If \(A\) is the areal velocity of a planet of mass \(M,\) then its angular momentum is:

The figure shows the elliptical orbit of a planet \(m\) about the sun \(\mathrm{S}.\) The shaded area \(\mathrm{SCD}\) is twice the shaded area \(\mathrm{SAB}.\) If \(t_1\) is the time for the planet to move from \(\mathrm{C}\) to \(\mathrm{D}\) and \(t_2\) is the time to move from \(\mathrm{A}\) to \(\mathrm{B},\) then:
           

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

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

Two planets are in a circular orbit of radius \(R\) and \(4R\) about a star. At a specific time, the two planets and the star are in a straight line. If the period of the closest planet is \(T,\) then the star and planets will again be in a straight line after a minimum time:
           

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. $6.48\times {10}^{23} \mathrm{kg}$
2. $6.48\times {10}^{25} \mathrm{kg}$
3. $6.48\times {10}^{20} \mathrm{kg}$
4. $6.48\times {10}^{21} \mathrm{kg}$

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. \(2Gm^2(\hat{i}+\hat{j})\)

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. \(F/9\)

If a particle is projected vertically upward with a speed \(u,\) and rises to a maximum altitude \(h\) above the earth's surface then (\(g=\) acceleration due to gravity at surface)