Question

a stone weighs 100n on earth's surface . The ratio of its weight at a height of half the radius of the earth to its weight at a depth of half the radius of the earth will be approximately

Answer 1

For distance between object and centre of the earth we have

Acceleration due to gravity at height  be gh = (R/R + h)^2

           mg1 = GMm / (R + h)^2

gh = GM/(R + h)^2

        Now gh/g = R^2/(R + h)^2

  or gh = g(R/R + h)^2

 given h = R/2

gh = g(R/R+R/2)^2

 gh = g(2R / 3R)^2

  = g x 4/9

Acceleration due to gravity at depth = R/2

gD = g(1 - D/R)

gD = g(1 - R/2/R)

  = g/2

Wh / WD = gh /gD

 = 4g/9 / g/2

= 4g/9 x 2/g = 8/9 = 0.8888

Answer 2

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 8 : 9