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 \(R\) is the radius of the orbit of a planet and \(T\) is the time period of the planet, then which of the following graphs correctly shows the motion of 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. \(n^2_1d_1= n^2_2d_2\)

The figure shows a planet in an elliptical orbit around the sun \((S).\) The ratio of the momentum of the planet at point \(A\) to that at point \(B\) is:

                           
1. \(\frac{r_1}{r_2}\)
2. \(\frac{r_{1}^{2}}{r_{2}^{2}}\)
3. \(\frac{r_2}{r_1}\)
4. \(\frac{r_{2}^{2}}{r_{1}^{2}}\)

Two satellites \(S_1\)​ and \(S_2\)​ revolve around a planet in coplanar and concentric circular orbits of radii \(R_1\) and \(R_2,\) respectively, both moving in the same direction. Their orbital periods are \(1~\text{hr}\) and \(8~\text{hr},\) respectively. If the orbital radius of \(S_1\)​ is \(10^4~\text{km},\) what will be 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. \(F/9\)