If $R$ is the radius of the earth and $g$ is the acceleration due to gravity on the earth surface. Then the me

Question

If $R$ is the radius of the earth and $g$ is the acceleration due to gravity on the earth surface. Then the mean density of the earth will be:

Accepted Answer

1. **Formula for g:** The acceleration due to gravity ($g$) on the surface of the Earth is related to its mass ($M$) and radius ($R$) by the formula:
   $$ g = \frac{GM}{R^2} $$
2. **Mass in terms of Density:** Assuming the Earth is a sphere of uniform mean density ($\rho$), its mass is given by volume times density:
   $$ M = \frac{4}{3}\pi R^3 \rho $$
3. **Substitution and Solving:** Substitute the expression for $M$ into the equation for $g$:
   $$ g = \frac{G}{R^2} \left( \frac{4}{3}\pi R^3 \rho \right) $$
   $$ g = \frac{4}{3} \pi G R \rho $$
   Rearranging to solve for density ($\rho$):
   $$ \rho = \frac{3g}{4 \pi G R} $$

Step-by-Step Solution

Practice All PHYSICS Questions