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为什么地球不是一个球?

2014 - 04 - 16 - t07:35:26z

我们都在学校学习,地球是一个球体。实际上,它更近一个稍扁球体——一个扁椭球面,也称为一个扁球。这是一个椭圆的短轴旋转。这一现象的物理原因是什么?

回答由Kenshin为什么地球不是一个球体吗?

2014 - 04 - 16 - t08:01:26z

通常没有旋转,重力自然租赁将地球在一个球体的形状。 然而事实上地球在赤道凸起,在赤道平面直径是42.72公里超过直径从南极到北极。 这是由于地球的旋转。 < img src = " https://i.stack.imgur.com/707k9.gif " alt = "在这里输入图像描述" > 在上面的图片中,我们可以看出旋转磁盘似乎凸出在磁盘上的点最远的旋转轴。 这是因为为了让磁盘的颗粒保持在轨道上,一定有一个内在的力量,被称为向心力,由: $ $ F = \压裂{mv ^ 2} {r}, $ $ F的力是美元,m美元的质量是旋转的身体,v是速度和r美元美元从转动轴粒子的半径。 如果磁盘旋转在给定的角速度,说\ω美元,然后切向速度v,美元是由v =ω\ r美元。因此, $ $ F = m \ω^ 2 r $ $ 因此粒子的半径越大,需要更多的力量来维护这样一个轨道。 因此粒子在地球赤道附近,这是最远的轴旋转,将buldge外,因为他们需要一个更大的内在力量来维持他们的轨道。 <人力资源> 更多细节更多数学文化现在启用mathjax: 合力在物体旋转的赤道半径r美元左右一颗行星的引力美元\压裂{Gm_1m_2} {r ^ 2} $的向心力, $ $ f{净}= \压裂{Gm_1m_2} {r ^ 2} - N = m \ω^ 2 r, $ $ $ N美元在哪里支持力。 重新排列上面的方程为: $ $ N = \压裂{Gm_1m_2} {r ^ 2} - m \ω^ 2 r $ $ 这里的法向力是向下的力,旋转的身体观察员。方程表明,感知到的向下的力降低是由于向心运动。典型的例子来说明这是有一个外观的0重力卫星绕着地球,因为在这种情况下,向心力是由重力平衡。然而,地球上的向心力远小于重力,所以我们把几乎整个mg美元的贡献。 现在我们将研究如何感知重力不同纬度的不同角度。 Let $\theta$ represent the angle of latitude. Let $F_G$ be the force of gravity.

In vector notation we will take the $j$-direction to be parallel with the axis of rotation and the $i$-direction to be perpendicular with the axis of rotation.

In the absence of the Earth's rotation,

$$F_G = N = (-\frac{Gm_1m_2}{r^2}\cos\theta)\tilde{i} + (-\frac{Gm_1m_2}{r^2}\sin\theta)\tilde{j}$$

It is easily seen that the above equation represents the perceived force of gravity in the absence of rotation. Now the centripetal force acts only in the i-direction, since it acts perpendicular to the axis of rotation.

If we let $R_{rot}$ be the radius of rotation, then the centripetal force is $m_1\omega^2R_{rot}$, which for an angle of latitude of $\theta$ corresponds to $m_1\omega^2r\cos{\theta}$

$$N = (-\frac{Gm_1m_2}{r^2} + m_1\omega^2r)\cos{\theta}\tilde{i} + (-\frac{Gm_1m_2}{r^2})\sin{\theta}\tilde{j}$$

By comparing this equation to the case shown earlier in the absence of rotation, it is apparent that as $\theta$ is increased (angle of latitude), the effect of rotation on perceived gravity becomes negligible, since the only difference lies in the $x$-component and $\cos\theta$ approaches 0 as $\theta$ approaches 90 degrees latitude. However it can also be seen that as theta approaches 0, near the equator, the $x$-component of gravity is reduced as a result of the Earth's rotation. Therefore, we can see that the magnitude of $N$ is slightly less at the equator than at the poles. The reduced apparent gravitational pull here is what gives rise to the slight bulging of the Earth at the equator, given that the Earth was not originally as rigid as it is today (see other answer).

