Question

Answer the following question in detail.

State the formula for the acceleration due to gravity at depth ‘d’ and altitude ‘h’. Hence show that their ratio is equal to `(("R - d")/("R - 2h"))` by assuming that the altitude is very small as compared to the radius of the Earth.

Answer 1

1. For an object at depth d, acceleration due to gravity of the Earth is given by,
   `"g"_"d" = "g"(1 - "d"/"R")`      ....(1)
2. Also, the acceleration due to gravity at smaller altitude h is given by,
   `"g"_"h" = "g"(1 - "2h"/"R")`      ....(2)
3. Hence, dividing equation (1) by equation (2), we get,
   `"g"_"d"/"g"_"h" = ("g"(1 - "d"/"R"))/("g"(1 - "2h"/"R")) = ("R - d")/"R" xx "R"/("R - 2h")`
   ∴ `"g"_"d"/"g"_"h" = ("R - d")/("R - 2h")`