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Questions and their answers are presented here in the module text format as if it were an extension of the theoretical treatment of the topic. The idea is to provide a verbose explanation of the solution, detailing the application of theory. Solution presented here, therefore, is treated as the part of the understanding process – not merely a Q/A session. The emphasis is to enforce ideas and concepts, which can not be completely absorbed unless they are put to real time situation.
We discuss problems, which highlight certain aspects of the calculation of moment of inertia of regularly shaped rigid bodies. For this reason, questions are categorized in terms of the characterizing features pertaining to the questions as :
Example 1
Problem : The moment of inertia of a straight wire about its perpendicular bisector and moment of inertia of a circular frame about its perpendicular central axis are ${I}_{1}$ and ${I}_{2}$ respectively. If the composition of wires are same and lengths of the wires in them are equal, then find the ratio $\raisebox{1ex}{${I}_{1}$}\!\left/ \!\raisebox{-1ex}{${I}_{2}$}\right.$ .
Solution : The MI of the straight wire about a perpendicular through the mid point is :
$$\begin{array}{l}{I}_{1}=\frac{M{L}^{2}}{12}\end{array}$$
The MI of the circular frame about its perpendicular central axis is :
$$\begin{array}{l}{I}_{2}=M{R}^{2}\end{array}$$
According to question, the lengths of the wires in them are equal. As the composition of the wires same, the mass of two entities are same. Also :
$$\begin{array}{l}L=2\pi R\\ \Rightarrow R=\frac{L}{2\pi}\end{array}$$
Substituting in the expression of MI of circular frame,
$$\begin{array}{l}{I}_{2}=\frac{M{L}^{2}}{4{\pi}^{2}}\end{array}$$
Now, the required ratio is :
$$\begin{array}{l}\frac{{I}_{1}}{{I}_{2}}=\frac{M{L}^{2}x4{\pi}^{2}}{12xM{L}^{2}}\end{array}$$
$$\begin{array}{l}\frac{{I}_{1}}{{I}_{2}}=\frac{{\pi}^{2}}{3}\end{array}$$
Example 2
Problem : The moments of inertia of a solid sphere and a ring of same mass about their central axes are same. If ${R}_{s}$ be the radius of solid sphere, then find the radius of the ring.
Solution : The MI of the solid sphere about its central axis is given as :
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