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Two uniform solid spheres of equal radii \({R},\) but mass \({M}\) and \(4M\) have a centre to centre separation \(6R,\) as shown in the figure. The two spheres are held fixed. A projectile of mass \(m\) is projected from the surface of the sphere of mass \(M\) directly towards the centre of the second sphere. The expression for the minimum speed \(v\) of the projectile so that it reaches the surface of the second sphere is:

1. \(\left(\dfrac{3 {GM}}{5 {R}}\right)^{1 / 2}\)2. \(\left(\dfrac{2 {GM}}{5 {R}}\right)^{1 / 2}\)

3. \(\left(\dfrac{3 {GM}}{2 {R}}\right)^{1 / 2}\)

4. \(\left(\dfrac{5 {GM}}{3 {R}}\right)^{1 / 2}\)

Subtopic: Gravitational Potential Energy |

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The planet Mars has two moons, Phobos and Delmos. Phobos has a period of \(7\) hours, \(39\) minutes and an orbital radius of \(9 . 4 \times 10^{3}\) km. The mass of mars is:

1. | \(6 . 48 \times 10^{23} \text{ kg}\) | 2. | \(6 . 48 \times 10^{25} \text{ kg}\) |

3. | \(6 . 48 \times 10^{20} \text{ kg}\) | 4. | \(6 . 48 \times 10^{21} \text{ kg}\) |

Subtopic: Satellite |

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Three equal masses of \(m\) kg each are fixed at the vertices of an equilateral triangle \(ABC.\) What is the force acting on a mass \(2m\) placed at the centroid \(G\) of the triangle?

(Take \(AG=BG=CG=1\) m.)

1. \(Gm^2(\hat{i}+\hat{j})\)

2. \(Gm^2(\hat{i}-\hat{j})\)

3. zero

4. \(2Gm^2(\hat{i}+\hat{j})\)

Subtopic: Newton's Law of Gravitation |

82%

From NCERT

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Let the speed of the planet at the perihelion \(P\) in figure shown below be \(v_{_P}\) and the Sun-planet distance \(\mathrm{SP}\) be \(r_{_P}.\) Relation between \((r_{_P},~v_{_P})\) to the corresponding quantities at the aphelion \((r_{_A},~v_{_A})\) is:

1. | \(v_{_P} r_{_P} =v_{_A} r_{_A}\) | 2. | \(v_{_A} r_{_P} =v_{_P} r_{_A}\) |

3. | \(v_{_A} v_{_P} = r_{_A}r_{_P}\) | 4. | none of these |

Subtopic: Kepler's Laws |

78%

From NCERT

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Two spheres of masses \(m\) and \(M\) are situated in air and the gravitational force between them is \(F.\) If the space around the masses is filled with a liquid of specific density \(3,\) the gravitational force will become:

1. \(3F\)

2. \(F\)

3. \(F/3\)

4. \(F/9\)

Subtopic: Newton's Law of Gravitation |

82%

From NCERT

AIPMT - 2003

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For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is:

1. $\frac{1}{\sqrt{2}}$

2. \(2\)

3. $\sqrt{2}$

4. $\frac{1}{2}$

Subtopic: Satellite |

81%

From NCERT

AIPMT - 2005

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If \(v_e\) is the escape velocity and \(v_0\) is the orbital velocity of a satellite for orbit close to the earth's surface, then these are related by:

1. | \(v_o=v_e\) | 2. | \(v_e=\sqrt{2v_o}\) |

3. | \(v_e=\sqrt{2}~v_o\) | 4. | \(v_o=\sqrt{2}~v_e\) |

Subtopic: Orbital velocity |

76%

From NCERT

AIPMT - 2012

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The time period of an earth satellite in circular orbit is independent of:

1. | the mass of the satellite |

2. | radius of the orbit |

3. | none of them |

4. | both of them |

Subtopic: Satellite |

78%

From NCERT

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Two satellites A and B move around the earth in the same orbit. The mass of B is twice the mass of A. Then:

1. | speeds of A and B are equal. |

2. | the potential energy of earth \(+\) A is same as that of earth \(+\) B. |

3. | the kinetic energy of A and B are equal. |

4. | the total energy of earth \(+\) A is same as that of earth \(+\) B. |

Subtopic: Satellite |

68%

From NCERT

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Which of the following quantities remain constant in a planetary motion (consider elliptical orbits) as seen from the sun?

1. speed

2. angular speed

3. kinetic energy

4. angular momentum

Subtopic: Kepler's Laws |

84%

From NCERT

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