Answer

Verified

412.8k+ views

**Hint:**To calculate the moment of inertia of a thin uniform square sheet, we can divide the square into two triangular laminas.

Now the moment of inertia of triangular lamina is

$ =\dfrac{1}{2}\text{M}{{\text{a}}^{2}} $ I

Here taking M $ =\dfrac{\text{m}}{2} $ and a $ =\dfrac{\text{a}}{\sqrt{2}} $ , we can find moment of inertia of triangular lamina and by multiplying it by 2, moment of inertia of uniform square sheet can be calculated.

**Complete step by step solution**

The moment of inertia of a triangular lamina is given by:

I $ =\dfrac{1}{2}\text{M }{{\text{h}}^{2}} $

Here we will put

M $ =\dfrac{\text{m}}{2} $ because mass of triangle is half of the total square

Also h $ =\dfrac{\text{a}}{\sqrt{2}} $ the reason is that the diagonal of square is $ \sqrt{{{\text{a}}^{2}}+{{a}^{2}}}=\sqrt{2{{\text{a}}^{2}}}=\sqrt{2}\text{a} $

Therefore, half of diagonal of square $ =\dfrac{\sqrt{2}\text{a}}{2} $

h $ =\dfrac{\text{a}}{\sqrt{2}} $

So

Moment of inertia of triangular lamina is

$ \text{I=}\dfrac{1}{2}\times \left( \dfrac{\text{m}}{2} \right)\times {{\left( \dfrac{\text{a}}{\sqrt{2}} \right)}^{2}} $

$ =\dfrac{1}{2}\times \dfrac{\text{m}}{2}\times \dfrac{{{\text{a}}^{2}}}{2}=\dfrac{1}{8}\text{m}{{\text{a}}^{2}} $

I $ =\dfrac{1}{8}\text{m}{{\text{a}}^{2}} $

So

Moment of inertia of square lamina $ =2\times $ moment of inertia of triangular lamina

I $ =2\times \dfrac{1}{8}\text{m}{{\text{a}}^{2}} $

I $ =\dfrac{1}{4}\text{m}{{\text{a}}^{2}} $ .

**Note**

Moment of inertia is a quantity expressing a body’s tendency to resist angular acceleration, which is the sum of the product of the mass of each particle in the body with the square of its distance from the axis of rotation.

Moment of inertia formulas for different shapes are:

Solid square $ =\dfrac{1}{2}\text{M}{{\text{R}}^{2}} $

Uniform sphere (through center) $ =\dfrac{2}{5}\text{M}{{\text{R}}^{2}} $

Thin loop (through center) $ =\text{M}{{\text{R}}^{2}} $

Thin loop (through central diameter) $ =\dfrac{1}{2}\text{M}{{\text{R}}^{2}}+\dfrac{1}{12}\text{M}{{\text{w}}^{2}} $

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

Difference Between Plant Cell and Animal Cell

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE