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从地心穿越地球有多长距离

How Long To Fall Through The Earth?

我最近在email收到这样一个问题
I recently received this question via email:
嗨 Henry
Hey Henry,
不久前你做了一个视频 讲述的“如果地球是空的会发生什么?”
A while ago you did a video… called “What if the Earth were Hollow?” where you showed
视频中你展示了穿过地球需要多长时间 我只是很想知道
how long it would take to fall through the earth. I was simply wondering how that was
你是如何计算由于你头上的质量不断增加而改变的重力的呢
even calculated since the force of gravity would constantly be changing due to the growing
————
amount of mass above you.
来自彼得
Peter
现在我们将在地球上打穿一个洞 从北极到南极
Ok, so we’ve got a hole through the earth, and it goes from the north pole to the south
这一步我们不考虑地球转动产生的科里奥利效应
pole – that way we don’t have to worry about the Coriolis effect from the earth’s spinning.
在洞里没有空气 不然你会快速到达另一端
There’s also no air in the hole, otherwise you’d reach terminal velocity pretty quickly
接着你的旅程变得无聊又冗长 有一些不同的方法来计算出
and your trip would be slow and boring. There are a few different ways to figure out how
你到达另一端的时间
long it’ll take for you to reach the other side.
一旦跳入 速度会越来越快 并且用数学物理能计算出
One is to jump in, but it’ll be faster – and probably a greater chance of survival – if
有很大的存活概率
we calculate using math & physics.
首先 再简化一点:假设地球是个完美球体 密度处处相同
First, more simplifications: assume the earth is perfectly spherical and has the same density
事实证明 如果球体的质量都集中在中心
everywhere. It turns out that the gravitational attraction from any spherically symmetric
那么来自对立面的球体引力都将相同
object is the same as if all its mass were concentrated at the center of that object
更紧密部分的引力比平均大 远的部分的引力比较小但依旧超过平均
– the closer parts attract more than average, the far away parts attract less, but over
同样 如果你在一个球壳里
the whole sphere it averages out. In a similar vein, if you’re _inside_ a spherical shell,
然后来自不同方向的重力相抵消
then the gravitational pulls from all the different parts cancel out and you experience
你的体验就是在球壳中的零效应
zero effect from the shell.
这意味着 地球内部 远离中心的任何部分
This means that, inside the earth, any parts that are farther away from the center than
力都被抵消 你没有任何感觉 因为这些力处于平衡
you are cancel out and have no effect – almost like they’ve been trimmed off and you’re temporarily
你暂时在一个更小的被削去一部分的地球表面上 因为我们假设密度处处相同
on the surface of a smaller, shaved earth. Since we assumed the same density everywhere,
被削去部分的质量和体积成正比
the shaved-earth’s mass is simply proportional to its volume, which is proportional to its
体积又与半径的立方成正比 因为是球体
radius cubed. And because it’s a sphere we get to pretend all that mass is actually concentrated
我们假设质量集中在中心点
at a single point in the middle.
那这个小点对你的拉力是多少?
So how much does the shaved-earth-point pull on you? Well, the gravitational attraction
两个物体之间的万有引力与质量成正比 与距离的平方成反比
between two objects is proportional to their masses but inversely proportional to the distance
所以 我们必须用被削去的地球质量除以
between them, squared, so we have to divide the mass of the shaved-earth by the square
你与中心距离的平方-就是被削地球的半径
of the distance you are from the center – which is just the radius of the shaved earth. R
原半径R立方除以削去的半径r立方 那么很容易求出拉力F
cubed divided by r squared is r, so the force on you is simply F equals some constant stuff
一个常熟乘以 你与中心距离r
times r, your distance from the center.
从本质上说 随着你下落质量会减小
Essentially, as you fall the mass beneath you decreases, while the **average gravitational
重力对你的平均拉力会增加
pull on you** from any bit of that mass increases, but the mass decreases **more than the average
但质量下降大于拉力增加
pull** of gravity increases.
所以当你接近地球中心 你会越来越快
So as you approach the earth’s center, you go faster and faster but the force pulling
把你拉向中心的力越来越小在中间 你就会体验
you towards the middle gets smaller and smaller. Exactly in the middle you experience zero
零合力 因为地球对你的拉力来自四面八方
