一根原长为 $a$、模量为 $2mg$ 的轻弹性绳，一端连接于粗糙水平面上的固定点 $A$。细绳另一端连接质量为 $m$ 的质点 $P$。质点和细绳均静止在平面上。$P$ 与平面间的摩擦系数为 $\mu$。初始时，质点 $P$ 在距 $A$ 为 $\frac43a$ 处被保持平衡，随后由静止释放。

(a) 已知细绳始终不会变松，求 $\mu$ 的最小值。

现已知 $\mu=\frac12$。

(b) 求质点停止运动时细绳的伸长量。

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。

临界情形是粒子恰好在绳达到自然长度时停下。将初始弹性势能与这段路程上摩擦力做的功比较。

The initial extension is $a/3$, so the initial elastic energy is

If $P$ moves to the point of natural length, friction does work of magnitude $\mu mg(a/3)$. For the string never to become slack, this must be at least the initial elastic energy. Equality corresponds to reaching the natural length with zero speed. Thus

The minimum is $\boxed{1/3}$.

设停下时的伸长为 $x$，实际滑动距离就是 $a/3-x$。把弹性势能的减少量等于克服摩擦所做的功。

The initial tension is $2mg/3$, greater than the friction force $mg/2$, so the particle moves. If the extension at rest is $x$, then

Since the particle has moved, $x<a/3$. Factorising the difference of squares and dividing by $mg(a/3-x)>0$ gives