**Q: 1. Suppose there existed a planet that went around the sun twice the sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?**

**Sol. From Kepler’s 3rd law ,**

**Here, T _{1} = 1 years, T_{2} = 1/2 year, R_{1} = 1A.U. , R_{2} = ?**

**R _{2} = 0.63 A.U**

**Q:2. One of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is 4.22 × 10 ^{8} m. show that mass of Jupiter is about one thousand times that of sun. (Take 1 year = 365.25 mean solar day)**

**Sol. For the satellite of Jupiter,**

**T _{1} = 1.769 days**

**= 1.769 × 24 × 60 × 60 s**

**R _{1} = 4.22 × 10^{8} m**

**Mass of Jupiter , **

** …(i)**

**For earth around the sun, T _{2} = 1 year = 365.25 × 24 × 60 × 60 s**

**R _{2} = 1 A.U = 1.496 × 10^{11} m**

**Mass of the sun , **

** …(ii)**

**Dividing (ii) by (i)**

**Q: 3.Let us consider that our galaxy consists of 2.5 × 10 ^{11} stars each of one solar mass. How long will this star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milky way to be 10^{5} ly. G = 6.67 × 10^{-11} Nm^{2}/kg^{2}**

**SOl : r = 50,000 ly**

**= 50,000 × 9.46 × 10 ^{15} ,**

**m = 4.73 × 10 ^{20} m**

**M = 2.5 × 10 ^{11} solar mass**

**= 2.5 × 10 ^{11} × 2 × 10^{30} kg**

**= 5 × 10 ^{41} kg**

**= 1.12 × 10 ^{16} s**

**Q: 4. A rocket is fired from the earth towards the sun. At what point on its path is the gravitational force on the rocket zero? Mass of sun = 2 × 10 ^{3} kg, Mass of earth = 6 × 10^{24} kg. Neglect the effect of the other planets. Orbital radius of the earth = 1.5 × 10^{11} m**

**Sol. M _{s} = 2 × 10^{30} kg.**

**M _{e} = 6 × 10^{24} kg , r = 1.5 × 10^{11} m**

**Let at a distance x from the earth the gravitational force on the rocket due to sun and the earth are equal and opposite.**

**Distance of rocket from the sun = r – x**

**If m is the mass of the rocket then**

**x = 2.59 × 10 ^{8} m**

**Q: 5.A Saturn year is 29.5 times the earth year. How far is the Saturn from the Sun if the Earth is 1.5 × 108 km away from the sun ?**

**Sol: T _{s} = 29.5 Te ; R_{e} = 1.5 × 10^{8} km;**

**From Kepler’s 3rd law**

**= 1.43 × 10 ^{9} km**