If you drilled a hole pole to pole and you descended through it with no brakes and hypothetically you don't die from the inhospitable conditions, would you stop accelerating once you reach the core of the Earth? Or would you be able to pass through the core and reach the other side?
Gravitational acceleration is higher as you get further down, but without brakes you would have enough acceleration that as you passed the centre, you would start shooting away toward the other pole albeit with less energy
Your acceleration would decrease as you approached the center, you'd pass right through the other side, and you'd keep slowing down as you rose until you emerged exactly on the other side of the Earth about 42.2 minutes later. Then you'd fall back down again, and neglecting air resistance, keep oscillating forever.
Fun fact: If you ignore friction and the Earth's rotation, then a gravity-powered ride (say a rolling ball or something) down *any* straight hole drilled between two points on Earth's surface will reach the other end in about 42.2 minutes. The reduced rate of acceleration is exactly canceled out by the reduced distance.
At the exact center of the Earth, there is zero gravity, because the pull of everything above you on one side is exactly canceled out by the pull of everything on the other side.
If you stood there, with your center of mass at the exact center, you would feel a very slight pull outwards on the entire rest of your body. However, it would be *very* slight, so you probably wouldn't actually feel it.
Also you would die instantly because of the heat, pressure, etc.
Because there's less mass underneath you to pull you down, and the deeper you go the more mass there is above you to pull you upwards. The result is a smaller net downwards force, until at the exact center there's zero acceleration at all.
(For more information, look up "Shell Theorem" on Wikipedia)
No mysterious anti-gravity force; just the fact that gravity doesn't actually emanate from the *center* of a planet, but from its entire material bulk. It just all approximately cancels out to acting as a single point mass under normal conditions.
You would probably already be traveling at such a speed that you'd blast right through the core. Whether or not you'd have the velocity to actually escape on the other side, I imagine would depend on the location and elevation.
I don't know if the Earth's rotation would factor in either.
You would stop accerating before you reached the center. You would end up about 100ft lower than you started. This would be your fate until your hot skeleton found equillibrium in the center.
Logic tells me that you'd fly toward the opposite surface because your acceleration exceeds gravitational pull, then fall back towards the center. This would continue repeatedly, with your distance traveled lessening each time as you lost momentum, like a playground swing that someone has stopped pushing. After some number of repetitions you'd settle at the center of the earth, suspended in the hypothetical empty space there due to relatively equal gravitational influences on your body.
But I don't know.
On second thought, I don't think you would settle to floating, as in zero G, due to asymmetric distribution of mass in the human body. I do think it would be similar to an extremely low gravity environment, though. You'd sort of drift toward whichever region most of your body mass was located in.
>If you stood there, with your center of mass at the exact center, you would feel a very slight pull outwards on the entire rest of your body.
Nope. It's not like half the Earth is pulling exclusively on one half of your body while the other half tugs on the other. If we persist in analyzing it as such, each half pulls on your ENTIRE BODY at once.
Is this a joke? Right in the fucking wikipedia article you were talking about it says there in no net gravitational force inside a hollow sphere.
If you jumped through the direct center of the hole you would end up at the other side about 42 minutes later. If you take into account air resistance you won't quite make it, but you'd be close.
This graph is the same thing my physics professor showed us in high school.
Also - en.wikipedia.org/wiki/Gravity_of_Earth#Depth
There's no joke; you've just misunderstood.
When you're inside the Earth at radius r, all the matter at radius>r forms a, hollow spherical shell and thus its gravity cancels. So you're effectively standing at the surface of a smaller Earth of radius r, consisting of all the matter between radius 0 and r, causing gravity to be correspondingly lower.
As a result, gravity decreases as you descend, and at the exact center all the matter is in the spherical shell above you and so g = 0.
In differential equations you also would treat it as a mass-spring system producing a second-order linear whose damping is a function of wind resistance.
I bet that fucker would be critically damped
Well, it's a pain, but it looks like this.
I'm not solving this
here's the deal: simplified high school version, mass-spring model, harmonic oscillation, hooke's law, no air. (w means omega)
w=sqrt(k/m), k=F/R=m*g/R -> w=sqrt(g/R) -> t=2π*sqrt(R/g)
R=6.4e6m, g = 10m/s^2 -> t = 2π*sqrt(6.4e6/10) = 2π*sqrt(64e4) = 2π*800 ≈ 5000s (that's about 84 minutes, full cycle))
and from y = R*cos(wt) -> ydot = -R*w*sin(wt) you get v_max = sqrt(g*R) ≈ 8000m/s
doesn't that look a lot like 7.9 km/s? let's see:
w^2*R = G*M/R^2 = g -> w = sqrt(g/R) -> t=2π*sqrt(R/g) -> bingo
so kinetic potential of a satellite orbiting close to the surface = gravitational potential of an object at the center of the earth
now wild conjecture:
a clock at the center of the earth and the on-board clock of said satellite will stay in syc
With air resistance I would expect the thing to quickly reach its terminal velocity and fall slower and slower because of the decreasing acceleration. Would that still qualify as a harmonic oscillation? How far would it overshoot the center, if at all?
Why 42 minutes? Once you are 24 minutes in, you are halfway through. You would have terminal velocity, but as you went on to escape the core of the earth, you would decelerate at -9.8ms for 24 extra minutes. I'm pretty sure -9.8m/s^2 of deceleration for 1440 minutes would be more than enough to stop terminal velocity, and you would fall back to the core before you ever reached the surface at all.