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Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below as well as above the surface of earth.

Calculate the distance from the surface of the earth at which the acceleration due to gravity is the same below as well as above the surface of earth.

Grade:11

3 Answers

sanatan sharma
26 Points
9 years ago

gh = gs ( R/(R+H))2                                      where s is surface and h is height

gd = gs ( (R-d)R)                                       where d is depth

gh = gs

gs ( R/(R+H))2 = gs ( (R-d)R)

since H = d 

therefore 

(R2)R = (R -d)((R+H)2)

R3 = R3 - RH2 + HR2 - H3

H3 = H( R2 - RH )

H2 = R2 - RH

H2 + RH - R2 = 0

Substitute the value of R as 6400 km

then the answer will come as 3200(√6 -1) km.


Ashwin Muralidharan IIT Madras
290 Points
9 years ago

Hi Arnab,

 

Let Re be the radius of the earth.

And let x be the distance at which the accn due to gvt is the same (both above and below)

 

For above,

g = gs*[ Re2/(Re+x)2 ] -----------(1)

 

For below

g = gs* [ (Re-x)/Re ] --------------(2)

 

where gs is the accn due to gvt at the surface of the earth.

Both are given to be equal. So (1) = (2) will imply,

 

[ Re2/(Re+x)2 ] = [ (Re-x)/Re ]

Hence x2 + Rex - Re2 = 0

 

Which will give x = [(√5 - 1)/2]*Re.

 

Considering Re to be 6400km, will yield x = 3200{√5 - 1}

 

Hope this helps.

 

Best Regards,

Ashwin (IIT Madras).

Arnab Mandal
18 Points
9 years ago

Please explain in detail how did you derive the second equation, i.e., no. 2.

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