一个物体由半径为 $3r$ 的均匀实心半球与半径为 $r$、高为 $x$ 的均匀实心圆柱连接而成。半球和圆柱由同一种材料制成，且它们的对称轴重合（见图）。

(a) 求物体的质心到圆柱底面的距离，用 $r$ 和 $x$ 表示。

(b) 点 $A$ 位于半球底面的圆周上。当物体从点 $A$ 自由悬挂时，半球的底面与经过 $A$ 的竖直向下方向成 $\theta$ 角。

已知 $x=r$。

求 $\theta$ 的值。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

Where a numerical value for the acceleration due to gravity ($g$) is needed, use $10\,\mathrm{m\,s^{-2}}$.

需要使用重力加速度 $g$ 的数值时，取 $10\,\mathrm{m\,s^{-2}}$。

两部分由同种材料制成，可用体积代替质量。把圆柱底面作为统一基准，注意半球重心到该基准的距离还要加上圆柱高度。

The cylinder has volume $\pi r^2x$ and its centre of mass is at height $x/2$ above its base. The hemisphere has volume $\frac23\pi(3r)^3=18\pi r^3$. Its centre of mass is $3(3r)/8=9r/8$ above its flat face, so its height above the cylinder’s base is $x+9r/8$.

By symmetry the combined centre of mass is on the common axis. Its height is

自由悬挂时，重心在 $A$ 的竖直下方。用底面内从 $A$ 到轴线的距离 $3r$，以及重心离半球底面的距离，组成直角三角形。

For $x=r$, part (a) gives $\bar z=r(2+72+81)/(4\times19)=155r/76$. The centre of mass is therefore $\bar z-r=79r/76$ from the plane of the hemisphere’s flat face. When suspended, the line from $A$ to the centre of mass is vertical. Its projection onto the flat face has length $3r$, so

Hence $\boxed{\theta\approx19.1^\circ}$.