Question:
The moment of inertia of a thin square plate $A B C D$, of uniform thickness about an axis passing through the centre $O$ and perpendicular to the plane of the plate is where $I_{1}, I_{2}, I_{3}$ and $I_{4}$ are respectively moments of inertia about axes $1,2,3$ and $4$ which are in the plane of the plate
The moment of inertia of a thin square plate $A B C D$, of uniform thickness about an axis passing through the centre $O$ and perpendicular to the plane of the plate is where $I_{1}, I_{2}, I_{3}$ and $I_{4}$ are respectively moments of inertia about axes $1,2,3$ and $4$ which are in the plane of the plate
Updated On: Jun 14, 2022
- $I_{1}+I_{2}$
- $I_3+I_4$
- $I_1+I_3$
- $I_1+I_2+I_3+I_4$
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The Correct Option is B
Solution and Explanation
Since, it is a square lamina
$I_{3}=I_{4}$
and $I_{1}=I_{2}$ (by symmetry)
From perpendicular axes theorem.
Moment of inertia about an axis perpendicular to square plate and passing from $O$ is
$I_{0}=I_{1}+I_{2}=I_{3}+I_{4}$
or $I_{0}=2 I_{2}=2 I_{3}$
Hence, $I_{2}=I_{3}$
Rather we can say $I_{1}=I_{2}=I_{3}=I_{4}$
Therefore, $I_{0}$ can be obtained by adding any two i.e.
$I_{0}=I_{1}+I_{2}=I_{1}+I_{3}$
$=I_{1}+I_{4}=I_{2}+I_{3} $
$=I_{2}+I_{4}=I_{3}+I_{4}$
$I_{3}=I_{4}$
and $I_{1}=I_{2}$ (by symmetry)
From perpendicular axes theorem.
Moment of inertia about an axis perpendicular to square plate and passing from $O$ is
$I_{0}=I_{1}+I_{2}=I_{3}+I_{4}$
or $I_{0}=2 I_{2}=2 I_{3}$
Hence, $I_{2}=I_{3}$
Rather we can say $I_{1}=I_{2}=I_{3}=I_{4}$
Therefore, $I_{0}$ can be obtained by adding any two i.e.
$I_{0}=I_{1}+I_{2}=I_{1}+I_{3}$
$=I_{1}+I_{4}=I_{2}+I_{3} $
$=I_{2}+I_{4}=I_{3}+I_{4}$
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Concepts Used:
System of Particles and Rotational Motion
- The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
- The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
- The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.