Leaping into the Endless Pit

Scientifically speaking it would be impossible to dig a tunnel through to the other side of the world, but what if we could, in theory?



If you attempted to dig a hole to the other side of the Earth, you would be digging through:

• More than 12 000 kilometres of solid rock and molten magma
• Rock reaching temperatures up to 6000 ºC and
• Extreme pressures up to 300 million times greater than the pressures we experience on the surface of the Earth!

Also, the Earth is not a perfect sphere. It is slightly flattened at the poles, and bulges a little at the equator due to the Earth’s spin. So technically, if you dig a tunnel through to the other side of the globe from New York, you would not find yourself in China.

If you did somehow manage to dig a hole to the other side of the Earth, would you fall through?

Again, theoretically no! The Earth continues to spin as you fall, gravity changes as you fall to the Earth’s centre, and friction would slow you down.

If you ignored all of these factors, how long would it take to fall through the tunnel?



A tunnel along Earth's diameter through it's centre. When an object with mass is release from one end of the tunnel, it would oscillate from one end to another , just like a yo-yo.

Note: In this case, we use centripedal acceleration α, instead of linear acceleration a, because it bounces back and forth. Much like how a sine waveform could be seen as a circular motion.

Acceleration of the object mass:
Gravitation force, F = -(GM'm)/r²
Mass of Earth, M' = (4/3)(πr³ρ)

mα = F
mα = -(GM'm)/r²

α = -[GM']/r²
α = -[G(4/3)(πr³ρ)]/r²

α = -[(4πGρ)/3]r
α = -ω²r

Note: Similar to a swinging pendulum or a weight on a spring

The object would move in simple harmonic motion:
Gravitational acceleration, g = -α
Period for one cycle, T = (2π)/ω

g = ω²L,
ω = (g/L)½

T = 2π[L/g]½
T = 2π[3/(4πGρ)]½

T = 2π[1/4π½][3/(Gρ)]½
T = [3π/(Gρ)]½

Calculations:
Universal gravitational constant, G = 6.67x10­­­­̄¹¹ Nm²/kg²
Average density of Earth, ρ = 5.48x10³ kg/m³

T = 5080 seconds
T = 84.7 minutes

Hence, it will take about 42 minutes 21 seconds to fall through the tunnel. However, you'll never come out to the other end.

Due to the pull of gravity, you'll swing back and forth from one end to the other till it slows down at the middle. A theorectically endless pit.