Question

(a) Calculate Earth’s mass given the acceleration due to gravity at the North Pole is ${\text{9.830 m/s}}^{\text{2}}$and the radius of the Earth is $6371km$from pole to pole. (b) Compare this with the accepted value of$M=5.979\times {10}^{24}kg$.

Short answer

(a) The calculated mass of the earth is$M=5.979\times {10}^{24}kg$

(b) The calculated value is matching with accepted value.

Step-by-step solution

Definition of Gravity

Gravity is a universal phenomenon and is introduced by Newton and Derived the expression for gravitational force.

Calculating the mass of Earth

Earth’s radius${\text{= R = 6371 x 10}}^{\text{3}}\text{m}$

Gravitational acceleration near the North Pole${\text{= a = 9.83 m/s}}^{\text{2}}$

Earth’s mass$\text{= M = ?}$

The gravitational force is calculated as follows:

$\text{F = G}\frac{\text{Mm}}{{\text{R}}^{\text{2}}}$

Here, $m$is object’s mass, $M$is Earth’s mass, R Earth’s radius is the gravitation constant and F is the gravitational force. The force acting on the object is calculated as follows:$F=ma$, here, is the object’s acceleration.

$\text{ma = G}\frac{\text{Mm}}{{\text{R}}^{\text{2}}}$

${\text{M = aR}}^{\text{2}}\text{/G}$

$M=\left(9.83\times {\left(6371\times {10}^3\right)}^2\right)/\left(6.637\times {10}^{-11}\right)$

$M=5.979\times {10}^{24}kg$

This is the calculated mass of the earth.

Determining difference between the calculated and accepted value

The calculated value of mass is $M=5.979\times {10}^{24}kg$which is exactly matching with the accepted value of the mass of the earth. The difference in percent between the computed and approved values is zero percent.