The semi-major axis of the orbit of Saturn is approximately nine times that of Earth. The time period of revolution of Saturn is approximately equal to

  • 81 years

  • 27 years

  • 729 years

  • 813 years


B.

27 years

Given,

Semi-major axis of the orbit of Saturn = 9 rE

where, rE = semi major axis of earth

According to Kepler's law,

       T2 ∝ r3

Let the time period of revolution of saturn around the sun is TS

  T2ST2E = 9 rErE3       T2S = T2E 93         TS = T2E 93              = 93/2 × 1 year               27 years


A comet orbits around the Sun in an elliptical orbit. Which of the following quantities remains constant during the course of its motion ?

  • Linear velocity

  • Angular velocity

  • Angular momentum

  • Kinetic energy


C.

Angular momentum

When comet orbits around Sun in an elliptical orbit, it is under action of a central force and its angular momentum remains constant.


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Suppose the gravitational force varies inversely as the nth power of distance. Then the time period planet in circular orbit of radius R around the sun will be proportional to

  • straight R to the power of open parentheses fraction numerator straight n plus 1 over denominator 2 end fraction close parentheses end exponent
  • straight R to the power of open parentheses fraction numerator straight n minus 1 over denominator 2 end fraction close parentheses end exponent
  • Rn

  • Rn


A.

straight R to the power of open parentheses fraction numerator straight n plus 1 over denominator 2 end fraction close parentheses end exponent

The necessary centripetal force required for a planet to move round the sun = gravitational force exerted on it

straight i. straight e space mv squared over straight R space equals space fraction numerator GM subscript straight e straight m over denominator straight R end fraction
straight v space equals space open parentheses GM over straight R to the power of straight n minus 1 end exponent close parentheses to the power of 1 divided by 2 end exponent
Now comma space straight T space equals space fraction numerator 2 πR over denominator straight v end fraction space equals space 2 πRx space open parentheses straight R to the power of straight n minus 1 end exponent over GM subscript straight e close parentheses to the power of 1 divided by 2 end exponent
space equals space 2 straight pi space open parentheses fraction numerator straight R squared space straight x space straight R to the power of straight n minus 1 end exponent over denominator GM subscript straight e end fraction close parentheses to the power of 1 divided by 2 end exponent
space equals space 2 straight pi space open parentheses fraction numerator straight R to the power of left parenthesis straight n plus 1 right parenthesis divided by 2 end exponent over denominator left parenthesis GM subscript straight e right parenthesis to the power of 1 divided by 2 end exponent end fraction close parentheses
straight T space proportional to space straight R to the power of left parenthesis straight n plus 1 right parenthesis divided by 2 end exponent

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Consider a satellite moving in a circular orbit around Earth. If K and V denote its kinetic energy and potential energy respectively, then (Choose the convention, where V = 0 as r → ∞)

  • K = V

  • K = 2V

  • V = − 2K

  • K = − 2V


C.

V = − 2K

When a satellite moves in a circular orbit around the earth its

(i) Potential Energy,

           V = - GMmr

(ii) Kinetic Energy,

               K = 12 mv2 = GMm2r       v = GMr  V = -2 K


Assuming the mass of Earth to be ten times the mass of Mars, its radius to be twice the radius of Mars and the acceleration due to gravity on the surface of Earth is 10 m/s2. Then the acceleration due to gravity on the surface of Mars is given by

  • 0.2 m/s2

  • 0.4 m/s2

  • 2 m/s2

  • 4 m/s2


D.

4 m/s2

Given, mass of earth = 10 x Mm

where, Mm = Mass of mars

Radius of earth = 2 Rm

where, Rm = radius of mass

and g = GMR2

Let gravity on the surface of mass is gm

 gmgE = MmME × RERm2gm = ge × MmME RERm2     = 10 × Mm10 Mm 2RmRm2     = 4 m/s2


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