Answer
444.9k+ views
Hint: First of all, we will find the moment of inertia of a disc having $9M$mass and radius R, then we will find the moment of inertia of the small disc which is removed. Then we will take the difference of moment of inertia, so, finally we will get the Moment of inertia of the remaining disc.
Formula used: $I=\dfrac{1}{2}M{{R}^{2}}$
Complete step by step answer:
In question we are given that From a circular disc of radius R and mass $9M$, a small disc of mass M and radius $\dfrac{R}{3}$ is removed concentrically and we are asked to find the moment of inertia of remaining disc along the axis perpendicular to its plane. So, first of all we will draw a figure for our simplicity
Now, let’s consider the mass of the original disc as ${{M}_{1}}$ and ${{M}_{2}}$ is the mass of the disc which is removed. Now, moment of inertia of disc rotating along the axis passing through its centre and perpendicular to the plane, can be given by the formula,
$I=\dfrac{1}{2}M{{R}^{2}}$
Now, moment of inertia of original disc can be given as,
${{I}_{1}}=\dfrac{1}{2}{{M}_{1}}{{R}^{2}}$
Where, ${{M}_{1}}=9M$ and radius is R, on substituting these values we will get,
${{I}_{1}}=\dfrac{9}{2}M{{R}^{2}}$
Now, the moment of inertia of remaining disc can be given as,
${{I}_{2}}=\dfrac{1}{2}{{M}_{2}}{{R}^{2}}$
where, ${{M}_{2}}=M$ and radius is $\dfrac{R}{3}$, on substituting these values we will get,
${{I}_{2}}=\dfrac{1}{2}M{{\left( \dfrac{R}{3} \right)}^{2}}$
\[{{I}_{2}}=\dfrac{1}{18}M{{R}^{2}}\]
Now, taking difference of moment of inertia of original and removed part we fill find the moment of inertia of remaining disc, which can be given as,
$I={{I}_{1}}-{{I}_{2}}$
On substituting the values, we will get,
$I=\dfrac{9}{2}M{{R}^{2}}-\dfrac{1}{18}M{{R}^{2}}$
$I=\dfrac{81-1}{18}M{{R}^{2}}\Rightarrow I=\dfrac{80}{18}M{{R}^{2}}$
$\Rightarrow I=\dfrac{40}{9}M{{R}^{2}}$.
Thus, moment of inertia of remaining disc is $\dfrac{40}{9}M{{R}^{2}}$
Hence, option (a) is the correct answer.
Note: Students might try to solve the sum in shorter way by directly deducting the masses and radius of the disc such as, $I=\dfrac{1}{2}\left( 9-1 \right)M{{\left( R-\dfrac{R}{3} \right)}^{2}}\Rightarrow I=\dfrac{8}{2}M\left( \dfrac{2R}{3} \right)=\dfrac{16}{9}M{{R}^{2}}$ but this will give wrong answer because the area of the disc also comes into consideration while solving in this way so, to avoid the errors students must solve using the above derived method.
Formula used: $I=\dfrac{1}{2}M{{R}^{2}}$
Complete step by step answer:
In question we are given that From a circular disc of radius R and mass $9M$, a small disc of mass M and radius $\dfrac{R}{3}$ is removed concentrically and we are asked to find the moment of inertia of remaining disc along the axis perpendicular to its plane. So, first of all we will draw a figure for our simplicity
![seo images](https://www.vedantu.com/question-sets/a7e6f4d5-7218-45cd-84ec-2fb78325f11c9032925548246512025.png)
Now, let’s consider the mass of the original disc as ${{M}_{1}}$ and ${{M}_{2}}$ is the mass of the disc which is removed. Now, moment of inertia of disc rotating along the axis passing through its centre and perpendicular to the plane, can be given by the formula,
$I=\dfrac{1}{2}M{{R}^{2}}$
Now, moment of inertia of original disc can be given as,
${{I}_{1}}=\dfrac{1}{2}{{M}_{1}}{{R}^{2}}$
Where, ${{M}_{1}}=9M$ and radius is R, on substituting these values we will get,
${{I}_{1}}=\dfrac{9}{2}M{{R}^{2}}$
Now, the moment of inertia of remaining disc can be given as,
${{I}_{2}}=\dfrac{1}{2}{{M}_{2}}{{R}^{2}}$
where, ${{M}_{2}}=M$ and radius is $\dfrac{R}{3}$, on substituting these values we will get,
${{I}_{2}}=\dfrac{1}{2}M{{\left( \dfrac{R}{3} \right)}^{2}}$
\[{{I}_{2}}=\dfrac{1}{18}M{{R}^{2}}\]
Now, taking difference of moment of inertia of original and removed part we fill find the moment of inertia of remaining disc, which can be given as,
$I={{I}_{1}}-{{I}_{2}}$
On substituting the values, we will get,
$I=\dfrac{9}{2}M{{R}^{2}}-\dfrac{1}{18}M{{R}^{2}}$
$I=\dfrac{81-1}{18}M{{R}^{2}}\Rightarrow I=\dfrac{80}{18}M{{R}^{2}}$
$\Rightarrow I=\dfrac{40}{9}M{{R}^{2}}$.
Thus, moment of inertia of remaining disc is $\dfrac{40}{9}M{{R}^{2}}$
Hence, option (a) is the correct answer.
Note: Students might try to solve the sum in shorter way by directly deducting the masses and radius of the disc such as, $I=\dfrac{1}{2}\left( 9-1 \right)M{{\left( R-\dfrac{R}{3} \right)}^{2}}\Rightarrow I=\dfrac{8}{2}M\left( \dfrac{2R}{3} \right)=\dfrac{16}{9}M{{R}^{2}}$ but this will give wrong answer because the area of the disc also comes into consideration while solving in this way so, to avoid the errors students must solve using the above derived method.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)