三个大小分别为 $40\,\mathrm{N}$、$30\,\mathrm{N}$ 和 $X\,\mathrm{N}$ 的共面力，沿图示方向作用于同一点。

已知这些力处于平衡状态，求 $\theta$ 和 $X$ 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 $10\,\mathrm{m\,s^{-2}}$。

把未知力沿水平、竖直方向分解，分别平衡另外两个力。由图可知角为锐角。

Resolving horizontally and vertically gives

Squaring and adding,

Since $0<\theta<90^\circ$,

平衡的三个力首尾相接形成闭合直角三角形，直接用勾股定理及三角比。

The three forces form a closed triangle. The $40\,\mathrm{N}$ and $30\,\mathrm{N}$ sides are perpendicular, so

The acute angle corresponding to the diagram satisfies

用 Lami 定理，各力除以其余两力夹角的正弦相等。先求角，再求力。

The angles between the other two forces are $90^\circ$, $180^\circ-\theta$ and $90^\circ+\theta$, respectively. By Lami’s theorem,

Thus

Therefore

Since $\theta$ is acute and $\tan\theta=3/4$, $\sin\theta=3/5$. Thus