Answer
Verified
391.8k+ views
Hint: You can start the solution by calculating the mass per unit cross section area. Then divide the cylinder into an inner and outer cylinder. Then find the mass of the inner and outer cylinder by using the equation $ \dfrac{M}{{\pi (R_2^2 - R_1^2)}} \times \pi {R^2} $ . Then use the equation $ I = \dfrac{{MR_{}^2}}{2} $ to find the moment of inertia of the inner and the outer cylinder. Then calculate the difference between the moment of the inertia of the outer and inner cylinder to reach the solution.
Complete step-by-step answer:
Here we are given a hollow cylinder with a mass $ M $ and inner radius $ {R_1} $ and outer radius $ {R_2} $ .
So the total area of cross section of the cylinder is
Area $ = \pi (R_2^2 - R_1^2) $
The mass $ M $ of the hollow cylinder is distributed over a cross section area of $ \pi (R_2^2 - R_1^2) $ .
So the mass per unit cross section area is $ \dfrac{M}{{\pi (R_2^2 - R_1^2)}} $
In this problem we have a hollow cylinder, let’s divide it into two parts: a bigger cylinder with a radius $ {R_2} $ and a smaller cylinder with a radius $ {R_1} $ .
The mass of the outer cylinder is
$ {M_{outer}} = \dfrac{M}{{\pi (R_2^2 - R_1^2)}} \times \pi R_2^2 $
$ {M_{outer}} = \dfrac{M}{{(R_2^2 - R_1^2)}} \times R_2^2 $
Similarly the mass of inner cylinder is
$ {M_{inner}} = \dfrac{M}{{\pi (R_2^2 - R_1^2)}} \times \pi R_1^2 $
$ {M_{inner}} = \dfrac{M}{{(R_2^2 - R_1^2)}} \times R_1^2 $
The moment of inertia of the outer cylinder is
$ {I_{outer}} = \dfrac{{{M_{outer}}R_2^2}}{2} $
$ {I_{outer}} = \dfrac{{\left( {\dfrac{M}{{R_2^2 - R_1^2}} \times R_2^2} \right)R_2^2}}{2} $
$ {I_{outer}} = \dfrac{{MR_2^4}}{{2(R_2^2 - R_1^2)}} $
The moment of inertia of the inner cylinder is
$ {I_{inner}} = \dfrac{{{M_{inner}}R_1^2}}{2} $
$ {I_{inner}} = \dfrac{{\left( {\dfrac{M}{{R_2^2 - R_1^2}} \times R_1^2} \right)R_1^2}}{2} $
$ {I_{inner}} = \dfrac{{MR_1^4}}{{2(R_2^2 - R_1^2)}} $
The net moment of inertia is the difference in the moment of inertia of the outer cylinder and the movement of inertia of the inner cylinder
$ {I_{net}} = {I_{outer}} - {I_{inner}} $
\[{I_{net}} = \dfrac{{MR_2^4}}{{2(R_2^2 - R_1^2)}} - \dfrac{{MR_1^4}}{{2(R_2^2 - R_1^2)}}\]
\[{I_{net}} = \dfrac{{M(R_2^4 - R_1^4)}}{{2(R_2^2 - R_1^2)}}\]
\[{I_{net}} = \dfrac{{M(R_2^2 - R_1^2)(R_2^2 + R_1^2)}}{{2(R_2^2 - R_1^2)}}\]\[[\because {A^2} - {B^2} = (A - B)(A + B)]\]
\[{I_{net}} = \dfrac{{M(R_2^2 + R_1^2)}}{2}\]
Hence, option B is the correct choice.
Note: In this problem we divided the hollow cylinder into an outer bigger cylinder and smaller cylinder, found out the moment of inertia of outer cylinder and inner cylinder individually. In this question we will not use the value of moment of inertia of a cylinder around its central diameter i.e. \[\dfrac{1}{4}M{R^2} + \dfrac{1}{{12}}M{L^2}\].
