Rotation 1 Question 21
24. A symmetric lamina of mass $M$ consists of a square shape with a semi-circular section over of the edge of the square as shown in figure. The side of the square is $2 a$. The moment of inertia of the lamina about an axis through its centre of mass and perpendicular to the plane is $1.6 Ma^{2}$. The moment of inertia of the lamina about the tangent $A B$ in the plane of the lamina is
(1997, 2M)
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Answer:
Correct Answer: 24. $4.8 M a^2$
Solution:
- Assuming the lamina to be in $x-y$ plane.
From, the perpendicular axis theorem,
$$ \begin{aligned} & I _x+I _y=I _z \\ & \text { but } \quad I _x=I _y \\ & \text { and } \quad I _z=1.6 Ma^{2} \\ & \therefore \quad I _x=\frac{I _z}{2}=0.8 M a^{2} \end{aligned} $$
Now, from the parallel axis theorem,
$I _{A B}=I _x+M(2 a)^{2}=0.8 M a^{2}+4 M a^{2}=4.8 M a^{2}$
