The distance of a planet from the sun is \(5\) times the distance between the earth and the sun. The time period of the planet is: 

A planet moves around the sun. At a point \(P,\) it is closest to the sun at a distance \(d_1\) and has speed \(v_1.\) At another point \(Q,\) when it is farthest from the sun at distance \(d_2,\) its speed will be:

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

Kepler's third law states that the square of the period of revolution (\(T\)) of a planet around the sun, is proportional to the third power of average distance \(r\) between the sun and planet i.e. \(T^2 = Kr^3\), here \(K\) is constant. If the masses of the sun and planet are \(M\) and \(m\) respectively, then as per Newton's law of gravitation, the force of attraction between them is \(F = \frac{GMm}{r^2},\) here \(G\) is the gravitational constant. The relation between \(G\) and \(K\) is described as:
1. \(GK = 4\pi^2\)
2. \(GMK = 4\pi^2\)
3. \(K =G\)
4. \(K = \frac{1}{G}\)

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

If \(R\) represents the orbital radius of a planet and \(T\) its orbital period, which of the following graphs correctly depicts the relationship between \(R\) and \(T\) for a planet revolving around the Sun?

1.		2.
3.		4.

If two planets are at mean distances \(d_1\) and \(d_2\) from the sun and their frequencies are \(n_1\) and \(n_2\) respectively, then:
1. \(n^2_1d^2_1= n_2d^2_2\)
2. \(n^2_2d^3_2= n^2_1d^3_1\)
3. \(n_1d^2_1= n_2d^2_2\)
4. \(n^2_1d_1= n^2_2d_2\)

A planet moves around the Sun \(S\) in an elliptical orbit, as shown in the figure. If its distances from the Sun at points \(A\) and \(B\) are \(r_1\)​ and \(r_2\)​ respectively, what is the ratio of its linear momentum at \(A\) to that at \(B\)?

                           
1. \(\dfrac{r_1}{r_2}\)

2. \(\dfrac{r_{1}^{2}}{r_{2}^{2}}\)

3. \(\dfrac{r_2}{r_1}\)

4. \(\dfrac{r_{2}^{2}}{r_{1}^{2}}\)

Two satellites \(S_1\)​ and \(S_2\)​ move in the same direction in coplanar, concentric circular orbits of radii \(R_1\)​ and \(R_2.\) Their orbital periods are \(1~\text{hr}\) and \(8~\text{hr}\) respectively. If \(R_1=10^4~\text{km},\) what is their relative speed when they are closest to each other?
1. \(2\pi \times 10^4~\text{kmph}\)

2. \(\pi \times 10^4~\text{kmph}\)

3. \(\dfrac{\pi}{2} \times 10^4~\text{kmph}\)

4. \(\dfrac{\pi}{3} \times 10^4~\text{kmph}\)

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. \(F/9\)