The weight of a body at a height h = R 3 above the surface of the earth is equal to its weight at a depth d below the surface of the earth. If R is the radius of the earth, what is the value of d ?

Step-by-step solution

The acceleration due to gravity at a height h = R 3 is given by the exact formula: g_h = g (1 + h R )^2 Substituting h = R 3 : g_h = g (1 + 1 3 )^2 = g ( 4 3 )^2 = 9g 16 The acceleration due to gravity at a depth d is given by: g_d = g (1 - d R ) Since the weight of the body is the same at both positions, g_h = g_d : 9g 16 = g (1 - d R ) 1 - d R = 9 16 d R = 1 - 9 16 = 7 16 d = 7R 16 Using the approximate formula for height g_h = g (1 - 2h R ) incorrectly yields d = 2R 3 . Answer: 7R 16