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. T²= Kr³, 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 = $\frac{\mathrm{GMm}}{{\mathrm{r}}^2}$, here G is the gravitational constant. The relation between G and K is described as:

1. GK = 4$\mathrm{\pi }$2

2. GMK = 4$\mathrm{\pi }$2

3. K = G

4. K = $\frac{\mathrm{I}}{\mathrm{G}}$

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:
           

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 n₁ and n₂ respectively, then:

1. ${n_1}^2{d_1}^2=n_2{d_2}^2$                                 

2. ${n_2}^2{d_2}^3={n_1}^2{d_1}^3$

3. $n_1{d_1}^2=n_2{d_2}^2$                                   

4. ${n_1}^2{d}_1={n_2}^2{d}_2$

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:

Two satellites S₁ and S₂ are revolving around a planet in coplanar and concentric circular orbits of radii R₁ and R₂ in the same direction respectively. Their respective periods of revolution are 1 hr and 8 hr. The radius of the orbit of satellite S₁ is equal to 10⁴ km. Find the relative speed when they are closest to each other. 

1. $2\mathrm{\pi }\times {10}^4$ $\mathrm{kmph}$

2. $\mathrm{\pi }\times {10}^4$ $\mathrm{kmph}$

3. $\frac{\mathrm{\pi }}{2}\times {10}^4$ $\mathrm{kmph}$

4. $\frac{\mathrm{\pi }}{3}\times {10}^4$ $\mathrm{kmph}$ 

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. \(F/9\)