Assuming the earth as a sphere of uniform density, the acceleration due to gravity half way towards the centre of the earth will be if it weighed 250N on the surface?

0.25 g

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0.50 g

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0.125 g

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0.75 g

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Solution

The correct option is C $0.125 g$
Suppose $g_{1} g_{a}$ be the acceleration due to gravity on earth's surface and at a depth $d$ from surface respectively. Also suppose $w$ and $w_{d}$ be weight of body on earth's surface at a depth $d$ .
$w = m g = 250 N ⟶ (1)$
$w_{d} = {m g}_{d} ⟶ (2)$
We know that $g_{d} = g (1 - \frac{d}{R}) ⟶ (3)$
Here $d_{2} = \frac{R}{2}$ $R =$ radius of earth
From equation $(3)$ and $(4)$
$g_{d} = g (1 - \frac{R / 2}{R})$
$= g (1 - 1 / 2) = g \times \frac{1}{2} = \frac{g}{2} ⟶ (5)$
$w_{d} = {m g}_{d} = \frac{m g}{2} = \frac{1}{2} m g = \frac{1}{2} w = \frac{1}{2} \times 250 = 125 N$ .

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