Question

Assuming the earth to be a sphere of uniform mass density, how much would a body weigh halfway down to the centre of the earth, if it is weight 250 N on the surface?

Answer 1

We know,

accⁿ due to gravity at depth h from earth's surface ,

g = go( 1 - h/r)

here, h = r/2

g = go(1 - r/2r) = go/2

Now, weight of body at the surface of the earth = 250N = mgo

hence, weight of body at depth R/2 = mgo/2 = 250/2 = 125 N

Answer 2

weight of body on the earth's surface is 250N.
we know, acceleration due to gravity on the earth's surface is g ≈ 10 m/s²

From Newton's 2nd law, F = mg
250 = m × 10 => m = 25kg

Now, we have to find acceleration due to gravity in depth R/2 from the earth's surface.
Use formula, $g=g_0\left(1-\frac{d}{R}\right)$
where $g_0$ is the acceleration due to gravity on the earth's surface, d is depth and R is the radius of the earth.

Here, $g_0$ = 10m/s², d = R/2

so, g' = g(1 - R/2/R) = g/2 = 5m/s²

Hence, weight of body = mg'
= 25 × 5 = 125N.

Hence, answer is 125N