Planet Earth

# What is the weight of the Earth?

## Weight of the Earth

The Earth is in orbit around the Sun, or in freefall, so

technically its weight is zero.

Weight depends on measuring the attraction between two objects,

such as a person

and the Earth. It is actually a mutual attraction between their

masses. The mass of

the Earth, however, can be calculated from its gravity.

## Mass of the Earth

Newton showed that, for spherical objects, you can make the

simplifying assumption

that all of the object's mass is concentrated at the center of

the sphere. The following

equation expresses the gravitational attraction that two

spherical objects have on one another:

F = G * M1 * M2 / R2

R is the distance separating the two objects.

G is a constant that is 6.67259x10-11m3/s2 kg.

M1 and M2 are the two masses that are attracting each other.

F is the force of attraction between them.

Assume that Earth is one of the masses (M1) and a 1-kg sphere is

the other (M2).

The force between them is 9.8 kg*m/s2 -- we can calculate this

force by dropping

the 1-kg sphere and measuring the acceleration that the Earth's

gravitational field

applies to it (9.8 m/s2).

The radius of the Earth is 6,400,000 meters (6,999,125 yards).

If you plug all of these

values in and solve for M1, you find that the mass of the Earth

is 6,000,000,000,000,000,000,000,000 kilograms (6 x 1024

kilograms), or

6 quintillion metric tons (6 x 1021

tonnes).

----

As odd as it sounds, the weight of Earth is exactly zero,

because the Earth is in

orbit around the sun, and as such, the Earth is free falling in

space around the

sun.1 Any object in free fall in space, including an object in

orbit, is weightless.

That is why astronauts are weightless when in orbit around the

Earth.

Weight is a characteristic of an object as it relates to the

gravitational field it is

resting in. You would have to take the earth to a much more

massive world, like

Jupiter, and ignoring the difficulties caused by the gaseous

make up of the

planet, put the Earth on a rather large scale to see what the

Earth "weighs"

there. Of course, its weight would change based on its distance

from the center

of gravity of the attracting object. On Earth, weight changes

negligibly at any

altitude within the atmosphere.

The mass of the earth is another matter. The mass of the earth

is 5.9736×1024 kg,

1 An orbit is a special case of a free fall condition. As the

orbiting object falls downwards, it

also travels transversely (sideways) at such a rate that its

falling trajectory projects a curve

that always remains the same distance from the planet's

surface.

======================================

From a different perspective:

The following experient was performed. Gravitational forces

always occur in pairs,

between the centers of two masses, and the two

forces are equal, so that the

force between me and the earth is what I call my "weight". If

this is generally

correct, then the weight of any object depends on the

other object to which

it is gravitationally attracted at the moment, and if

that's true, then I can weigh

the earth on me.

In my laboratory, I placed a tiny mirror on the floor. I then

took a bathroom scale

out of a cabinet, inverted it, and placed it top-down on the

floor, with its digital

display visible in the tiny mirror. I then alighted upon the

scale, placing my full

body upon the surface that is normally the bottom of the scale

... the surface

with the label, the rubber feet, and the battery door on it. In

this way, I was

able to weigh the earth in my gravitational

field, and (just as Sir Isaac might

have predicted) it was precisely equal to my

weight when measured in the

earth's gravitational field.

It would seem that in order to accurately quote the earth's

weight, the question

must specify the other object to which the earth is being

gravitationally attracted,

and must also specify the other object's mass, and the distance

between the

centers of mass of the earth and the other object. As any of

these details

changes, so too does the earth's 'weight' change!

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