net force because the earth is pulling you equally in all directions, though since you’re
不过这时候你速度很快 当你到了另一半的时候速度会减慢
going so fast you’ll continue to speed towards the other side, gradually slowed by the now
拉力会把你拉向中间 F等于
increasing force pulling you back towards the middle. F equals some constant stuff times
常熟乘以距离
distance.
相同的方程-常熟乘以距离-
The exact same equation – some constant stuff times a distance – also describes
也描述了重力的波动 像单摆或抛物线中的猫
a mass on a spring, or simple pendulum, or a cat in a parabola. And from studying _those_
从研究方程中 我们发现一个运动物体
equations we know that **the time taken by the moving object – whatever it is –
从一边到达另一边所花费的时间有一个简单的公式:π乘以 开根号下
to go from** one side to the other has a simple formula: pi times the square root of the mass
质量除以常熟 因为是从地球中间下落
divided by the “constant stuff”. In the case of falling through the earth, your mass cancels
质量抵消 所以方程中只要地球的密度
out of the equation so we just need to put in numbers for the density of the earth and
和引力常熟 最后得到答案 在地球中间下落花费42分钟
the gravitational constant to get the answer – 42 minutes to fall through the earth.
这个时间和绕地球半圈的时间一样
This turns out to be exactly the same as the time it takes to fall _around_ the earth to
你会发现这个数字经常在网上被提到
the other side, and it’s the number you’ll find commonly mentioned on the internet. Even
更令人吃惊的是 时间计算中 地球的半径没有被带入过
more surprising, the radius of the earth didn’t factor into the time calculation – it predicts
所以预计42分钟就可以通过或者绕半圈任何球体
you’ll take 42 minutes to fall through or orbit around to the other side of ANY sphere
只要这个球体密度和地球一样
with the same density as the earth.
但地球密度不是处处相等
But the earth isn’t exactly the same density throughout – we know from seismology that
从地震学中可知 地核密度比地壳和地幔都大
the earth’s core is much denser than its mantle and crust. So as you begin to fall, most of
所以你下落时 大部分质量还是往下拉你 重力不会减少
the mass is still below you, pulling, so the pull of gravity doesn’t decrease as much as
和我们简单的模型预测是一样的 事实上 力是相对恒定的
our simple model predicted. In fact, the force is actually pretty constant until about halfway
直到到达中心 力会快速降低
to the center, at which point it starts quickly decreasing as more and more of the earth is
随着你掉的越来越深
“above” you.
这里的计算更烦人 因为我们必须拼凑出两种不同的部分
The calculations here are a bit more annoying because we have to piece together two different
在下降加速度恒定的情况下 这是很简单的
parts – the falling with constant acceleration part, which is easy, and the falling with
随着下降减少的重力与半径成正比 这和之前的一样
decreasing gravity proportional to your radius part, which is the same thing we did before,
你到了地球中间的一半位置 速度是17000米/时
except now you’re starting out halfway to the middle of the earth with a speed of 17
和地表的零速度不一样
thousand miles per hour, instead of on the surface with no speed. Once our mathemagical
一旦计算尘埃落定 我们就把两部分相加 乘以2
dust settles, we combine the two parts and multiply by two to get the total time back
得到了从表面到另一边的总时间:37分钟
to the surface on the other side: 37 minutes.
当然 这只是一个比较相近的数据-稍微比之前的真实一些
Of course, this is still just an approximation – slightly more realistic than before, but
但离完美还很远 当你参考更详细的地球密度比如《地球模型初步参考》
far from perfect. If you carefully piece together the time for a falling-through-the-earth trip
来仔细拼凑掉入地球旅行的时间
based on a more detailed density profile of the earth, like maybe the Preliminary Reference
你可以得到更精准的时间-38分6秒
Earth Model, you can can be even more precise – 38 minutes and 6 seconds from pole to
从地球一极到另一极
pole.
但无论如何 假如你从视频的开始都没有计算跳入洞的时间
But either way, if instead of calculating you jumped into the hole at the start of this
那么在你到达地球另一端之前 你还有很长的路要走 旅行安全!
video, you still have a long ways to go before reaching the other side
of
the earth. Safe travels!

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https://www.youtube.com/watch?v=urQCmMiHKQk

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