一个质量为 $m$ 的质点连接于一根长为 $8a$ 的不可伸长轻绳的一端，另一端固定于桌子光滑水平桌面上的 $O$ 点。$E$ 点位于桌边，$OE=5a$，且 $OE$ 垂直于桌边。将质点保持在 $O$ 正上方的 $A$ 点，绳绷紧。

如图 5 所示，将质点从 $A$ 沿 $OE$ 方向以速率 $\sqrt{8ag}$ 水平发射。

质点位于 $OE$ 所在水平面上方时，沿半径为 $8a$ 的竖直圆周运动。

已知绳与过 $O$ 的竖直向上方向成 $\theta$ 角时，绳中的张力为 $T$，

(a) 证明 $T=3mg(1-\cos\theta)$。

绳水平的瞬间，质点经过 $B$ 点。

(b) 求质点经过 $B$ 时绳中张力的瞬时变化。

质点撞向桌子的竖直侧面 $EF$ 后反弹。碰撞使质点损失碰撞前瞬间动能的三分之一。

在随后的运动中，绳与过 $O$ 的竖直向上方向成 $\alpha$ 角时开始松弛。

(c) 证明 $\cos\alpha=\dfrac7{12}$。

先从 A 到一般位置用能量求速率，再沿指向 O 的径向写牛顿第二定律。角度从竖直向上量起，因此重力指向圆心的分量为 mg cosθ。

Take the level of $O$ as the zero of gravitational potential energy. At $A$ the height is $8a$, and at angle $\theta$ it is $8a\cos\theta$. Conservation of mechanical energy gives

Hence

Resolving towards $O$ along the radius,

Substituting the energy result,

通过 B 时绳开始绕桌边 E 弯折，有效圆心改为 E，半径从 8a 变为 3a。瞬时速率连续，但所需向心力改变，所以张力跃增。

At $B$, $\theta=\pi/2$, so the speed immediately before passing $B$ satisfies

Immediately before $B$, the radius is $8a$. Since weight has no radial component there,

As the string bends round the edge at $E$, the free length becomes

The speed is continuous at $B$, but the radius immediately afterwards is $3a$. Therefore

Thus the instantaneous change is

先算绕 E 下落到竖直桌边时的速率，再将动能乘 2/3 得反弹后的速率。返回桌面上方后，圆心又是 O、半径为 8a；松绳时张力为零。

Below the desk level the particle moves on a circle of radius $3a$ about $E$. It strikes the wall vertically below $E$, at height $-3a$ relative to $O$. Let its speeds immediately before and after the impact be $v_1$ and $v_2$. The total fall from $A$ to the impact point is $11a$, so conservation of energy before impact gives

The collision removes one third of the kinetic energy, leaving two thirds. Hence

After rebounding the particle rises, and above the desk level the string again describes a circle of radius $8a$ about $O$. When the string makes angle $\alpha$ with the upward vertical, let the speed be $v$. Energy from immediately after impact gives

Therefore

At the instant the string becomes slack, $T=0$. The inward radial equation for the circle about $O$ is consequently

Substituting,