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ques17

Asked by rbhatt16 27th December 2017, 11:52 PM
Answered by Expert
Answer:
Since the acceleration due to gravity varies with depth from the earths surface, we have to calculate the gravitational force experienced at the center of the sphere.
 
As shown in figure, let us consider a disc shped earth crust, at a distance h from center. let dh be the thickness of the crust.
 
gravitational force of this disc is given by 
 
begin mathsize 12px style d f space equals space pi open parentheses R squared minus h squared close parentheses d h space rho g h over R end stylebegin mathsize 12px style d f space equals space pi open parentheses R squared minus h squared close parentheses d h space rho g h over R end style
where ρ is uniform density of earth's crust and acceleration due to gravity varies with depth as g×(h/R)
 
total force due to hemishpere is given by
begin mathsize 12px style F space equals space integral d f space equals space fraction numerator pi rho g over denominator R end fraction integral subscript 0 superscript R left parenthesis R squared minus h squared right parenthesis h d h space equals space 1 fourth pi rho g R cubed space................... left parenthesis 1 right parenthesis
sin c e space g space equals space fraction numerator G M over denominator R squared end fraction space equals space G over R squared 4 over 3 pi R cubed rho space equals space 4 over 3 pi G R rho

s u b s t i t u t e space f o r space g space a s space g i v e n space b y space e q n. left parenthesis 2 right parenthesis space i n space e q n. left parenthesis 1 right parenthesis space t o space g e t space F space equals space 1 third pi squared rho squared G R to the power of 4
h e n c e space s t r e s s space equals space F over A space equals space fraction numerator 1 third pi squared rho squared G R to the power of 4 over denominator pi R squared end fraction space equals space 1 third pi G rho squared R squared end style
Answered by Expert 29th December 2017, 4:26 PM
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