question_answer

Assuming earth to be a sphere of a uniform density. What is the value of gravitational acceleration in a mine 100 km below the earth's surface? (given R = 6400 km)

A)  3.9m/s2                              

B)  9.65 m/s2

C)  7.75 m/s2                          

D)  5.25 m/s2

Correct Answer: B

Solution :

                 Key Idea: As we go below the surface of the earth the value of acceleration due to gravity goes on decreasing and becomes zero at the centre of the earth. Value of g at a depth h below the surface of earth is given by \[g'=g\left( 1-\frac{h}{{{R}_{e}}} \right)\] where \[{{R}_{e}}\] is radius of earth. Given,    \[h=100\,km={{10}^{5}}m\] \[{{R}_{e}}=6400\,km=64\times {{10}^{5}}m\] \[\therefore \]                  \[g'=g\left( 1-\frac{h}{{{R}_{e}}} \right)\]                       \[=9.8\left( 1-\frac{{{10}^{5}}}{64\times {{10}^{5}}} \right)\]                          \[=9.8\left( 1-\frac{1}{64} \right)=9.65\,m/{{s}^{2}}\]