Gravitation Question 66

Question:At what depth below the surface of the earth, acceleration due to gravity g will be half itsvalue 1600 km above the surface of the earth[Pb. PMT 2004]

Options:

A) $ [4.2\times 10^{6}m] $

B) $ [3.19\times 10^{6}m] $

C) $ [1.59\times 10^{6}m] $

D) None of these

Show Answer

Answer:

Correct Answer: A

Solution:

Radius of earth R = 6400 km \ $ [h=\frac{R}{4}] $

Acceleration due to gravity at a height h

$ [g_{h}=g{{\left( \frac{R}{R+h} \right)}^{2}}] $

$ [=g{{\left( \frac{R}{R+\frac{R}{4}} \right)}^{2}}] $

$ [=\frac{16}{25}g] $ At depth ’d’ value of acceleration due to gravity $ [g_{d}=\frac{1}{2}g_{h}] $

(According to problem) Þ $ [g_{d}=\frac{1}{2}\left( \frac{16}{25} \right)g] $

$ [\Rightarrow g\left( 1-\frac{d}{R} \right)] $

$ [=\frac{1}{2}\left( \frac{16}{25} \right)\ g] $

By solving we get$ [d=4.3\times 10^{6}m] $



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