The moment of inertia of a square loop made of four uniform solid cylinders, each having radius and length about an axis passing through the mid points of opposite sides, is [2026]

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Q.
The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$ about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

1 $\frac{3}{8} M R^{2} + \frac{1}{6} M L^{2}$

2 $\frac{3}{8} M R^{2} + \frac{7}{12} M L^{2}$

3 $\frac{3}{4} M R^{2} + \frac{7}{12} M L^{2}$

4 $\frac{3}{4} M R^{2} + \frac{1}{6} M L^{2}$

Ans.
(1)

$I_{net} = 2 (I_{1} + I_{2})$

$= 2 (\frac{M' R^{2}}{4} + \frac{M' ℓ^{2}}{12}) + 2 (\frac{M' R^{2}}{2} + M' {(\frac{ℓ}{2})}^{2})$

$= \frac{M' R^{2}}{2} + \frac{M' R^{2}}{6} + M' R^{2} + \frac{M' ℓ^{2}}{2}$

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

$Given masses M' = \frac{M}{4}$

$So, I = \frac{3 (M / 4) R^{2}}{2} + \frac{2 (M / 4) ℓ^{2}}{3}$

$I = \frac{3}{8} M R^{2} + \frac{M ℓ^{2}}{6}$

Solution

Q.
The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$ about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

1 $\frac{3}{8} M R^{2} + \frac{1}{6} M L^{2}$

2 $\frac{3}{8} M R^{2} + \frac{7}{12} M L^{2}$

3 $\frac{3}{4} M R^{2} + \frac{7}{12} M L^{2}$

4 $\frac{3}{4} M R^{2} + \frac{1}{6} M L^{2}$

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Solution

Q. The moment of inertia of a square loop made of four uniform solid cylinders, each having radius R and length L (R<L) about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

1 38MR2+16ML2

2 38MR2+712ML2

3 34MR2+712ML2

4 34MR2+16ML2

Ans. (1) Inet=2(I1+I2) =2(M'R24+M'ℓ212)+2(M'R22+M'(ℓ2)2) =M'R22+M'R26+M'R2+M'ℓ22 =3M'R22+2M'ℓ23 Given masses M'=M4 So, I=3(M/4)R22+2(M/4)ℓ23 I=38MR2+Mℓ26

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Q.
The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L$ $(R < L)$ about an axis passing through the mid points of opposite sides, is ______. (Take the mass of the entire loop as M). [2026]

1 $\frac{3}{8} M R^{2} + \frac{1}{6} M L^{2}$

2 $\frac{3}{8} M R^{2} + \frac{7}{12} M L^{2}$

3 $\frac{3}{4} M R^{2} + \frac{7}{12} M L^{2}$

4 $\frac{3}{4} M R^{2} + \frac{1}{6} M L^{2}$