📚 Combining Forces: Resultant, Equilibrium and Resolution | 力的合成：合力、平衡与分解
Combining forces is one of the most fundamental skills in CIE A-Level Physics. Whether you analyse a suspended mass, an object on a slope, or a bridge in equilibrium, you must be able to replace several forces with a single resultant force, break forces into perpendicular components, and apply the condition that the vector sum of forces is zero when acceleration is zero. This article develops those ideas from basic vector language to worked numerical examples.
力的合成是 CIE A-Level 物理中最基本的技能之一。无论是分析悬挂的物体、斜面上的物体，还是处于平衡状态的桥梁，你都必须能够用一个合力替代多个力，将力分解为垂直分量，并在加速度为零时应用合力矢量和为零的条件。本文将从基本矢量语言出发，逐步发展到数值例题。
1. Scalars, Vectors and Force as a Vector | 标量、矢量与力作为矢量
A scalar quantity has magnitude only: examples include mass, time, temperature and energy. A vector quantity has both magnitude and direction: examples include displacement, velocity, acceleration, momentum and force. Force is a vector because its effect depends on direction as well as size.
标量只有大小，例如质量、时间、温度和能量。矢量既有大小又有方向，例如位移、速度、加速度、动量和力。力是矢量，因为它的作用效果既取决于大小，也取决于方向。
In SI units the unit of force is the newton (N). One newton is the force required to give a mass of 1 kg an acceleration of 1 m s⁻², so the unit relationship is exact.
在国际单位制中，力的单位是牛顿（N）。一牛顿是使 1 kg 质量产生 1 m s⁻² 加速度所需的力，因此该单位关系是精确的。
1 N = 1 kg m s⁻²
When several forces act on a body, their directions must be included before any addition is carried out. The definition of the resultant follows from this requirement.
当多个力作用在物体上时，在进行任何加法之前都必须考虑它们的方向。合力的定义正是由此而来。
| Quantity | Type | SI unit |
| Mass, time, energy, temperature, density | Scalar | kg, s, J, K, kg m⁻³ |
| Displacement, velocity, acceleration, momentum, force | Vector | m, m s⁻¹, m s⁻², kg m s⁻¹, N |
2. Representing Forces and Free-Body Diagrams | 力的表示与受力分析图
A force is drawn as an arrow. The length of the arrow represents the magnitude to scale, the arrowhead shows the direction, and the starting point shows the point of application. Clear diagrams reduce mistakes in multi-force problems.
力用箭头表示。箭头的长度按比例表示大小，箭头指向表示方向，起点表示作用点。清晰的示意图可以减少多力问题中的错误。
A free-body diagram isolates a single object and shows every external force acting on it. Common forces include the weight acting vertically downwards, the normal contact force perpendicular to a surface, tension along a string or cable, friction opposing sliding, and applied forces such as pushes or pulls.
受力分析图将单个物体隔离出来，并画出作用在它上面的所有外力。常见的力包括：竖直向下的重力 W = mg、垂直于表面的法向接触力 N、沿绳或缆绳方向的张力 T、阻碍滑动的摩擦力 f，以及推力或拉力等外力。
Weight: W = m g, always vertically downwards, acting at the centre of gravity.
重力：W = m g，始终竖直向下，作用在重心处。
Normal contact force N: perpendicular to the surface, never automatically equal to mg.
法向接触力 N：垂直于表面，绝不自动等于 mg。
Tension T: along the string, pulling away from the object at the point of attachment.
张力 T：沿绳方向，在连接点处背离物体向外拉。
Friction f: along the surface, opposing relative sliding motion.
摩擦力 f：沿表面方向，阻碍相对滑动。
Always check that no force is counted twice, especially when a force has already been resolved into components for a calculation.
务必检查没有力被重复计算，尤其是在计算中已经把某个力分解为分量之后。
3. Adding Forces: Triangle and Parallelogram Rules | 力的加法：三角形法则与平行四边形法则
Two forces acting at a point can be added geometrically. In the triangle rule, the first force is drawn as an arrow, and the second force is drawn starting from the head of the first. The resultant is the arrow drawn from the tail of the first force to the head of the second.
作用在同一点的两个力可以用作图法相加。在三角形法则中，先画出第一个力的箭头，再从第一个力的箭头末端画出第二个力。合力就是从第一个力的起点指向第二个力的终点的箭头。
In the parallelogram rule, the two forces are drawn from a common point and the diagram is completed into a parallelogram. The resultant is the diagonal through the common point. The magnitude and direction of the resultant are found by scale drawing or by calculation.
