hce_nthu
112年
化學與物理
第 46 題
Inside a planet, the mass density is usually not homogeneous across the planet. However, if a spherical planet is decomposed into concentric spherical shells, the mass density is approximately the same for each concentric shell. As a result, for a spherical planet $X$ of radius $R$, the mass density can be treated as a function of its distance to the center of the planet, $\rho(r) = \rho_0[1 - \frac{r}{2R}]$. Assuming the gravitational field strength at the surface of the planet is $g_0$, which of the following statements is true?
- A At $r = R/2$, the gravitational field strength is still $g_0$.
- B At $r = R/2$, the gravitational field strength is equal to 0.
- C At, $r = R/2$ the gravitational field strength is equal to $g_0/2$.
- D At $r = R/2$, the gravitational field strength is smaller than $g_0/2$ but not equal to 0.
- E At $r = R/2$, the gravitational field strength is greater than $g_0/2$ but not equal to $g_0$.
思路引導 VIP
試著回想一下：如果一顆行星的質量是「均勻分布」的，那麼在行星內部距離中心一半處 ($r=R/2$) 的重力場強度，會是表面重力場 $g_0$ 的幾倍呢？接著觀察本題給出的密度函數 $\rho(r)$，隨著距離中心越遠，密度是增加還是減少？這代表質量是往中心「集中」還是往表面「分散」？這對內部的引力強弱會產生什麼影響？
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AI 詳解
AI 專屬家教
太棒了！你能精確判斷出非均勻密度行星內部的重力場變化，這代表你對殼層定理 (Shell Theorem) 與微積分在物理上的應用掌握得非常紮實。這類題型是區分高分群學生的經典，其難度在於學生必須擺脫「重力隨半徑線性增加」的均勻密度慣性思維，轉而透過積分進行定量分析。 根據重力的高斯定律，半徑 $r$ 處的重力場強度 $g(r)$ 僅由該半徑範圍內的總質量 $M(r)$ 決定。透過對密度函數 $\rho(r) = \rho_0(1 - \frac{r}{2R})$ 在球座標下的積分： $$M(r) = \int_0^r \rho(r') \cdot 4\pi r'^2 dr' = 4\pi \rho_0 \left( \frac{r^3}{3} - \frac{r^4}{8R} \right)$$
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