Question

Assuming the Earth to be a sphere of uniform density, how much would a body weigh one fourth down to the centre of the Earth, if it weighed 250 N on the surface?

Answer 1

let the mass of earth be M and radius be r, density be d , and volume be v
mass of body be m
its weight on surface be W₁= 250 N...1
acceleration due to gravity be g
now, W₁=mg
        W₁=m × $\frac{GM}{ r^{2} }$
since density =mass/volume
therefore mass=density  × volume
        W₁=m × $\frac{Gvd}{ r^{2} }$
        W₁=m × $\frac{G d 4/3 \pi r^{3} }{ r^{2} }$
        W₁=m × G × 4/3 π r...................2
now on going 1/4 down to center of earth its radius become 3r/4
  then W₂=m × g' 
         W₂=m × G × d × 4/3 π × 3r/4
         W₂=m × G × d × 4/3 π × r × 3/4
from 2
         W₂=W₁ × 3/4
         W₂=250 × 3/4
         W₂=187.5 ans.......