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 |
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Let the speed of the planet at the perihelion $$P$$ in the figure shown below be $$v_p$$ and the Sun-planet distance $$SP$$ be $$r_p$$. Will the planet take equal time to traverse $$BAC$$ and $$CPB?$$

1. no
2. yes
3. depends on the mass of the planet
4. we can't say anything

Subtopic:  Kepler's Laws |
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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 |
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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 if the mass at the vertex A is doubled? Take AG = BG = CG = 1 m.

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

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

3.   0

4.   $$2Gm^{2} \hat{j}$$

$\stackrel{}{}$

Subtopic:  Newton's Law of Gravitation |
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The potential energy of a system of four particles placed at the vertices of a square of side $$l$$ (as shown in the figure below) and the potential at the centre of the square, respectively, are:

1. $$- 5 . 41 \dfrac{Gm^{2}}{l}$$ and $$0$$

2. $$0$$ and $$- 5 . 41 \dfrac{Gm^{2}}{l}$$

3. $$- 5 . 41 \dfrac{Gm^{2}}{l}$$ and $$- 4 \sqrt{2} \dfrac{Gm}{l}$$

4. $$0$$ and $$0$$

Subtopic:  Gravitational Potential Energy |
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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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You are given the following data: $$g = 9.81~\text{m/s}^{2}$$, $$R_{E} = 6 . 37 \times 10^{6}~\text m$$, the distance to the moon, $$R = 3 . 84 \times 10^{8}~\text m$$ and the time period of the moon’s revolution is $$27.3$$ days. Mass of the Earth $$M_{E}$$ in two different ways is:
1. $$5 . 97 \times 10^{24} ~ \text{kg and }6 . 02 \times 10^{24} \text{ kg}$$
2. $$5 . 97 \times 10^{24} \text{ kg and } 6 . 02 \times 10^{23} \text{ kg}$$
3. $$5 . 97 \times 10^{23} ~ \text{kg and }6 . 02 \times 10^{24} \text{ kg}$$
4. $$5 . 97 \times 10^{23} \text{ kg and } 6 . 02 \times 10^{23} \text{ kg}$$
Subtopic:  Satellite |
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Constant $$k = 10^{- 13} ~ \text s^{2}~ \text m^{- 3}$$ in days and kilometres is?
1. $$10^{- 13} ~ \text d^{2} ~\text{km}^{- 3}$$
2. $$1 . 33 \times 10^{14} \text{ dkm}^{- 3}$$
3. $$10^{- 13} ~ \text d^{2} ~\text {km}$$
4. $$1 . 33 \times 10^{- 14} \text{ d}^{2} \text{ km}^{- 3}$$
Subtopic:  Satellite |
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The moon is at a distance of $$3.84\times10^5~\text{km}$$ from the earth. Its time period of revolution in days is:
$$\left(\text{Given: }k=\dfrac{4\pi^2}{GM_E}=1.33\times10^{-14}~\text{days}^{2}\text-\text{km}^{-3}\right)$$
1. $$17.3$$ days
2. $$33.7$$ days
3. $$27.3$$ days
4. $$4$$ days
Subtopic:  Satellite |
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