At a high-altitude research station situated at an unknown height above sea level

PHYSICS · Multi-part · CBSE · Physics · Gravitation · Gravitation

Problem

At a high-altitude research station situated at an unknown height

h

above sea level, the local value of

g

is found to be

0.2\,\%

less than its sea-level value. A tunnel is later drilled vertically downward from the station to a depth

d

(measured from the station) where

g

is again equal to its sea-level value. (a) Obtain expressions for

g

at the station and at the bottom of the tunnel in terms of

h

d

and

R

(where

R \gg h,\,d

). (b) Hence derive a relation between

h

and

d

that restores the original value of

g

. (c) Calculate the value of

d

when

h = 1.5\,\text{km}

and

R = 6.4 \times 10^{6}\,\text{m}

. (d) Comment on whether the tunnel needs to be longer or shorter than the height of the station and provide a physical explanation.