# Mass of the Earth

hasan_researc

## Homework Statement

"From your rough ideas or knowledge of the density and radius of the earth, estimate its mass to an order of magnitude. "

## Homework Equations

I don't know the density or the radius of the earth, but I do know that it takes 8 minutes for light from the Sun to reach the earth.

## The Attempt at a Solution

So, the radius of the earth's orbit around the sun = (8*60)s * (3*108)m/s
= 1.4*1011m.
[ My estimate of the time is crude, so I've kept the radius to one s.f.]

Assuming that the radius of the earth << the radius of its orbit around the sun, I am estimating the radius to be on an order of magnitude of 3 to 6. Is this reasonable?

Next, let's estimate the mass of the Earth. According to the theories of classical mechanics,
g = GM/R2, where g = 9.8 m/s2, M = mass of the earth and R = radius of the earth.

R = 106 implies M = 1.5*1023.
R = 105 implies M = 1.5*1021.
R = 104 implies M = 1.5*1019.
R = 103 implies M = 1.5*1017.
[All estimates are in SI units.]

But I have not used density to find my answer. Moreover, my answer has a very large error range.

Tusike
You're idea seems good, but I don't think that's what the problem wants you to do.
For the radius of Earth, I suggest using how 1km was defined in the old days as the circumference at the equator / 40000. The google the average density of Earth... I don't know what it is, but my guess would be near 4g/cm3. From there all you need to use is m = V*q.

hasan_researc
You're asking me to google the average density of the earth, but can't we get that value based alone on logical arguments.

Jobrag
I'm not sure why you are concerned with the radius of Earth's orbit round the sun.
If you have a value for G (universal gravitational constant) and an approximation of the earths radius just plug the values into g=MG/R*R.

Tusike
@Jobrag: how could he use that he doesn't know the mass of Earth, M...

@OP: here's a logical argument: the density probably isn't higher x*10^2. I would say that could be a "rough idea" or "knowledge". Seeing as you have a real precise value for the radius of the Earth, and so it's volume (from the post I've given above), the order of magnitude you'll get for Earth using either q=x*10^0 or q=x*10 will only have an error range of two magnitudes, much less than your original answer (since m=q*V).

theJorge551
@Jobrag: how could he use that he doesn't know the mass of Earth, M...

Jobrag is correct. If you have a somewhat accurate number for the radius of Earth, and you know the mass of something that falls at an average 9.8 m/s2, you then plug it into the equation here and should come out with a good estimation of Earth's mass.

Tusike
@theJorge551, @Jobrag: Sorry, from the way I read the post I just thought he was trying to get the radius of Earth, not it's mass:) And yeah that would work as well, I was just trying to incorporate the density as the problem asked.