A rod of linear mass density '' and length 'L' is bent to form a rig of radius 'R'. Moment of inertia of ring about any of its diameter is: [2025]

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Solution

Q.
A rod of linear mass density ' $λ$ ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is: [2025]

1 $\frac{λ L^{3}}{16 π^{2}}$

2 $\frac{λ L^{3}}{12}$

3 $\frac{λ L^{3}}{4 π^{2}}$

4 $\frac{λ L^{3}}{8 π^{2}}$

Ans.
(4)

$L = 2 π R \Rightarrow R = \frac{L}{2 π}$

$I = \frac{M R^{2}}{2} = \frac{λ \times L}{2} \times {(\frac{L}{2 π})}^{2} = \frac{λ L^{3}}{8 π^{2}}$

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