Can we go through the center of the Earth

[Tiome_nguyen]

Hello, i need help on this question, i have been thinking about it
assume there is a hole through the center of the Earth , and oxygen is exist so we can breath , if we drop the ball into the hold, is it possible for ball go through the center of the Earth to other side of the hole ?
so what i think is we can't go through the center to other side of the Earth, because gravitational force will pulling ball the center of the Earth so the ball will be stuck in the center of the Earth forever .

 

[mathman]

Ball will go through to the other side. Gravity at the center of the earth=0. F ~ m/r². However (assuming uniform density) m ~ r³ (volume within r), so that F ~ r. r is the distance of the ball from the center.

 

[Superstring]

I just posted this recently in another thread:

For a hole running through the diameter, using Gauss' Law I got:

$$g = -k^2r$$

Where:

$$k^2 = \frac{4}{3} \pi G \rho$$

With ρ being the average density of the earth.since g=a:

$$\ddot{r}+k^2r=0$$

Which is the equation of a simple harmonic oscillator.This means that:

$$r=r_0~cos(kt+\phi)$$Solving for the period:

$$T=\frac{2\pi}{k}=\sqrt{\frac{3\pi}{G\rho }}$$The time it takes to fall completely through the Earth is half the period, or:

$$t=\frac{\pi}{k}=\sqrt{\frac{3\pi}{4G\rho }}$$

So yes, you would be able to fall completely through a theoretical hole and arrive on the other side of the earth, assuming ideal conditions.

 

[HallsofIvy]

"oxygen is exist so we can breath". If there is air in the hole, there will be air resistance so the ball loses energy as it falls. It will not have enough energy to reach the other side of the Earth and will eventually stop at the center of the earth.

Now, if there is no air resistance (the ball doesn't need to breathe!), we have a "conservation of energy" situation. As the ball falls, it will lose potential energy and gain kinetic energy. As it passes the center of the Earth all of its potential energy will be converted to kinetic energy. As it goes up again, its kinetic energy will be convert back to potential energy. With no air resistance or other friction, the total of potential energy and kinetic energy is constant and, due to the symmetry, all kinetic energy will have converted back to potential energy, and the ball will stop, with it gets back to the same level at the other end of the hole. Theoretically, with no air resistance or friction, would continue back and forth indefinitely.

 

[Tiome_nguyen]

thank you so much for ur the explanations :)