Calculating the Earth's Mass

The mass of the Earth is calculated mathematically by combining the formula for gravitational force with the formula for acceleration due to gravity. By setting the two force equations equal to each other, the mass of an object on the surface cancels out, leaving an equation that directly yields the mass of the planet.

Understanding the two gravity formulas

Gravitational force increases when the masses of two interacting bodies increase, and it decreases in proportion to the square of the distance between them. This relationship is written as force equals the gravitational constant multiplied by the two masses divided by the distance squared. At the same time, any object pulled downward by Earth experiences a force equal to its mass multiplied by its acceleration. Because this acceleration comes from gravity, the general acceleration term is replaced by acceleration due to gravity, giving the formula force equals mass times acceleration due to gravity.

Deriving the formula for Earth's mass

• Both equations describe the exact same gravitational force acting on an object, so they are set equal as mass times acceleration due to gravity equals the gravitational constant times both masses divided by the radius squared.
• Dividing both sides by the smaller mass eliminates the object from the calculation, leaving acceleration due to gravity equals the gravitational constant times the Earth's mass divided by radius squared.
• In this derived formula, small g is acceleration due to gravity, capital G is the gravitational constant, capital M is the mass of the Earth, and r is the radius of the Earth.
• Rearranging the formula to isolate the mass of the Earth gives capital M equals small g times r squared divided by capital G.

Substituting the values

To solve for Earth's mass, the known physical measurements are inserted into the rearranged equation. The acceleration due to gravity is 9.8 meters per second squared, which means a falling object speeds up by 9.8 meters per second every second. The radius of the Earth is 6,356 times 10 to the power of 3 meters, and the gravitational constant is 6.67430 times 10 to the power of minus 11 Newton meter squared per kilogram squared. Multiplying and dividing these values gives Earth's mass as 5.972 times 10 to the power of 24 kilograms.

Practical applications in space science

• Knowing the gravitational force between Earth and an orbiting satellite helps engineers use thrusters to apply an opposing force and keep the satellite stable.
• These gravity calculations allow space scientists to track the exact path of a satellite and understand how it moves in orbit.
• The same mathematical principles can be used to calculate where a falling meteor will land on Earth.