Complete step-by-step answer:
Here we are given a hollow cylinder with a mass $ M $ and inner radius $ {R_1} $ and outer radius $ {R_2} $ .
So the total area of cross section of the cylinder is
Area $ = \pi (R_2^2 - R_1^2) $
The mass $ M $ of the hollow cylinder is distributed over a cross section area of $ \pi (R_2^2 - R_1^2) $ .
So the mass per unit cross section area is $ \dfrac{M}{{\pi (R_2^2 - R_1^2)}} $
In this problem we have a hollow cylinder, let’s divide it into two parts: a bigger cylinder with a radius $ {R_2} $ and a smaller cylinder with a radius $ {R_1} $ .
The mass of the outer cylinder is
$ {M_{outer}} = \dfrac{M}{{\pi (R_2^2 - R_1^2)}} \times \pi R_2^2 $
$ {M_{outer}} = \dfrac{M}{{(R_2^2 - R_1^2)}} \times R_2^2 $
Similarly the mass of inner cylinder is
$ {M_{inner}} = \dfrac{M}{{\pi (R_2^2 - R_1^2)}} \times \pi R_1^2 $
$ {M_{inner}} = \dfrac{M}{{(R_2^2 - R_1^2)}} \times R_1^2 $
The moment of inertia of the outer cylinder is
$ {I_{outer}} = \dfrac{{{M_{outer}}R_2^2}}{2} $
$ {I_{outer}} = \dfrac{{\left( {\dfrac{M}{{R_2^2 - R_1^2}} \times R_2^2} \right)R_2^2}}{2} $
$ {I_{outer}} = \dfrac{{MR_2^4}}{{2(R_2^2 - R_1^2)}} $
The moment of inertia of the inner cylinder is
$ {I_{inner}} = \dfrac{{{M_{inner}}R_1^2}}{2} $
$ {I_{inner}} = \dfrac{{\left( {\dfrac{M}{{R_2^2 - R_1^2}} \times R_1^2} \right)R_1^2}}{2} $
$ {I_{inner}} = \dfrac{{MR_1^4}}{{2(R_2^2 - R_1^2)}} $
The net moment of inertia is the difference in the moment of inertia of the outer cylinder and the movement of inertia of the inner cylinder
$ {I_{net}} = {I_{outer}} - {I_{inner}} $
\[{I_{net}} = \dfrac{{MR_2^4}}{{2(R_2^2 - R_1^2)}} - \dfrac{{MR_1^4}}{{2(R_2^2 - R_1^2)}}\]
\[{I_{net}} = \dfrac{{M(R_2^4 - R_1^4)}}{{2(R_2^2 - R_1^2)}}\]
\[{I_{net}} = \dfrac{{M(R_2^2 - R_1^2)(R_2^2 + R_1^2)}}{{2(R_2^2 - R_1^2)}}\]\[[\because {A^2} - {B^2} = (A - B)(A + B)]\]
\[{I_{net}} = \dfrac{{M(R_2^2 + R_1^2)}}{2}\]
Hence, option B is the correct choice.
Note: In this problem we divided the hollow cylinder into an outer bigger cylinder and smaller cylinder, found out the moment of inertia of outer cylinder and inner cylinder individually. In this question we will not use the value of moment of inertia of a cylinder around its central diameter i.e. \[\dfrac{1}{4}M{R^2} + \dfrac{1}{{12}}M{L^2}\].
Recently Updated Pages
Basicity of sulphurous acid and sulphuric acid are
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
What is the stopping potential when the metal with class 12 physics JEE_Main
The momentum of a photon is 2 times 10 16gm cmsec Its class 12 physics JEE_Main
Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
Which type of bond is stronger ionic or covalent class 12 chemistry CBSE
What organs are located on the left side of your body class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
When people say No pun intended what does that mea class 8 english CBSE