By whom and how was the mass of the earth discovered?

Answer 1

Sir Isaac newton, Henry Cavendish, Galileo Galilei and Eratosthenes contributed to the calculation of earth's mass.

The mass of the earth was calculated by using the Newton's Law of Gravity which states that the gravitational force between two objects is directly proportional to the product of their masses of these objects and is inversely proportional to the square of distance between the two objects.

Mathematically: F = GmM/r2

Where F is gravitational force, G is proportionalty canstant, m and M are masses of the objects and r is distance between them.

This law can be used for the gravitational force between the earth and the object near its surface. Then the mass of the earth can be calculated by using the formula: M= Fr2 / Gm where M is the mass of the earth, F is gravitational force, r is radius of the earth and G is gravitational constant.

The circuference of earth was calculated by a Greek mathematician Eratosthenes thousands of years ago. Eratosthenes knew that on the summer solstice at local noon sun appears directly overhead on the Tropic of Cancer. He calculated his local noon (half-way between sunrise and sunset) in Alexandria. He erected a vertical stick and measured the length of its shadow. After this he obtained the angle of elevation of sun. Then he calculated the zenith angle by subtracting the angle of elevation of sun from 90°. He calculated the earth circumference using the formula:

Angle of the sun / 360° = Distance to Tropic of Cancer / Earth circumference

When the circumference is known, radius can be calculated. So Eratosthenes was the first person to measure the radius of the earth.

The value of g (acceleration due to gravity) was measured by Galileo and it is used for the calculation of earth's mass. Aristotle's concept of physics was that the lighter objects fall slowly while the heavier objects fall faster, later Galileo discovered that acceleration towards the earth due to gravity is constant for all objects. He dropped the heavier and lighter object from the same height at the same time and noticed that all bodies accelerate towards the earth in the same way.

Newton's law of Gravitational force: F=GMm/r2

Newton's second law of motion: F= mg

mg = GMm/r2 or g = GM/r2 where g is gravity, G is gravitational constant, M is mass of earth and r is distance of falling object from earth. Acceleraton due to gravity (g) does not depend on the mass of falling object m. All objects fall at the same rate.

Gallileo showed downward motion of a ball due to gravity and calculated the value of g. Galileo could not measure the vertical fall because it was too fast. Instead he measured the acceleration by rolling balls down an inclined board. He measure the distance covered by the balls and measured the times in clicks.

Galileo could not measured the exact value of acceleration of gravity because :

1: he measured g by rolling balls down the inclined planes (angle of inclined board affected the acceleration of balls)

2: the ball was rolling (he could not measure motion of falling balls directly to the earth due to gravity)

So he could not measured the right value of g but he had a right idea. The actual value of g is 9.8 m/s2

　

In 1798 Henry Cavendish was the first person to measure the mass of earth and for this purpose he calculated the value of constant G (gravitational constant) because the value of g and the radius of earth had already been discovered. After discovering the value of G he putted the values in the formula of Newton's Law of Gravitation and discovered the mass of the earth. The Cavendish's apparatus consist of two wooden rods with the lead spheres attached to each end (two small balls and two larger balls). The rod with the small balls was suspended from a wire and was free to rotate. A mirror was attached to the wire. The two large balls were placed near the small balls. The attraction between the large balls and small balls caused the wooden rod to rotate, twisting the wire through a small angle. Henry Cavenih measured the amount of twist by the position of reflected light spot from the mirror. He measured the force F between the balls by measuring the oscillation period of wire.

He measured the value of G using the formula: F = GMm/r2 where M is mass of large ball, m is mass of small ball and r is distance between balls.

Calculations

F=GMm/r2 where F is gravitational force, G is gravitational constant, M is mass of earth, m is mass of another object near the surface of earth, r is the radius of the earth.

According to Newton's second law of motion F=ma where a = g (acceleration due to gravity)

mg = GMm/r2 or g = GM/r2 (mass of object m is canceled out)

M = ar2 / G (value of g is 9.8m/s2, G= 6.67*10 -11 m3/kg.s2)

M = 6.0*10 24 kg

Answer 2

1. Cavendish was the first person to determine Newton's gravitational constant and accurately measured the mass and density of the earth. Eratosthenes was a Greek scholar who was the first person to determine the circumference of the earth. The mass of the earth is 5.972×10^24kg. The density of the earth is 5.51g/cm3. The circumference of the earth is 40075 km.

Answer 3

The earth is so large that it has never been lost.