of the FH axis is least. So, Moment of inertia is less about the FH axis. In the given diagrams below, we can clearly see that in FH,
$\text{r = }\dfrac{\text{l}}{\text{4}}$

Here, when we analyze the diagrams, we can easily understand which axis is having the least distance. Hence, FH being the lowest among them.
Therefore, the correct answer is FH.

So, the correct answer is “Option D”.

Note: Moment of inertia is a quantity which determines the torque that is needed for a desired angular acceleration about the rotational axis. It is also known as the mass moment of inertia and mass of rotational inertia. It is just a measure of the resistance of a body to its angular acceleration about an axis. Moment of Inertia can be denoted as:
\[\left( {{m}_{1}}\text{ }\times \text{ }{{\text{m}}_{\text{2}}}\ldots \times \text{ }{{\text{m}}_{\text{n}}} \right)\text{ * }{{\text{d}}^{^{2}}}\]
Where,
m is the mass of each element in the body.
d is the distance of the element from the axis.