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

Question

A stone weighs 100 N on the surface of the Earth. The ratio of its weight at a height of half the radius of the Earth to a depth of half the radius of the earth will be approximately (A) 3.6 (B) 2.2 (C) 1.8 (D) 0.9

Tardigrade Admin · Accepted Answer

Weight of a mass depends on the acceleration due to gravity

(g)

.
Acceleration due to gravity at height

h = (= \frac{R}{2})

from the surface of earth

g_{h} = \frac{g}{( 1 + \frac{\frac{R}{2}}{R} ) ^{2}} = \frac{4}{9} g

....(i)
Acceleration due.to gravity at depth

d (= \frac{R}{2})

from the surface of earth

g_{d} = g (1 - \frac{\frac{R}{2}}{R}) - g (1 - \frac{1}{2}) = \frac{g}{2}

.....(ii)
Required ratio =

\frac{W _{h}}{W _{d}} = \frac{m g _{h}}{m g _{d}} = \frac{\frac{4}{9} g}{\frac{g}{2}} = \frac{8}{9} = 0.9

Q. A stone weighs 100N on the surface of the Earth. The ratio of its weight at a height of half the radius of the Earth to a depth of half the radius of the earth will be approximately

3.6

2.2

1.8

0.9

Q. A stone weighs $100 N$ on the surface of the Earth. The ratio of its weight at a height of half the radius of the Earth to a depth of half the radius of the earth will be approximately