均匀薄片 $OABCD$ 由矩形 $OBCD$ 与四分之一圆 $OAB$ 沿边 $OB$ 拼接而成。$DO$ 的长度为 $ka$，$OB$ 的长度为 $a$。薄片位于竖直平面内，其边 $CB$ 放在水平面上（见图）。

(a) 用 $k$、$a$ 和 $\pi$ 表示质心高于水平面的距离。

[可不加证明地使用公式表（MF19）中给出的圆扇形质心结果。]

薄片正要绕 $B$ 点倾倒。

(b) 求 $k$ 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.

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

需要重力加速度的数值时，取 $g=10\,\mathrm{m\,s^{-2}}$。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。

把薄片分成矩形和四分之一圆。扇形重心在角平分线上，先求其相对 $O$ 的竖直分量，再换成离地高度。最后用面积加权。

The rectangle has area $ka^2$ and its centre of mass is at height $a/2$. The quarter circle has area $\pi a^2/4$. For a sector of half-angle $\alpha=\pi/4$, its centre of mass is at distance $2a\sin\alpha/(3\alpha)$ along the bisector from $O$. The vertical component downward from $O$ is

Since $O$ is at height $a$, the quarter circle’s centre of mass is at height $a-4a/(3\pi)$. Therefore the combined height is

将要绕 $B$ 倾覆时，整体重心在 $B$ 的正上方。比较 $OB$ 两侧的面积矩，扇形重心到 $OB$ 的水平距离与它到 $OA$ 的竖直距离相同。

At the point of toppling, the combined centre of mass lies vertically above $B$. The rectangle’s centre of mass is $ka/2$ to the left of $OB$, while the sector’s is $4a/(3\pi)$ to its right. Balancing area moments gives

Thus $k^2=2/3$, and since $k>0$ we obtain $\boxed{k=\sqrt{2/3}}$.

也可从 $D$ 向右量整体质心的水平坐标 $X$。矩形与扇形质心的坐标分别为 $ka/2$ 和 $ka+4a/(3\pi)$。临界倾覆时，$X$ 等于 $B$ 的水平坐标 $ka$。

Let $X$ be the horizontal distance of the combined centre of mass from $D$, measured to the right. Area-weighting gives

At the point of toppling about $B$, $X=ka$. Thus

Since $k>0$, $\boxed{k=\sqrt{2/3}}$.