Q 1 :    

The minimum energy required to launch a satellite of mass m from the surface of Earth of mass M and radius R in a circular orbit at an altitude of 2R from the surface of the Earth is               [2024]
 

  • 5GmM6R

     

  • 2GmM3R

     

  • GmM2R

     

  • GmM3R

     

(1)

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Energy of satellite on surface of earth,

Ei=-GmMR

The energy at distance (2R+R=3R),

Ef=-GMm2(3R)=-GMm6R

Energy required to launch the satellite,

       E=Ef-Ei=-GMm6R-(-GMmR)

                              =5GMm6R



Q 2 :    

A satellite of mass m is orbiting the earth (of radius R) at a height h from its surface. The total energy of the satellite in terms of g0, the value of acceleration due to gravity at the earth’s surface, is                  [2016]
 

  • mg0R22(R+h)

     

  • -mg0R22(R+h)

     

  • 2mg0R2R+h

     

  • -2mg0R2R+h

     

(2)

Total energy of satellite at height h from the earth's surface,

        E=PE+KE=-GMm(R+h)+12mv2                             ...(i)

Also, mv2(R+h)=GMm(R+h)2

or,    v2=GMR+h                                                                     ...(ii)

From eqns. (i) and (ii),

      E=-GMm(R+h)+12GMm(R+h)=-12GMm(R+h)

          =-12GMR2×mR2(R+h)=-mg0R22(R+h)