114 學年度臺灣大學轉學生招生考試｜第 69 題
普通物理學(A)
- h / 4
- h / 3
- 2h / 3
- h / 5
- 2h / 5
提示
Integrate using disks of radius that shrinks linearly from the base to the apex. The centroid of a solid cone lies one-quarter of the height up from the base.
正確答案
正解：A
詳解
A. (A) h/4 — correct. Slicing the cone into disks whose radius scales linearly with height and computing z_cm = ∫z dm / ∫dm gives the centroid at h/4 measured from the base (equivalently 3h/4 from the apex). The mass concentrates toward the wide base, pulling the center of mass low.
B. (B) h/3 — incorrect. h/3 is the centroid of a triangular lamina or the volume factor (⅓πr²h), not the solid cone's center of mass.
C. (C) 2h/3 — incorrect. This is the distance from the base to the apex centroid measured the wrong way, or a confusion with 3h/4 from the apex.
D. (D) h/5 — incorrect. Too low; the integration of linearly-shrinking disks yields 1/4, not 1/5, of the height.
E. (E) 2h/5 — incorrect. This is the result for a solid paraboloid-type profile, not a cone with linearly tapering radius.
