# if you had to calculate the mass of the earth, how would you do it?

### Asked by Navavidha Kale | 24th Dec, 2013, 07:43: PM

###

Newton, Galileo, Henry Cavendish, and Eratosthenes contributed to this amazing calculation.

This calculation is done using Newton's Law of Gravity, which formulates the attractive force (gravitational force) that two masses exert on each other:

F=GmM/r^{2}

In Newton's equation, F is the gravitational force, G is a constant of proportionality, M and m are the two masses exerting the forces, and r is the distance between the two objects.

G was calculated by Henry Cavendish in 1798, and was determined to be

6.67 x 10-^{11}m^{3}/(kg sec^{2}).

Also needed is Newton's second law of motion, F=ma, where F is the force applied to an object, m is the mass of the object, and a is its acceleration due to the force.

Galileo determined that the acceleration due to the force of gravity of Earth was a constant equal to 9.8 m/sec^{2} near the surface of the Earth.

Lastly, you need to know the radius of the Earth; this was first calculated by the Greek Eratosthenes.

Calculation:

1. F = GmM/r^{2}= ma, where F is the gravitational force, G is the gravitational constant, M is the mass of the Earth, r is the radius of the Earth, and m is the mass of another object (near the surface of the Earth).

2. GM/r^{2}= a (The m's canceled out.) Now solve for M, the mass of the Earth.

3. M = ar^{2}/G, where a=9.8m/sec^{2}, r=6.4 x 10^{ 6}m, and G=6.67 x 10^{-11}m^{3}/(kg sec^{2}).

4. M = 9.8 x (6.7 x 10^{6})2/6.7 x 10^{-11}= 6.0 x 10^{24}kg

^{ }

Newton, Galileo, Henry Cavendish, and Eratosthenes contributed to this amazing calculation.

This calculation is done using Newton's Law of Gravity, which formulates the attractive force (gravitational force) that two masses exert on each other:

F=GmM/r^{2}

In Newton's equation, F is the gravitational force, G is a constant of proportionality, M and m are the two masses exerting the forces, and r is the distance between the two objects.

G was calculated by Henry Cavendish in 1798, and was determined to be

6.67 x 10-^{11}m^{3}/(kg sec^{2}).

Also needed is Newton's second law of motion, F=ma, where F is the force applied to an object, m is the mass of the object, and a is its acceleration due to the force.

Galileo determined that the acceleration due to the force of gravity of Earth was a constant equal to 9.8 m/sec^{2} near the surface of the Earth.

Lastly, you need to know the radius of the Earth; this was first calculated by the Greek Eratosthenes.

Calculation:

1. F = GmM/r^{2}= ma, where F is the gravitational force, G is the gravitational constant, M is the mass of the Earth, r is the radius of the Earth, and m is the mass of another object (near the surface of the Earth).

2. GM/r^{2}= a (The m's canceled out.) Now solve for M, the mass of the Earth.

3. M = ar^{2}/G, where a=9.8m/sec^{2}, r=6.4 x 10^{ 6}m, and G=6.67 x 10^{-11}m^{3}/(kg sec^{2}).

4. M = 9.8 x (6.7 x 10^{6})2/6.7 x 10^{-11}= 6.0 x 10^{24}kg

^{ }

### Answered by | 26th Dec, 2013, 01:06: PM

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