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.

Also, don't forget to follow us on FacebookTwitter, and Google+.
 


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11/03/2015 4:15am

Magnetic power of earth is very high that attract anything. We can get very interesting and beneficent knowledge about falling to earth by the way i am thankful for this beneficent post.

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I enjoy computing every number for my grades. I know that computing is every part of what I do. Computing is a discipline that offers rewarding and challenging possibilities for a wide range of people. I learned a lot in computing. I have physic subject in our classes. I compute the capacity of man to fall to the earth. I’m a first year physics student. According to my professor, the assumed role that the acceleration depends only on the mass below me and the radius only works with a solid sphere.

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