Question

A stone weighs 100n on surface of earth. What will be ratio of its weight at height of half the radius of earth to its weight at a depth of half the radius of the earth?

Answer 1

weight of stone on earth's surface = 100N

Let mass of stone is M

then, weight of stone = Mg [ were g is acceleration due to gravity at earth's surface ]

100N = M × 10m/s²

M = 10kg,

case1 :- acceleration due to gravity at a height of h of the radius of earth, R is given by,

here, h = R/2

so, g' = g/(1 + R/2R)² = g/(1 + 1/2)² = 4g/9

so, weight of stone at half of the radius of the earth, W' = M × g' = 10 × 4g/9 = 10 × 40/9 = 400/9 N

case2 :- acceleration due to gravity at a depth h of half of the radius of the earth R is given by,

here, h = R/2

so, g" = g(1 - R/2R) = g/2 = 5m/s²

so, weight of stone at a depth half of the radius of the earth, W" = M × g" = 10 × 5 = 50N

now ratio, W'/W" = 400/(9 × 50) = 40/45 = 8/9

hence, answer should be 8 : 9