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Calculate the moment of inertia of a hollow cylinder of mass M and radius R about a line parallel to the axis of the cylinder and on the surface of the cylinder.

(A). $M{{R}^{2}}$
(B). $\dfrac{2}{3}M{{R}^{2}}$
(C). $\dfrac{5}{3}M{{R}^{2}}$
(D). None of the above.

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Last updated date: 22nd Mar 2024
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MVSAT 2024
Answer
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Hint: We will first draw the diagram of the hollow cylinder as per the given conditions and then we will use the parallel axis theorem to find the moment of inertia by the help of parallel axes theorem. And then we will consider a tangent in the hollow cylinder through which we can measure the moment of inertia.

Complete step-by-step answer:
Here in the diagram we see a hollow cylinder with an axis in the middle , now we consider an imaginary line AB, on the right of the cylinder, which is parallel to the axis of the cylinder, The radius of the cylinder is R and the mass of the cylinder is M.
Now, we know that along the axis if the hollow cylinder moves then the moment of inertia is,
 $M{{R}^{2}}$, where M is the Mass and R is the radius.
Now using the parallel axes theorem,
The theorem of parallel axis states that the moment of inertia of a body about an axis parallel to axis passing through centre of mass is equal to the sum of the moment of inertia of body about an axis passing through centre of mass and product of mass and square of distance between the two axes.
I=I$_{0}$+$M{{R}^{2}}$,
I=$M{{R}^{2}}$+$M{{R}^{2}}$,
I=2$M{{R}^{2}}$.
Hence we can say that,
Option D is the correct answer.

Note: If the mass of solid cylinder and hollow cylinder are the same, then we can say that the distribution of mass for hollow cylinder is much farther from the axis of cylinder as compared to a solid cylinder. This is the reason why the moment of inertia for a hollow cylinder is greater than that of a solid cylinder. The parallel axis theorem is used to find the moment of inertia for the area of a rigid body whose axis is parallel to the axis of the moment body that is known and the moment is through the center of gravity of the object.



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