Estimating Mass of Earth: 8 min Light to Reach 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.

Can anyone help, please?

 

[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 Earth's 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.