Question

What is the average density of earth in terms of g , G and R

Answer 1

g = GM / R²
g = G × (4/3 π R³ ρ) / R²
[∵ M = Volume × Density = (4/3) πR³ × ρ]
g = 4πGρR / 3
ρ = 3g / (4πGR)

Average density of Earth is 3g / (4πGR)

Answer 2

Answer:

The average density of earth in terms of g, G and R is D = $\bold{=\frac{3 g}{4 \pi R G}}$

Explanation:

The “acceleration due to gravity”, g is given by the formula:

$\begin{array}{l}{g=\frac{G M}{R^{2}}} \\ {\text { or } M=\frac{g R^{2}}{G}}\end{array}$

Where G is the “universal gravitational constant” with value $\bold{6.67 \times 10^{-11} \mathrm{m}^{3} / \mathrm{s}^{2} \mathrm{kg}}$

M, mass of the earth= $\bold{5.98 \times 10^{24} k g}$

R is the radius of the earth with value $\bold{=6.37 \times 10^{6} \mathrm{m}}$

Density $=\frac{\text {Mass}}{\text {Volume}}$

Now since earth is a sphere, the volume of given by the formula

$V=\frac{4}{3} \pi r^{3}$

Now, putting all the values in the formula

Generally the density is ratio of mass and volume,

Density $=\frac{\text {Mass}}{\text {Volume}}$

We get,

$\begin{array}{l}{\text { Density }=\frac{\frac{g R \epsilon^{2}}{G}}{\frac{4}{3} \pi R^{3}}} \\ {=\frac{3 g}{4 \pi R G}}\end{array}$