Question

13. Let l1 and I2 be the moment of inertia of a uniform square plate about axes shown in the figure. Then the ratio I1:I2 is​

Answer 1

$</p><p>\tt{\purple{\huge{Hello!!! }}}$

$</p><p>\tt{\boxed{\boxed{\orange{\huge {Concept \: Used \: - }} Parallel \: Axis \: Theorem }}}$

By using the parallel axis theorem we can easily state the following :

$</p><p>\tt{\purple{\implies{I_{1} = \dfrac{M{a}^2}{12} }}}$

$</p><p>\tt{\blue{\inplies{I_{2} = \dfrac{M{a}^2}{12} + \dfrac{M{a}^2}{2} = \dfrac{7M{a}^2}{12} }}}$

To Find :

The required ratio of I1 and I2 :

$</p><p>\tt{\red{\therefore{\boxed{\boxed{\dfrac{I_{1}} {I_{2}} = \dfrac{\dfrac{M{a}^2}{12}}{\dfrac{7M{a}^2}{12}} = 1 : 7 }}}}}$

Hence Option 4 is the correct option.

$</p><p>\tt{\green{Additional \: Information \: On \: Parallel \: Axis \: Theorem \: - }}$

The parallel axis theorem, also known as Huygens–Steiner theorem, or just as Steiner's theorem,is named after Christiaan Huygens and Jakob Steiner.

It can be used to determine the moment of inertia or the second moment of area of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of gravity and the perpendicular distance between the axes.

Answer 2

Answer:

Given :

I1 and I2 are the moment of Inertia of a uniform square plate about different axes .

To find:

Ratio of the moment of Inertia along different axes.

Concept:

We first need to find the Moment of Inertia along the centre of mass of the square (Geometric axis). Then we will apply Perpendicular theorem to get MI along the diagonal . Finally apply parallel theorem to get the moment of Inertia along the desired axis.

Calculation:

We know that for a square , The moment of Inertia along Centre of Mass (geometric axis) will be :

$I = \dfrac{m}{12} ( {l}^{2} + {l}^{2} ) = \dfrac{m {l}^{2} }{6}$

Now applying perpendicular theorem to get MI along diagonal :

$I1 = \dfrac{1}{2} \times \dfrac{m {l}^{2} }{6} = \dfrac{m {l}^{2} }{12}$

Finally applying Parallel theorem :

$I2 = \dfrac{m {l}^{2} }{12} + m { (\dfrac{l}{ \sqrt{2} }) }^{2} = \dfrac{7m {l}^{2} }{12}$

So required ratio will be :

$\boxed{ \red{ \huge{ \bold{I1 : I2 = 1 : 7}}}}$