Research topic: What would happen if there was a hole through the Earth?

Introduction: This is an idea or thought that we have all had before. What if I could go through the earth and come out on the other side? This concept has also been presented through cartoons but scientists have generally not entertained this idea and many believe this is impossible but I believe it is possible.

Aim: To determine what would happen if someone / something was to fall through a hole through the Earth and how long it would take them/it to reach the other side.

Hypothesis: The person or object would fall all the way to the opposite side of earth and would take approximately 20-30 minutes to get to the other side.

Background; The farthest you can travel from home ,and still remain on Earth,is about 12,700 kilometers straight down, but you’ll have to journey the long way round to get there:20,036 kilometers over land and sea.

Why not take a shortcut, straight down?

A whole through the earth would have many benefits to humans and would revolutionise the way we live.It would present a faster way to travel between countires. But for now, it is impossible (Source 1). The deepest hole ever dug is approximately 12km in length and would cost 1 billion dollars just to reach the mantle (Source 2)So odviously this sort of technology would require a lot of capital to get it going and also money to maintain it. The temperature at the core of the earth is estimated to be around 6000C which is as hot as the surface of the sun (impossible to survive) (Source 3).

Image 2

Formulas/constants needed:

F = ma

F = g×(M×m)/d^2

Volume of sphere = 4/3×?r^3 (r is the distance from the centre of the earth)

p=m/v (Density)

t=?(2?;m/k) (Time)

Earth’s Diameter = 12742km (Source 4)

Gravitational constant = 6.67×?10?^(-11) m^3 ?kg?^(-1) s^(-2) (Source 5)

Mass of Earth = 5.972×?10?^24 kg (Source 6)

Image 3

How it works:

Many factors must be assumed and/or ignored for this investigation.

Firstly, the hole must be connecting the North Pole and the South Pole exactly so that the Coriolis Effect or the rotating of the earth can not affect the experiment. Otherwise a straight fall would not be possible as the object/person would be hitting all sides of the Earth and would arrive damaged or hurt. (Source 7).

Air resistance and the temperature is also ignored. The temperature at the core of the Earth is approximately 6000C which is as hot as the sun (Source 3). The air resistance will cause friction and the object/person is most likely to burn

Lastly, assume that the Earth has the same density throughout. It is known that the density of the earth increases as it gets closer to the core which means that when you are falling most of the earths mass will be still below you and the force would not change much (Source 7). Therefore, density must be constant so that the decrease in the force of gravity decreases constantly.

Image 4

Method/Calculations:

The Earth is 12742km in diameter and its mass is

5.97×?10?^24Kg

4/3×?r^3, p=m/v (the volume and density formulas are manipulated to determine the equation for mass)

Mass is therefore 4/3×?r^3×p

The value for mass can now be used in the equation for the gravitational force (Source 7) which will result:

F = g×(4×?r^3×p×m)/(3d^2 )

Since r and d represents the equal value which is the distance of the object from the centre of the earth, it can be cancelled out.

The average weight of a human is 62kgs (Source 8), which will replace the “m” in the equation.

The final equation for the gravitational force is:

-g×(248 ×?r×p)/3

Density is mass divided by volume, therefore the equation for density is:

m/(4/3×?r^3 )

Density is now substituted into the equation.

-g×(248 ×?r)/3×M/(4/3×?r^3 ) = -9.556×?10?^(-5) downwards

Allow outward to be positive, therefore the force is negative as the earth pulls downwards.

As this falling and return motion repeats itself, the equation of a simple harmonic oscillator for time is used (Source 9).

t=?(2?&m/k)

Which in this case, the constant is the gravitational force. Therefore the radius is not included in the equation.

t=2??(62/(9.552×?10?^(-5) ))

= 5062.07s

Therefore, the fall and back are 82 minutes. The fall by itself before the person comes back up would be 42 minutes.

Conclusion:

It would take 82 minutes for a person to fall into a hole through the earth and come back. The person would not be stuck in the middle or fall one way, the motion would repeat itself back and forth if all the variables and conditions are met(as stated above). However, this conclusion is not very reliable as there are many variables (which were ignored in the beginning) that must be taken into account before calculation (E.g. The earth does not have the same density throughout.)

Sources/bibliography:

Source 1 –

Source 2 –

https://edition.cnn.com/2012/10/01/tech/mantle-earth-drill-mission/index.html

Source 3 – https://www.livescience.com/29054-earth-core-hotter.html

https://www.livescience.com/50312-how-long-to-fall-through-earth.html

Source 4 –

https://www.google.co.za/search?q=the+diameter+of+the+earth&ie=&oe

Source 5 –

https://www.google.co.za/search?ei=6ORcW4yHA8nMgAbl_KjoBQ&q=gravitational+constant&oq=gravitational+constant&gs_l=psy-ab.3..0i7i30k1l10.2556.3005.0.3106.2.2.0.0.0.0.311.311.3-1.1.0….0…1c.1.64.psy-ab..1.1.311….0.utjeBMtwKD4

Source 6 –

https://www.google.co.za/search?q=mass+of+earth&ie=&oe=

Source 7 –

Source 8 –

https://brightside.me/wonder-curiosities/how-much-do-people-weigh-in-different-countries-around-the-world-211605/

Source 9 –

https://study.com/academy/lesson/simple-harmonic-motion-shm-definition-formulas-examples.html

Image 1, 3 –

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/earthole.html

Image 2 –

https://www.livescience.com/29054-earth-core-hotter.html

image 4 –

https://upload.wikimedia.org/wikipedia/commons/8/89/RadialDensityPREM.jpg

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Research topic: What would happen if there was a hole through the Earth.

Formulas/values needed:

F = ma

F = g×(M×m)/d^2

Volume of sphere = 4/3×?r^3 (r is the distance from the centre of the earth)

p=m/v (density)

t=?(2?&m/k) (time)

Earth’s Diameter = 12742km (Source 4)

Gravitational constant = 6.67×?10?^(-11) m^3 ?kg?^(-1) s^(-2) (Source 5)

Mass of Earth = 5.972×?10?^24 kg (Source 6)

= 5062.07s

Therefore, the fall and back are 82 minutes. The fall by itself before the person comes back up would be 42 minutes.

Aim: To determine what would happen if someone or something was to fall through a hole in the Earth and how long it would take to reach the other side.

Hypothesis: The person/object would fall all the way to the opposite side of earth and would take 20 minutes to fall through.

Conclusion:

It would take 82 minutes for a person to fall into a hole through the earth and come back. The person would not be stuck in the middle or fall one way, the motion would repeat itself back and forth if all the variables and conditions are met. However, this conclusion is not very reliable as there are many variables (which were ignored in the beginning) that must be taken into account before calculation (E.g. The earth does not have the same density throughout.)