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Moment of Inertia of Hollow Sphere

The moment of...

Question

The moment of inertia of a thin square plate ABCD as shown in the fig. of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is
(Where $I_{1}$ , $I_{2}$ , $I_{3}$ , $I_{4}$ are the moments of inertia respectively about axis 1, 2, 3 and 4 which are in the plane of the plate.)

I_{1}

I_{2}

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I_{3}

I_{4}

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I_{1}

I_{3}

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I_{1}

I_{2}

I_{3}

I_{4}

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Solution

The correct options are
A $I_{1}$ + $I_{2}$
B $I_{3}$ + $I_{4}$
C $I_{1}$ + $I_{3}$
Let the MI about an axis passing through O and $⊥$ to the plane of the square plate ABCD is $I_{O}$
The axes designated as 3 and 4 are mutually perpendicular.
Applying perpendicular axis theorem,
$I_{O} = I_{3} + I_{4}$
But axes 3 and 4 are equivalent by symmetry.
$∴ I_{3} = I_{4}$
The axes designated as I and 2 are mutually perpendicular.
Applying perpendicular axis theorem,
$I_{O} = I_{1} + I_{2}$
But axes 1 and 2 are equivalent by symmetry.
$∴ I_{1} = I_{2}$
Since $I_{O} = I_{1} + I_{2} = I_{3} + I_{4}$
$\Rightarrow I_{O} = 2 I_{1} = 2 I_{2} = 2 I_{3} = 2 I_{4}$
$\Rightarrow I_{O} = I_{1} + I_{2} = I_{3} + I_{4} = I_{1} + I_{3}$

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Similar questions

The moment of inertia of thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is not equal to (Given $I_{1}, I_{2}, I_{3} a n d I_{4}$ respectively, are the moments of inertia about axes 1,2,3and 4 which are in the plane of the plate)

Q. A thin square plate

A B C D

of uniform thickness is shown in the figure.

I_{1}, I_{2}, I_{3}

and

I_{4}

are respectively the moments of inertia about axis

1, 2, 3

and

4

which are in the plane of the square plate. Then, the moment of inertia

I_{0}

of the plate about an axis passing through its centre

O

and perpendicular to the plane of the plate will be equal to : (Choose the correct option(s))

A B C D

is a square plate with centre

O

. The moment of inertia of the plate about the axes

1, 2, 3

and

4

are

I_{1}, I_{2}, I_{3} & I_{4}

respectively. It follows that

Q. The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is:
a)

I_{1} + I_{3}

I_{2} + I_{4}

2 I_{1} + I_{3}

I_{1} + 2 I_{3}

The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate can NOT be given by :

where, $I_{1}, I_{2}, I_{3}, I_{4}$ are the moments of inertia of the plates about axes 1,2,3,4 respectively.

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