why does the value of acceleration due to gravity decrease on moving towards the center of the planet as well?
Answers
Answer:
It's actually not entirely true that the strength of the Earth's gravitational field decreases as a function of depth. It is true for certain regions in the Earth, but it's untrue for others because of the non-trivial dependence of the Earth's density on depth.
To see what's going on, assume that the Earth is a sphere whose density is spherically symmetric.
Now consider a mass mm at some radius rr from the center of the Earth. Using Newton's Law of Gravitation, one can show that that given the spherical symmetry, the gravitational attraction on mm of all mass with radii greater than rr exert no net force on it. It follows that only the mass with radii less than or equal to rr contribute to the gravitational force on mm, which, by the Law of Gravitation is
F(r)=GM(r)mr2
F(r)=GM(r)mr2
where M(r)M(r) is the mass of stuff at radii less than or equal to rr. Notice, then, that F(r)F(r) will be an increasing function of rr (and will decrease as r→0r→0), provided M(r)/r2M(r)/r2 is an increasing function of rr.
Now, If the Earth were uniformly dense with density ρ0ρ0, then the mass within a radius rr would be
M(r)=43πr3ρ0
M(r)=43πr3ρ0
namely just the density times the volume of a sphere of radius rr, and in this case the strength of the gravitational field as a function of radius would be
g(r)=F(r)m=G1r243πr3ρ0=(43πgρ0)r
g(r)=F(r)m=G1r243πr3ρ0=(43πgρ0)r
So in this case, it would be true that the strength of the gravitational field would decrease with increasing depth.