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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Shell theorem ： ウィキペディア英語版 Shell theorem In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy. Isaac Newton proved the shell theorem and said that: # A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its centre. # If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell. A corollary is that inside a solid sphere of constant density the gravitational force varies linearly with distance from the centre, becoming zero by symmetry at the centre of mass. This is easy to see: take a point within such a sphere, at a distance $r$ from the centre of the sphere, then you can ignore all the shells of greater radius by the shell theorem. So, the remaining mass $m$ is proportional to $r^3$, and the gravitational force exerted on it is proportional to $m/r^2$, so to $r^3/r^2\; =r$, so is linear in $r$. These results were important to Newton's analysis of planetary motion; they are not immediately obvious, but they can be proven with calculus. (Alternatively, Gauss's law for gravity offers a much simpler way to prove the same results.) In addition to gravity, the shell theorem can also be used to describe the electric field generated by a static spherically symmetric charge density, or similarly for any other phenomenon that follows an inverse square law. The derivations below focus on gravity, but the results can easily be generalized to the electrostatic force. Moreover the results can be generalized to the case of general ellipsoidal bodies. == Outside a shell == A solid, spherically symmetric body can be modelled as an infinite number of concentric, infinitesimally thin spherical shells. If one of these shells can be treated as a point mass, then a system of shells (i.e. the sphere) can also be treated as a point mass. Consider one such shell: :''Note: dθ in the diagram refers to the small angle, not the arclength. The arclength is R dθ.'' Applying Newton's Universal Law of Gravitation, the sum of the forces due to mass elements in the shaded band is :$dF\; =\; \backslash frac.$ However, since there is partial cancellation due to the vector nature of the force, the leftover component (in the direction pointing toward ''m'') is given by :$dF\_r\; =\; \backslash frac\; \backslash cos\backslash phi.$ The total force on ''m'', then, is simply the sum of the force exerted by all the bands. By shrinking the width of each band, and increasing the number of bands, the sum becomes an integral expression: :$F\_r\; =\; \backslash int\; dF\_r$ Since ''G'' and ''m'' are constants, they may be taken out of the integral: :$F\_r\; =\; Gm\; \backslash int\; \backslash frac\; .$ To evaluate this integral one must first express ''dM'' as a function of ''dθ'' The total surface of a spherical shell is :$4\backslash pi\; R^2\; \backslash ,$ while the surface of the thin slice between ''θ'' and ''θ'' + ''dθ'' is :$2\backslash pi\; R^2\backslash sin\backslash theta\; \backslash ,d\backslash theta$ If the mass of the shell is ''M'' one therefore has that :$dM\; =\; \backslash frac\; M\backslash ,d\backslash theta\; =\; \backslash textstyle\backslash frac\; M\backslash sin\backslash theta\; \backslash ,d\backslash theta$ and :$F\_r\; =\; \backslash frac\; \backslash int\; \backslash frac\; \backslash ,d\backslash theta$ By the law of cosines, :$\backslash cos\backslash phi\; =\; \backslash frac$ :$\backslash cos\backslash theta\; =\; \backslash frac.$ These two relations link the 3 parameters ''θ'', ''s'' and ''φ'' that appear in the integral together. When ''θ'' increases from 0 to π radians ''φ'' varies from the initial value 0 to a maximal value to finally return to zero for ''θ'' = π. ''s'' on the other hand increases from the initial value ''r'' − ''R'' to the final value ''r'' + ''R'' when ''θ'' increases from 0 to π radians. This is illustrated in the following animation To find a primitive function to the integrand one has to make ''s'' the independent integration variable instead of ''θ'' Performing an implicit differentiation of the second of the "cosine law" expressions above yields :$\backslash sin\backslash theta\; \backslash ;d\backslash theta\; =\; \backslash frac\; ds.$ and one gets that :$F\_r\; =\; \backslash frac\; \backslash int\; \backslash frac\; \backslash ,ds$ where the new integration variable ''s'' increases from ''r'' − ''R'' to ''r'' + ''R''. Inserting the expression for cos(''φ'') using the first of the "cosine law" expressions above one finally gets that :$F\_r\; =\; \backslash frac\; \backslash int\; \backslash left(\; 1\; +\; \backslash frac\; \backslash right)\; \backslash ,\; ds.$ A primitive function to the integrand is :$s\; -\; \backslash frac$ and inserting the bounds ''r'' − ''R'', ''r'' + ''R'' for the integration variable ''s'' in this primitive function one gets that :$F\_r\; =\; \backslash frac,$ saying that the gravitational force is the same as that of a point mass in the centre of the shell with the same mass. Finally, integrate all infinitesimally thin spherical shell with mass of ''dM'', and we can obtain the total gravity contribution of a solid ball to the object outside the ball :$F\_\; =\; \backslash int\; dF\_r\; =\; \backslash frac\; \backslash int\; dM.$ Between the radius of ''x'' to ''x'' + ''dx'', ''dM'' can be expressed as a function of ''x'', i.e., :$dM\; =\; \backslash frac\; \backslash pi\; R^3\}\; M\; =\; \backslash frac$ Therefore, the total gravity is :$F\_\; =\; \backslash frac\; \backslash int\_^\; x^2\; dx\; =\; \backslash frac$ which suggests that the gravity of a solid spherical ball to outer object can be simplified as that of a point mass in the centre of the ball with the same mass. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Shell theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.