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Gravitation

Question
CBSEENPH11020477

A satellite of mass m revolves around the earth of radius R at a height x from its surface. If g is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is

  • fraction numerator gR over denominator straight R minus straight x end fraction
  • gx

  • fraction numerator gR squared over denominator straight R space plus straight x end fraction
  • open parentheses fraction numerator gR squared over denominator straight R space plus straight x end fraction close parentheses to the power of 1 divided by 2 end exponent

Solution

D.

open parentheses fraction numerator gR squared over denominator straight R space plus straight x end fraction close parentheses to the power of 1 divided by 2 end exponent

The gravitational force exerted on satellite at a height x is
straight F subscript straight G space equals space fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight x right parenthesis squared end fraction
where Me = mass of earth Since, gravitational force provides the necessary centripetal force, so,
fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight x right parenthesis squared end fraction space equals space fraction numerator mv subscript 0 superscript 2 over denominator left parenthesis straight R plus straight x right parenthesis end fraction
rightwards double arrow space fraction numerator GM subscript straight e straight m over denominator left parenthesis straight R plus straight x right parenthesis end fraction space equals space mv subscript 0 superscript 2
rightwards double arrow space fraction numerator begin display style gR squared straight m end style over denominator left parenthesis straight R plus straight x right parenthesis end fraction space equals space mv subscript 0 superscript 2
rightwards double arrow space fraction numerator begin display style gR squared straight m end style over denominator left parenthesis straight R plus straight x right parenthesis end fraction space equals space mv subscript 0 superscript 2 space
open parentheses because space straight g space equals space fraction numerator begin display style GM subscript straight e end style over denominator straight R end fraction close parentheses
straight v subscript straight o space equals space square root of open square brackets fraction numerator gR squared over denominator left parenthesis straight R plus straight x right parenthesis end fraction close square brackets end root
space equals space open square brackets fraction numerator gR squared over denominator left parenthesis straight R plus straight x right parenthesis end fraction close square brackets to the power of 1 divided by 2 end exponent