114 學年度臺灣大學轉學生招生考試｜第 60 題
普通物理學(A)
- -3GM2/5R
- -GM2/5R
- -3GM2/4R
- -2GM2/3R
- -3GM2/3R
提示
This is the gravitational self-energy of a uniform solid sphere, found by integrating the work to assemble it shell by shell. The standard result carries a coefficient of 3/5 and is negative.
正確答案
正解：A
詳解
A. (A) -3GM²/5R — correct. Assembling a uniform sphere shell by shell and integrating gives the self-energy U = -(3/5)GM²/R. It is negative because gravity is attractive (bound system), and the 3/5 factor is the standard uniform-density result.
B. (B) -GM²/5R — incorrect. Missing the factor of 3 in the numerator from the integration.
C. (C) -3GM²/4R — incorrect. The denominator coefficient is wrong; the correct factor is 3/5, not 3/4.
D. (D) -2GM²/3R — incorrect. Neither the numerator nor denominator matches the uniform-sphere integral.
E. (E) -3GM²/3R — incorrect. This simplifies to -GM²/R, dropping the characteristic 3/5 coefficient.
