固定点 $A$、$B$ 位于光滑水平面上，$AB=10a$。

质点 $P$ 的质量为 $m$。

一根原长为 $5a$、弹性模量为 $\dfrac13mg$ 的弹性轻绳的一端连接于 $P$，另一端连接于 $A$。

另一根原长为 $3a$、弹性模量为 $\dfrac25mg$ 的弹性轻绳的一端连接于 $P$，另一端连接于 $B$。

如图 6 所示，质点 $P$ 静止平衡于该平面上的 $E$ 点。

(a) 证明 $EB=\dfrac{11a}3$。

现在将质点 $P$ 保持在平面上位于 $A$、$B$ 之间的 $C$ 点，其中 $CB=3a$，然后由静止释放。

(b) 证明 $P$ 绕 $E$ 做简谐运动。

(c) 证明该简谐运动的周期为 $2\pi\sqrt{\dfrac{5a}g}$。

任意一次完整简谐振动中，质点距 $E$ 小于 $\dfrac13a$ 的持续时间为 $S$。

(d) 用 $\pi$、$a$ 和 $g$ 表示 $S$。

两个张力相等，但因原长和弹性模量不同，伸长量不相等。设右绳伸长为 $e$，则左绳伸长为 $2a-e$，分别使用各自的胡克定律。

Let the extension of the string attached to $B$ be $e$. Then

so the extension of the string attached to $A$ is $2a-e$. In horizontal equilibrium the tensions are equal:

Multiplying by $15a/(mg)$ gives

Therefore

取从 $E$ 向 $B$ 的位移为正，分别写两根绳随位移变化的伸长，合力为右张力减左张力。还要检查释放后的振幅范围是否导致绳出现一段松弛运动。

Let $x$ be the displacement from $E$ towards $B$. The extensions of the left and right strings are respectively

Taking towards $B$ as positive, Newton’s second law gives

The constant terms cancel:

Hence

The acceleration is proportional to the displacement from $E$ and directed towards $E$, so the motion is SHM with $\omega^2=g/(5a)$.

The release point has $CB=3a$, so

Thus the amplitude is $2a/3$. Throughout $-2a/3\le x\le2a/3$, the left extension is at least $2a/3>0$, and the right extension is nonnegative. The right string is at natural length only at the release endpoint; the resultant force there is towards $E$, so it immediately stretches again. There is no interval of slack-string motion, and the SHM equation applies throughout the oscillation.

从恢复加速度的系数读取角频率，再用 $2\pi/\omega$ 求周期。

From (b),

Therefore the period is

半径为 $a/3$ 的中心区间，其边界恰好是半振幅。重设过中心的时刻为零，求中心到边界的时间；一周期累计四段，而不是只算首次到达。

The amplitude is $b=2a/3$. Reset time to $\tau=0$ when the particle passes through $E$ towards $B$. Then

The first arrival at $x=a/3=b/2$ satisfies

There are two passages through $|x|<a/3$ in a complete oscillation, each with equal times on either side of $E$. Thus the total duration is four times the time just found:

The isolated instants at the two boundary positions do not affect this duration.