质量分别为 $m$ 和 $km$ 的两个质点 $A$ 和 $B$ 由一根长为 $a$ 的不可伸长轻绳连接。将两个质点放在粗糙水平圆形转台上，绳保持绷紧并沿转台的一条半径放置。质点 $A$ 到转台中心的距离为 $a$，质点 $B$ 到转台中心的距离为 $2a$。每个质点与转台之间的摩擦系数均为 $\frac15$。

当转台以角速率 $\frac25\sqrt{\frac{g}{a}}$ 旋转时，系统处于极限平衡状态。

(a) 用 $m$ 和 $g$ 表示绳中的张力。

(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.

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 $10\,\mathrm{m\,s^{-2}}$。

沿径向向内列式。绳对内侧质点 $A$ 向外拉，所以摩擦力既要提供向心力，也要抵消这部分张力。

The angular speed satisfies $\omega^2=4g/(25a)$. Let $T$ be the tension. The inward friction required for $A$ is

Hence the maximum permitted tension is $\frac15mg-\frac4{25}mg=\frac1{25}mg$. For $B$, the inward force required is $\frac8{25}kmg$, and friction can supply at most $\frac15kmg$. The string must supply at least $\frac3{25}kmg$. Since the system is at the limit of equilibrium, the lower bound from $B$ coincides with the upper bound from $A$. Otherwise an intermediate tension would allow both friction forces to be below their limits. Thus $F_A=mg/5$, and the limiting radial equation for $A$ is

对外侧质点 $B$，张力和极限摩擦力都向内。代入上一问的张力，求质量比。

For $B$ at the limiting state,

Cancelling $mg/25$ gives $1+5k=8k$, so $k=1/3$.