Value of gravitational constant at the centre of the sphere
Answers
What is the value of G at the centre of Earth?
G represents the ‘gravitational constant’, which is approximately
6.674×10−11N.kg−2m2
G has the same value at the centre of the Earth as it does on the tip of my nose, the surface of Jupiter, a point 533 light-years away in any direction you care to choose or anywhere else in the Universe.
Whereas g is the acceleration due to gravity at sea-level on the surface of the Earth. This is approximately
9.8ms−2
Perhaps what you are really interested in is the acceleration due to gravity towards the centre of the Earth as we move through the Earth towards the centre?
From Newton’s Law of Universal Gravitation, the force, F, acting between two objects is given by:
F=Gm1m2r2 [Eq.1]
where m1 and m2 are their respective masses and r is the distance between the centres of the two masses and G is, of course, the gravitational constant.
From Newton’s Second Law of Motion, Force=Mass×Acceleration, thus:
Acceleration=ForceMass
Combining this with Eq.1, the acceleration on object 2 (which has mass m2) is therefore:
Gm1m2m2r2=Gm1r2 [Eq.2]
From the , when calculating the force acting on an object that is a distance r from the centre of a sphere of radius R>r, one can safely ignore the effect of the mass of that part of the sphere that is further from its centre than r.
So, if object 1 is the Earth, the force acting on object 2 at a point r from the centre of the Earth would be:
Gmrr2
where mr is the mass of that part of the Earth that is no more than r from the centre of the Earth.
If the Earth was a sphere of uniform density ρ, then:
mr=43πρr3
Substituting this into Eq.2, we can express the acceleration as:
43πρr3×Gr2=43πρGr
So, the acceleration would be proportional to the distance from the centre of the Earth. At a point half-way to the centre, r=0.5R, so the acceleration is 0.5g; at a point nine-tenths of the way to the centre of the Earth, r=0.1R, so the acceleration is 0.1g; and at the centre, r=0, so the acceleration is 0.
Of course, the Earth isn’t a sphere (its close); more importantly, it isn’t of uniform density.
The ‘average’ density of the Earth’s crust is around 3 tonnes per cubic metre, the mantle ranges from 3.3 to 5.7 tonnes per cubic metre (increasing with depth) and the core ranges from 10 to 13 tonnes (again increasing with depth).
While the acceleration still decreases to zero, it is no longer a linear decrease. In fact, as you will see from the graph below, the acceleration is actually higher near the bottom of the mantle than it is at the surface!