Question

a stone weighs 100 N on earth surface. the ratio of its wait at the height of half the radius of earth to its weight at a depth of half the radius of the earth will be approximately equal to


A. 3.6 B. 2.2 C. 1.8 D. 0.9

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, 
$g'=\frac{g}{\left(1+\frac{h}{R}\right)^2}$
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,
$g"=g\left(1-\frac{h}{R}\right)$
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 0.9.
option (D) is correct.