You're walking down the street, stumble on a rock, and fall head first into an open manhole. Unfortunately for you, this manhole leads to a tunnel that goes straight through the Earth. As you begin your descent into the dark abyss you figure that you probably have quite some time before you emerge on the other side. But how long do you have? When should you get ready to brace yourself?

This is where EQNS comes in! Let's calculate just how much time you'll have.

But before we can delve any deeper we have to make a few

Assumptions


  • The Earth is a sphere of radius 6.38 E+006 meters
  • The Earth has a constant density of 5,510 kg m^-3
  • Temperature doesn't vary (for your own safety)



Computation


Let's start by determining the gravitational force of the Earth:
Let m_1 be defined as the mass of sphere between you and the center of the Earth, m_2 be defined as your mass, and r be defined as the radius of the Earth. Because m_1 changes as you fall deeper into the Earth we should rewrite the above formula in terms of constants. Let's rewrite it in terms of density. Recall that:
Plug equation [3] into equation [2]:
Solve for m:
Plug equation [5] into equation [1]:
Notice that F is proportional to r, or in other words the force increases linearly with depth. This is analogous to the force exerted by a spring!
The period of oscillation by a mass on a spring can be described by the following equation:
Since we don't want to return to where we are originally, we actually care about half the period. So, let's divide T  by 2 and plug in :
Notice that your mass cancels out:
Finally, plug in the values as solve:
So you have a whole 42 minutes - plenty of time! Take out your phone and start playing around on it. And don't worry about "dropping it". It'll fall at the same rate you do - so you can't even lose it!

If you want to go into more detail on anything we discussed here, please be sure to read Wired's article on this same subject matter.

Be sure to stay tuned to more blog posts and to check out some interesting physics equations on EQNS.

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Much like a needy ex-girlfriend, Pluto has resurfaced once more - and this time more beautiful than ever before. New Horizons recently flew by Pluto, and over 5 hours later we were finally graced with Pluto's intricate contours and features. 

Over 85 years ago, when Pluto was first discovered, we knew very little about this dwarf-planet - whose status, in recent years, has changed more than Kim Kardashian's relationship status. Back in 1930, this is all we had to go off of: 
Picture
Photographic plates depicting the discovery of Pluto (http://www.planetary.org/)
Nearly 70 years later, we were able to get an image of Pluto that was larger than a handful of pixels across. Although this image was still extremely blurry by today's standards, it served as a good first order approximation of Pluto's topography. The observations are listed in the small squares below, the larger images depict rendered computer models that are based on the pixeleted observations in the small squares.  
Picture
1996 Observations and renderings of Pluto (www.hubblesite.org)
Following these observations, Pluto stayed out of the public eye for nearly a decade while scientists were developing a probe that would give us a better understanding of this far away world. Google Trends actually serves as a good barometer for the popularity of Pluto, following 2004. Below is a graph of depicting the popularity of the search term "Pluto" normalized by the total number of searches made during that point in time. 
Picture
Popularity of the keyword "Pluto" over time. Letters depict news headlines that featured the keyword "Pluto" (www.google.com/trends)
The graph depicts many interesting points in time - at least from Pluto's perspective. Here, we will only discuss the three highest peaks. We'll start with peak "I". 

Peak "I"

As many of you may recall, there were many very upset school children in 2006. They were so upset, in fact, that they ended up sending letters to everyone from NASA to Neil deGrasse Tyson. The controversy, if it can even be called that, lay in the fact that we never had a formal definition for a planet and as such have been "winging it" when it came to determining what was a planet and was a asteroid, comet, or Kuiper Belt Object.

By 2006, several other planet-like objects, like Ceres and Eris, had been discovered and astronomers realized that a formal definition was required. During one IAU, or International Astronomical Union, meeting in 2006 a formal definition was established. Unfortunately, this definition demoted Pluto from the status of "planet" to the status of "dwarf planet".

The new definition of a planet is:

"A planet is a celestial body that 
     (a) is in orbit around the Sun,
     (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and
     (c) has cleared the neighbourhood around its orbit. "

Unfortunately, Pluto did not conform to the third requirement and therefore lost its status as a "planet". A formal definition for a "dwarf planet" was also established during the same meeting:

"A 'dwarf planet' is a celestial body that
     (a) is in orbit around the Sun,
     (b) has sufficient mass for self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape,
     (c) has not cleared the neighbourhood around its orbit, and
     (d) is not a satellite. "

This made it incontrovertible that Pluto was no longer a planet, although even in the incontrovertible is often argued.


Peak "J"

Moving backwards in time, we can see that a few months before Pluto was demoted, it was abnormally popular. The reason for this actually ties in quite nicely with the recent observations of Pluto. In January of 2006, NASA launched the New Horizons probe to gain a better understanding of Pluto. The same New Horizons that sent us close-up images of Pluto just a few days ago. 

Peak "A"


The final peak obviously corresponds to the recent images sent by New Horizons. After reading through all of this, potentially boring, history I'll reward you with the beautiful images that were recently received. Enjoy!

Be sure to stay tuned to more blog posts and to check out some interesting astronomy equations on EQNS.

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