[ 可直接使用高为 $h$、底面半径为 $r$ 的圆锥的体积为 $\dfrac13\pi r^2h$。 ]

均匀实心正圆锥 $C$ 的顶点为 $V$，底面半径为 $r$，高为 $h$。

(a) 证明 $C$ 的质心到 $V$ 的距离为 $\dfrac34h$。

从 $C$ 中挖去高为 $\dfrac13h$、顶点为 $V$ 的实心正圆锥 $C'$，形成下方图 4 所示的立体 $F$。

(b) 证明 $F$ 的质心到其较大平面端面的距离为 $\dfrac3{13}h$。

立体 $F$ 静止平衡，其曲面与水平面接触。

(c) 证明 $13r^2\leqslant17h^2$。

沿圆锥对称轴从顶点量 x，用相似三角形表示截面半径。圆盘体积矩除总体积，必须展示积分与代限。

Measure $x$ along the axis from the vertex $V$, with $0\leq x\leq h$. By similar triangles, a perpendicular circular section has radius

Its area is $\pi r^2x^2/h^2$. Since the density is uniform,

Evaluating both limits,

小圆锥与大圆锥相似，线性比例 1/3，所以质量比例为 1/27。把被挖去的小圆锥作负质量，并统一从大端面量质心距离。

The removed cone is similar to the original cone with linear scale factor $1/3$, so the mass ratio is

The original cone’s centre of mass is $h-3h/4=h/4$ from the larger plane face. The removed cone’s centre of mass is $3(h/3)/4=h/4$ from $V$. Its distance from the larger plane face is therefore $h-h/4=3h/4$. Let $\bar y$ be the distance of the centre of mass of $F$ from its larger plane face. Taking signed mass moments,

Thus

侧放时，重力的竖直作用线必须落在接触母线段之内，不能越过小端触地点。求该触地点上方竖直线与轴线的交点，再与质心轴向位置比较，保留正确的不等号方向。

Consider the axial section in the plane of contact. Let $A$ and $C$ be the centres of the smaller and larger plane faces, respectively, and extend the axis backwards to the original vertex $V$. Let $N$ be the contact point at the smaller rim, and let the vertical through $N$ meet the axis at $B$.

The smaller radius is $AN=r/3$, the removed cone’s height is $AV=h/3$, and $AC=2h/3$. The contact generator $VN$ is horizontal, and the radius $AN$ is perpendicular to the axis. The right triangles formed by the axis, radius and vertical give

Hence

For equilibrium, the vertical line through the centre of mass must lie within the contact segment. In particular, the centre of mass must be on the larger-end side of $B$; otherwise the solid tips about $N$. Therefore its distance from $C$ must satisfy

(The other end gives no extra restriction, since the centre of mass lies between the two face centres.) Using part (b),

Multiplying by $39h>0$,

Equality corresponds to limiting equilibrium with the vertical through the centre of mass passing through $N$.