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Gravitation

Question
CBSEENPH11020480

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

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

Solution

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