Gravity Force Inside a Spherical Shell

For application of the law of gravity inside a uniform spherical shell of mass M, a point is chosen on the axis of a circular strip of mass. The problem is envisioned as dividing an infinitesemally thin spherical shell of density σ per unit area into circular strips of infinitesemal width. The choice of such a point involves no loss of generality because for any point inside the shell, the mass elements could be chosen so that the point is on their symmetry axis.

在应用质量为M的均匀球壳内的万有引力定律时，选择轴线上的一点，该点位于一个质量为的圆环的轴线上。问题被设想为将一个无限薄的球壳，其单位面积的密度为σ，分割成宽度无限小的圆环。这样的点的选择不会造成任何损失，因为对于壳体内的任何一点，都可以选择质量元素，使得该点位于它们的对称轴上。

To set up the necessary integral, the triangle above is used to take advantage of the symmetry of the system. All components of the gravity force perpendicular to r will cancel by symmetry, and all components along r will sum. The differential element of force on mass m can be written

为设置必要的积分，利用上面的三角形来利用系统的对称性。所有与r垂直的重力力分量将因对称性而抵消，而沿r方向的分量将相加。质量m上的力微元可以表示为

其中质量的微元dM由

The force from the entire spherical shell can be expressed as an integral over the angle θ

整个球壳产生的力可以表示为对角度θ的积分

To evaluate the integral, the variables s and α must be expressed in terms of the angle θ . Using the Law of Cosines with the interior angle θ gives

为计算这个积分，变量s和α必须用角度θ来表示。利用包含角θ的余弦定理可得

并可以求导得到

现在使用余弦定理和外角 α：

With these relationships, we can now express the integral in terms of s instead of θ. Up to this point, the treatment is the same at that for a point outside the shell, but now the form of the limits is different. For θ = 0, s = R - r and s = r + R for θ = π.

利用这些关系，我们现在可以将积分表示为关于s而不是θ。到目前为止，处理方式与点电荷外侧的情况相同，但极限形式不同。当θ=0时，s=R - r；当θ=π时，s=r + R。

利用面积密度表达式 σ = M/4πR²，该积分可以写成

现在这些部分被评估为多项式积分并简化。

The net gravitational force on a point mass inside a spherical shell of mass is identically zero! Physically, this is a very important result because any spherically symmetric mass distribution outside the position of the test mass m can be build up as a series of such shells. This proves that the force from any spherically symmetric mass distribution on a mass inside its radius is zero. If a given mass m is inside a spherically symmetric distribution of mass, that part of the mass outside its radius does not contribute to the net force on it.

一个位于球壳内的质点所受的净引力是零！从物理上来说，这是非常重要的结果，因为任何球对称的质量分布，只要在测试质量m的位置之外，都可以看作是由许多这样的壳组成的。这证明了，对于位于其半径内的质量来说，任何球对称的质量分布对其产生的力都是零。如果一个给定的质量m位于一个球对称的质量分布内，那么位于其半径之外的部分不会对它产生净力。