A thin wire of length and uniform linear mass density is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is [2000]

Question

A thin wire of length $L$ and uniform linear mass density $ρ$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is [2000]

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(4)

About the diameter of the circular loop (ring)

$I = \frac{1}{2} M R^{2}$

Using parallel axis theorem

Moment of inertia of the loop about XX' axis

$I_{X X'} = \frac{M R^{2}}{2} + M R^{2} = \frac{3}{2} M R^{2}$

Here mass $M = L ρ$ and radius $R = \frac{L}{2 π}$

$∴$ $I_{X X'} = \frac{3}{2} (L ρ) {(\frac{L}{2 π})}^{2} = \frac{3 L^{3} ρ}{8 π^{2}}$

Solution

Q.
A thin wire of length $L$ and uniform linear mass density $ρ$ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is [2000]

1 $\frac{ρ L^{3}}{8 π^{2}}$

2 $\frac{ρ L^{3}}{16 π^{2}}$

3 $\frac{5 ρ L^{3}}{16 π^{2}}$

4 $\frac{3 ρ L^{3}}{8 π^{2}}$

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Solution

Q. A thin wire of length L and uniform linear mass density ρ is bent into a circular loop with centre at O as shown. The moment of inertia of the loop about the axis XX' is [2000]

1 ρL38π2

2 ρL316π2

3 5ρL316π2

4 3ρL38π2