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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A rod of weight \(w\) is supported by two parallel knife edges, A and B, and is in equilibrium in a horizontal position. The knives are at a distance \(d\) from each other. The centre of mass of the rod is at a distance \(x \) from A. The normal reaction to A is:

1. \(wx \over d\)

2. \(wd \over x\)

3. \(w(d-x) \over x\)

4. \(w(d-x) \over d\)

Subtopic: Torque |

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A mass m moves in a circle on a smooth horizontal plane with velocity v_{0} at a radius R_{0}. The mass is attached to a string that passes through a smooth hole in the plane as shown.

The tension in the string is increased gradually and finally m moves in a circle of radius$\frac{{R}_{0}}{2}$. The final value of the kinetic energy is:

The tension in the string is increased gradually and finally m moves in a circle of radius$\frac{{R}_{0}}{2}$. The final value of the kinetic energy is:

1. $m{v}_{0}^{2}$

2. $\frac{1}{4}m{v}_{0}^{2}$

3. 2$m{v}_{0}^{2}$

4. $\frac{1}{2}m{v}_{0}^{2}$

Subtopic: Angular Momentum |

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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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A force $\overrightarrow{F}=\alpha \hat{i}+3\hat{j}+6\hat{k}$ is acting at a point $\overrightarrow{r}=2\hat{i}-6\hat{j}-12\hat{k}$. The value of $\alpha $ for which angular momentum about the origin is conserved is:

1. -1

2. 2

3. zero

4. 1

Subtopic: Angular Momentum |

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An automobile moves on a road with a speed of \(54\) kmh^{-1}. The radius of its wheels is \(0.45\) m and the moment of inertia of the wheel about its axis of rotation is \(3\) kg-m^{2}. If the vehicle is brought to rest in \(15\) s, the magnitude of average torque transmitted by its brakes to the wheel is:

1. \(6.66\) kg-m^{2}s^{-2}

2. \(8.58\) kg-m^{2}s^{-2}

3. \(10.86\) kg-m^{2}s^{-2}

4. \(2.86\) kg-m^{2}s^{-2}

Subtopic: Torque |

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A solid cylinder of mass 50 kg and radius 0.5 m is free to rotate about the horizontal axis. A massless string is wound around the cylinder with one end attached to it and the other end hanging freely.
The tension in the string required to produce an angular acceleration of 2 revolutions s^{-2} will be:

1. 25 N

2. 50 N

3. 78.5 N

4. 157 N

Subtopic: Rotational Motion: Dynamics |

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The ratio of the acceleration for a solid sphere (mass 'm' and radius 'R') rolling down an incline of angle 'θ' without slipping and slipping down the incline without rolling is:

1. 5:7

2. 2:3

3. 2:5

4. 7:5

Subtopic: Rolling Motion |

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A small object of uniform density rolls up a curved surface with an initial velocity ‘ν’. It reaches up to a maximum height $\frac{3{\mathrm{v}}^{2}}{4\mathrm{g}}$ with respect to the initial position. The object is:

1. Solid sphere

2. Hollow sphere

3. Disc

4. Ring

2. Hollow sphere

3. Disc

4. Ring

Subtopic: Rolling Motion |

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A rod PQ of mass M and length L is hinged at end P. The rod is kept horizontal by a massless string tied to point Q as shown in the figure. When the string is cut, the initial angular acceleration of the rod is:

1. g/L

2. 2g/L

3. 2g/3L

4. 3g/2L

Subtopic: Torque |

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