回答由Gaialogist为什么地球不是一个球体吗?

2014 - 04 - 23 - t09:33:47z

其实,地球不是一个球体的原因是双重的: < ol > <李>地球是旋转,旋转了很长一段时间< /李> <李>地球不是完全刚性,它甚至可以被视为一种粘性流体在长时间尺度李< / > < / ol > 如果地球没有旋转,这将是一个球体。如果地球已经开始旋转最近,它不会平衡,因此可能不是我们熟悉的椭球面。最后但并非最不重要,如果地球是完全刚性的,它不会被任何变形过程,包括旋转,因此仍然有它的初始形状。 我们可以认为地球是一个流体在流体静力学平衡(即静止流体)在每一个点,考虑重力的影响和离心(伪)由于旋转力。然后,如果我们寻找地球表面的形状,在这种情况下,解决方案是一个椭球面。非常接近实际的地球表面,是一个很好的证明了我们最初的假设——旋转流体在流体静力学平衡——长时间是合理的时间表。 这个问题的研究与著名的克莱劳方程从法国著名科学家的名字出版了专著理论de la图de la特的最后18世纪。 注:如果我们只是解释在赤道隆起指伪离心力的作用而忽略了流体静力学平衡的问题,我们应该得出这样的结论:极地半径是相同的或没有旋转。不过,这是小:约6357公里和6371公里的球形地球体积相等。

答案由大卫Hammen为什么地球不是一个球体吗?

2014 - 04 - 28 - t13:07:33z

地球大约是一个扁球是最好的解释为能量。 大理石在碗里。不管你在哪里的地方,它最终会在碗的底部。这个位置的总能量最小化大理石在碗中受到的约束。暂停两个职位之间的链。链来的时候需要在一个著名的形状,悬链线的曲线。这是最小化的能量链的形状,受约束的两篇文章之间的暂停。 如果你把大理石远离底部将前一段时间来休息。如果你拉链式远离悬链线形状会来回摆动一段时间来休息之前的稳定形状。不平衡的大理石和平面外链比他们做的更大的潜在能量,稳定的配置。如果可能的话,自然将试图最小化总势能。这是热力学第二定律的结果。

In the case of the Earth, that minimum energy configuration is a surface over which the sum of the gravitational and centrifugal potential energies are constant. Something that makes the Earth deviate from this equipotential surface will result in an increase in this potential energy. The Earth will eventually adjust itself back into that minimum energy configuration. This equipotential surface would be an oblate spheroid were it not for density variations such as thick and light continental crust in one place, thin and dense oceanic crust in another.

In terms of force, the quantity we call g is the gradient of the gravitational and centrifugal potential energies (specifically, $\vec g = -\nabla \Phi$). Since the Earth's surface is very close to being an equipotential surface and since that surface in turn is very close to being an oblate spheroid, gravitation at the poles is necessarily slightly more than it is at the equator.

This gravitational force will not be normal to the surface at places where the surface deviates from the equipotential surface. The tangential component of the gravitational force results in places where water flows downhill and in stresses and strains in the Earth's surface. The eventual responses to these tangential forces are erosion, floods, and sometimes even earthquakes that eventually bring the Earth back to its equilibrium shape.

Update: Why is this the right picture?

Based on comments elsewhere, a number of people don't understand why energy rather than force is the right way to look at this problem, or how the second law of thermodynamics comes into play.

There are a number of different ways to state the second law of thermodynamics. One is that a system tends to a state that maximizes its entropy. For example, put two blocks at two different temperatures in contact with one another. The cooler block will get warmer and the warmer block will get cooler until both blocks are at the same temperature, thanks to the second law of thermodynamics. That uniform temperature is the state that maximizes the entropy of this two block system.

Those two blocks only have thermal energy. What about a system with non-zero mechanical energy? Friction is almost inevitably going to sap kinetic energy from the system. That friction means the system's mechanical energy will decrease until it reaches a global minimum, if any. For a rotating, dissipative, self-gravitating body, that global minimum does exist and it is a (more or less) oblate spheroid shape.