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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When a mass is rotating in a plane about a fixed point, its angular momentum is directed along:

1. a line perpendicular to the plane of rotation

2. the line making an angle of 45^{o} to the plane of rotation

3. the radius

4. the tangent to the orbit

Subtopic: Angular Momentum |

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Two persons of masses 55 kg and 65 kg respectively, are at the opposite ends of a boat. The length of the boat is 3.0 m and weighs 100 kg. The 55 kg man walks up to the 65 kg man and sits with him. If the boat is in still water, the center of mass of the system shifts by:

1. 3.0 m

2. 2.3 m

3. zero

4. 0.75 m

Subtopic: Center of Mass |

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A solid cylinder of mass 3 kg is rolling on a horizontal surface with a velocity of 4 ms^{-1}. It collides with a horizontal spring of force constant 200 Nm^{-1}. The maximum compression produced in the spring will be:

1. 0.5 m

2. 0.6 m

3. 0.7 m

4. 0.2 m

Subtopic: Rolling Motion |

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ABC is an equilateral triangle with O as its centre. F_{1}, F_{2,} and F_{3} represent three forces acting along the sides AB, BC and AC respectively. If the total torque about O is zero, then the magnitude of F_{3} is:

1. ${\mathrm{F}}_{1}+{\mathrm{F}}_{2}$

2. ${\mathrm{F}}_{1}-{\mathrm{F}}_{2}$

3. $\frac{{\mathrm{F}}_{1}+{\mathrm{F}}_{2}}{2}$

4. $2\left(\mathrm{F}+{\mathrm{F}}_{2}\right)$

Subtopic: Torque |

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The instantaneous angular position of a point on a rotating wheel is given by the equation,

$\theta \left(t\right)$ = 2t^{3} – 6t^{2}

The torque on the wheel becomes zero at:

$\theta \left(t\right)$ = 2t

The torque on the wheel becomes zero at:

1. t= 0.5 s

2. t= 0.25 s

3. t= 2 s

4. t= 1 s

Subtopic: Rotational Motion: Kinematics |

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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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A circular disk of a moment of inertia I_{t} is rotating in a horizontal plane, about its symmetric axis, with a constant angular speed ${\mathrm{\omega}}_{\mathrm{i}}$. Another disk of a moment of inertia I_{b} is dropped coaxially onto the rotating disk. Initially, the second disk has zero angular speed. Eventually, both the disks rotate with a constant angular speed ${\mathrm{\omega}}_{\mathrm{f}}$. The energy lost by the initially rotating disc due to friction is:

1. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}^{2}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega}}_{\mathrm{i}}^{2}$

2. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{t}}^{2}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega}}_{\mathrm{i}}^{2}$

3. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}-{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega}}_{\mathrm{i}}^{2}$

4. $\frac{1}{2}\frac{{\mathrm{I}}_{\mathrm{b}}{\mathrm{I}}_{\mathrm{t}}}{\left({\mathrm{I}}_{\mathrm{t}}+{\mathrm{I}}_{\mathrm{b}}\right)}{\mathrm{\omega}}_{\mathrm{i}}^{2}$

Subtopic: Angular Momentum |

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Two particles that are initially at rest, move towards each other under the action of their mutual attraction. If their speeds are v and 2v at any instant, then the speed of the centre of mass of the system will be:

1. 2v

2. 0

3. 1.5v

4. v

Subtopic: Center of Mass |

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