Question

If the distance between earth and sun were one-third of its present value, the number of days in a year would be?
a) 75 days b) 70 days c) 67days d) 68.5 days

Answer 1

we know that
  time period T  = 1 year of a planet    and radius = R a re related  as :
          T²    α   R³
       T2²   = ( R2 / R1 )³  * T1²

          R2 = R1/3  given.
      =>    T2² = 1/27 * 365² = 4934

             T2  = 70.24 days

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velocity of a planet of mass m in orbit around the Sun of mass M :  v
  let time period = T = 2π/ω
   we have  v = r ω,              ω = angular velocity of the rotation of the planet
   let R be the radius of the orbit of the planet.
   G = universal gravitational constant

     centripetal force = gravitational force between Earth and Sun
                 m v²/R = G M m / R²
                      v² = G M / R
                     v = R ω

       =>  ω² = G M / R³
             ω = √(GM/R³)

                 T = 2 π / ω = 2π /√(GM )  * √R³
           or T² = 4π² / (GM)    * R³