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Atoms And Molecules

Question
CBSEENSC9006148

At what height above earth's surface would the value of acceleration due to gravity be half of what it is on the surface? Take the radius of earth to be R.

Solution

A t space t h e space s u r f a c e space o f space t h e space e a r t h comma space

straight g space equals space fraction numerator G M over denominator straight R squared end fraction
A t space h e i g h t space straight h space a b o v e space t h e space e a r t h apostrophe straight s space s u r f a c e comma space

space space straight g subscript straight h space equals space fraction numerator G M over denominator left parenthesis straight R plus straight h right parenthesis squared end fraction

therefore space space space straight g subscript straight h over straight g space equals space open parentheses fraction numerator straight R over denominator straight R plus straight h end fraction close parentheses squared space space space space

rightwards double arrow space space space space straight g subscript straight h space equals straight g open parentheses fraction numerator straight R over denominator straight R plus straight h end fraction close parentheses squared

B u t comma space space space space space straight g subscript straight h space equals space straight g over 2 
therefore space space space space space space straight g over 2 space equals space straight g open parentheses fraction numerator straight R over denominator straight R plus straight h end fraction close parentheses squared space space

rightwards double arrow space space space space space open parentheses fraction numerator straight R over denominator straight R plus straight h end fraction close parentheses squared space equals space 1 half

rightwards double arrow space space fraction numerator straight R plus straight h over denominator straight R end fraction space equals space square root of 2 space space space
rightwards double arrow space space space space space space straight R plus straight h space equals space square root of 2 space straight R

rightwards double arrow space space space space space space space straight h space equals space left parenthesis square root of 2 minus 1 right parenthesis straight R comma space w h e r e space straight R space i s space t h e space r a d i u s space o f space t h e space e a r t h.