Point masses m_{1} and m_{2}, are placed at the opposite ends of a rigid rod of length L and negligible mass. The rod is set into rotation about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\mathrm{\omega}}_{0}$ is minimum, is given by:

1. $x=\frac{{m}_{1}L}{{m}_{1}+{m}_{2}}$

2. $x=\frac{{m}_{1}}{{m}_{2}}L$

3. $x=\frac{{m}_{2}}{{m}_{1}}L$

4. $x=\frac{{m}_{2}L}{{m}_{1}+{m}_{2}}$

Subtopic: Moment of Inertia |

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Three identical spherical shells, each of mass m and radius r are placed as shown in the figure. Consider an axis XX', which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the XX' axis is:

1. $\frac{11}{5}m{r}^{2}$

2. $3m{r}^{2}$

3. $\frac{16}{5}m{r}^{2}$

4. $4m{r}^{2}$

Subtopic: Moment of Inertia |

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The moment of inertia of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through -

1. C

2. D

3. A

4. B

Subtopic: Moment of Inertia |

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The moment of inertia of a thin uniform rod of mass M and length L about an axis passing through its mid-point and perpendicular to its length is I_{0}. Its moment of inertia about an axis passing through one of its ends and perpendicular to its length is:

1. ${\mathrm{I}}_{0}+{\mathrm{ML}}^{2}/4$

2. ${\mathrm{I}}_{0}+2{\mathrm{ML}}^{2}$

3. ${\mathrm{I}}_{0}+{\mathrm{ML}}^{2}$

4. ${\mathrm{I}}_{0}+{\mathrm{ML}}^{2}/2$

Subtopic: Moment of Inertia |

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From a circular disc of radius R and mass 9M, a small disc of mass M and radius R / 3 is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is -

1. $M{R}^{2}$

2. 4$M{R}^{2}$

3. $\frac{4}{9}$$M{R}^{2}$

4. $\frac{40}{9}$$M{R}^{2}$

Subtopic: Moment of Inertia |

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Four identical thin rods, each of mass M and length l, form a square frame. The moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is:

1. $\frac{4}{3}M{l}^{2}$

2. $\frac{2}{3}M{l}^{2}$

3. $\frac{13}{3}M{l}^{2}$

4. $\frac{1}{3}M{l}^{2}$

Subtopic: Moment of Inertia |

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The ratio of the radii of gyration of a circular disc to that of a circular ring, each of the same mass and radius, around their respective axes is:

1. $\sqrt{3}:\sqrt{2}$

2. $1:\sqrt{2}$

3. $\sqrt{2}:1$

4. $\sqrt{2}:\sqrt{3}$

Subtopic: Moment of Inertia |

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A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is 90^{o}. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is:

1. $\frac{{\mathrm{ML}}^{2}}{24}$

2. $\frac{{\mathrm{ML}}^{2}}{12}$

3. $\frac{{\mathrm{ML}}^{2}}{6}$

4. $\frac{\sqrt{2}{\mathrm{ML}}^{2}}{24}$

Subtopic: Moment of Inertia |

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The moment of inertia of a uniform circular disc of radius *R* and mass *M* about an axis touching the disc at its diameter and normal to the disc is:

1. $M{R}^{2}$

2. $\frac{2}{5}M{R}^{2}$

3. $\frac{3}{2}M{R}^{2}$

4. $\frac{1}{2}M{R}^{2}$

Subtopic: Moment of Inertia |

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