Earth and Light Speed
 

If the Earth is spinning at the same rate of speed (angular velocity) that it is right now, how large would the Earth have to be (diameter) for a person who is standing on the surface of the Earth to be considered moving at the speed of light?

That's easy! It takes a day to revolve, so just convert that value to radians per second and divide the speed of light by it.

We must make our units consistent, so we'll use meters and seconds, since we know that the speed of light is about 3(10^8) m/s.

So, 2pi radians divided by 24 hours, is about 0.000007272 radians per second.

If wr = c, then r = c / w = 3(10^8) / 7.272(10^-5) = 4.125(10^12).

So, the Earth would need a diameter of about 8.2506(10^12) meters! That's an awfully big planet but, then again, planets don't revolve too fast, so this was not entirely unexpected...

Short answer: never. You can't be moving at the speed of light.

Long answer: if you ignored relativity
circumfrence / day = c
circumfrence = radius * 2 * pi
radius = c * 2 * pi * 1 day
Using google as a calculator:
http://www.google.ca/search?q=c*pi*2*60*60*24+seconds
I get:
1.62747495 × 10^14 meters

or, roughly 25 516 610.8 times larger (in every dimension) than it is right now.

This could be fun...

Let's see how quickly the Earth would have to rotate, in order for the surface to reach near-light speed. (Not taking in account for equitorial expansion)

Earth Diameter = 127,563,000M
Earth Circumference = 127,563,000M * Pi = 400,750,984M

Speed of Light in a Vacuum = 299,792,458 M/sec.

For fun, we'll take 1M/sec; I'd like to stay below the cosmic speed limit. ;)

And finally, 400,750,984M / (299,792,458M/sec) would tell us that the earth would have to complete 1 revolution in 1.337 "Standard" Earth seconds.

If in fact this were possible, our Earth would complete 64,634 revolutions per 24 "Standard" Earth hours.

"Standard" Earth seconds/hours is just my fancy way of saying, a second, as we know it today.

-SF

keep in mind also that, as it rotates faster and faster, it will start looking like a football to the outside observer. and probably like some kind of pancake to the observer on earth.

Not in this universe. Remember relativity doesn't exist if the Earth can rotate at light speed.

Im too lazy to do the calculations myself, but I couldn't help noticing there was a difference of two orders of magnitude between two of the answers. That's a lot.

I think KnifeMissile has it here as Yakk multiplied instead of dividing (a frustratingly easy mistake to make if my experience is anything to go by.)

I added that so we wouldn't have to worry about the earth expanding at higher speeds. It just adds to the confusion...

-SF

Ayep, off by (2*pi)^2.

Correction:

circumfrence / day = c
circumfrence = radius * 2 * pi
radius * 2 * pi = c* 1 day
radius = c * day / 2 / pi
Using google as a calculator:
http://www.google.ca/search?q=c*60*6...conds%2Fpi%2F2
I get:
4.12244222 × 10^12 meters

Or, 646 343.302 larger than the real earth.

In reality, the earth would eventually be ripped apart if it were spun faster and faster. At some point, objects on the surface would be moving at orbital velocity, and then things go nuts.

Btw, rigid bodies are not compatible with causality even in principle.

There is some interesting work in what would happen if you tried to spin up a black hole. All evidence indicates that there is a limit to the angular momentum it can have no matter what you do.

Assuming the rotation of the earth could approach the speed of light, I don't think that oribital velocity would come into play. Rememer, according to Relativity, as the speed of an object increases, so does its mass.

More mass means more gravity, which raises the escape velocity.

No, the earth would rip itself apart long before reaching the speed of light.

First of all, think of the scalings involved. The mass would not increase enough to keep things together. The surface will become weightless at a speed of 3.1 km/s, which is much less than the speed of light. That assumes that the earth is increasing radius without adding more material.

If the earth kept the same density as you increased its size, then the orbital velocity problem never comes up. Instead, it would collapse into a black hole long before reaching 4x10^12 meters in radius.

Next, you're applying a concept which has no validity here. Special relativity does not apply to gravitational problems, and you have to be extremely careful in applying it to anything involving acceleration at all. Analyzing exactly what would happen requires general relativity, and is an extremely difficult problem.

Thanks for all the great answers, people!
I must confess, I asked the question more for the discussion it might provoke rather than the actual desire for the answer.
I can't even begin to comprehend this concept of relativity and space and time being related and all that good stuff! But every little bit helps.