(a) 用代数积分证明：半径为 $a$ 的均匀实心半球的质心到其平面面的圆心的距离为 $\dfrac38a$。

[ 可直接使用半径为 $r$ 的球的体积为 $\dfrac43\pi r^3$。 ]

一个均匀实心半球的质量为 $m$，半径为 $a$。将一个质量为 $km$ 的质点连接于半球平面面的圆周上的 $A$ 点，形成加载物体 $S$。如图 4 所示，半球平面面的圆心为 $O$。

将加载物体 $S$ 放在水平面上，$S$ 的曲面与水平面接触。$S$ 处于平衡，$OA$ 与水平面成 $\alpha$ 角，其中 $\tan\alpha=\sqrt3$。

(b) 求 $k$ 的精确值。

将实心半球沿对称轴切成薄圆盘。用圆盘质量乘轴向位置作质量矩积分，再除以整个半球质量；这里不是半圆薄片的质心公式。

Take the origin at the centre of the plane face, and let $y$ measure distance into the hemisphere along its symmetry axis, with $0\leq y\leq a$. A thin disc at coordinate $y$ has radius satisfying

For constant density $\rho$, its mass is

The hemisphere has mass

By symmetry its centre of mass lies on the axis. Taking mass moments about the plane face,

Evaluating the integral,

Therefore

平面支持力沿接触半径通过 O，所以绕 O 取矩时不出现支持力。半球重力和附重的力臂分别由半球轴向质心距离及 OA 的水平投影得到。

Let $G$ be the centre of mass of the hemisphere. Then $OG=3a/8$, perpendicular to the plane face. The vertical normal reaction at the spherical contact point passes through $O$, so it has zero moment about $O$. Since $OA$ makes angle $\alpha$ with the horizontal, the horizontal moment arms of the particle and hemisphere weights about $O$ are respectively

These weights act on opposite sides of the vertical through $O$. Taking moments about $O$,

Thus

也可先求加载后整体质心相对于半球的两个分量，再要求整体质心位于 O 的正下方。

Use perpendicular axes fixed in the solid: $x$ along $OA$ towards $A$, and $y$ along the hemisphere’s axis into the solid. The hemisphere’s centre is at $(0,3a/8)$ and the particle is at $(a,0)$. For the combined centre of mass,

Therefore

The normal reaction passes through $O$, so equilibrium requires the combined centre of mass to be vertically below $O$. When $OA$ is inclined at $\alpha$ to the horizontal, cancellation of its horizontal component gives

Consequently