在平行四边形法则中，两个力从同一点画出，并补成平行四边形。合力就是过公共点的那条对角线。合力的大小和方向可以通过按比例作图或计算得到。
Choose a clear scale, for example 1 cm represents 2 N, and state it on the diagram.
选择清晰的标度，例如 1 cm 代表 2 N，并在图上标明。
Draw the arrows accurately with a sharp pencil and a ruler, and measure the resultant with the same scale.
用削尖的铅笔和直尺准确画出箭头，并用同一标度测量合力。
State the direction as an angle measured from a clearly identified reference line.
方向要用相对于明确标注的参考线的角度来表示。
4. Resultant of Two Forces: Pythagoras, Cosine Rule and Direction | 两个力的合力：勾股定理、余弦定理与方向
If two forces are perpendicular, the resultant is found with Pythagoras' theorem, because the two forces and the resultant form a right-angled triangle.
如果两个力互相垂直，合力可用勾股定理求出，因为这两个力与合力构成一个直角三角形。
R = √A² + B²
tan θ = B / A
Here A and B are the magnitudes of the perpendicular forces and θ is the angle between the resultant and the force A. For example, 3 N and 4 N at right angles give a resultant of 5 N at 53.1° to the 3 N force.
其中 A 与 B 是相互垂直的两个力的大小，θ 是合力与力 A 之间的夹角。例如 3 N 与 4 N 互相垂直时，合力为 5 N，与 3 N 的力成 53.1°。
For two forces at any angle θ measured between them, the cosine rule gives the resultant directly.
对于夹角为 θ 的任意两个力，可以直接用余弦定理求出合力。
R² = A² + B² + 2 A B cos θ
sin α / B = sin θ / R
reduces the cosine rule to Pythagoras, while θ = 0° gives R = A + B and θ = 180° gives R = |A − B|.
在正弦定理形式中，α 是合力与力 A 之间的夹角。一个有用的检验是：θ = 90° 时余弦定理化为勾股定理，θ = 0° 时 R = A + B，θ = 180° 时 R = |A − B|。
5. Resolving a Force into Perpendicular Components | 把力分解为垂直分量
Resolution is the reverse of combining. A single force F at an angle θ to the horizontal is replaced by a horizontal component and a vertical component acting at the same point.
分解是合成的逆过程。与水平方向成 θ 角的单个力 F，可以用作用在同一点的水平分量和竖直分量来替代。
Fₓ = F cos θ
Fy = F sin θ
The component adjacent to the angle always uses cos θ, and the component opposite the angle always uses sin θ. Write the two components as positive magnitudes first, then assign + or − signs according to the chosen x and y directions.
与角度相邻的分量总是用 cos θ，与角度相对的分量总是用 sin θ。先写出两个分量的正值大小，再根据选定的 x、y 正方向赋予 + 或 − 号。
zontal has components 50 cos 30° = 43.3 N horizontally and 50 sin 30° = 25.0 N vertically.
例如，一个 50 N 的力与水平方向成 30° 向上，其分量为水平方向 50 cos 30° = 43.3 N，竖直方向 50 sin 30° = 25.0 N。
Resolution is only valid for a chosen pair of perpendicular axes; state the axes before resolving.
分解只对选定的一对垂直坐标轴成立；分解前要说明所选的坐标轴。
The original force must not be used again in the same direction once it has been resolved.
力一旦被分解，就不能在同一方向上再重复使用原来的力。
Choose axes along the surface or along the string whenever possible to reduce the number of components.
尽可能让坐标轴沿表面或沿绳方向，以减少需要分解的力的数目。
6. Resultant of Many Forces by Components: Worked Example | 用分量法求多个力的合力：例题
For three or more forces, the component method is faster and more accurate than drawing. Resolve every force into x and y components, sum each direction separately, then recombine the two totals.
对于三个或更多力，分量法比作图更快、更准确。把每个力分解为 x 与 y 分量，分别求和，再把两个总量合起来。
ΣFₓ = F₁ cos θ₁ + F₂ cos θ₂ + F₃ cos θ₃ + …
ΣFy = F₁ sin θ₁ + F₂ sin θ₂ + F₃ sin θ₃ + …
Worked example. Three horizontal coplanar forces act on a small ring: 6.0 N acting east (0°), 5.0 N acting north (90°), and 4.0 N acting west (180°). Find the resultant.
例题。 三个共面水平力作用在一个小圆环上：6.0 N 向东（0°），5.0 N 向北（90°），4.0 N 向西（180°）。求合力。
| Force | x-component / N | y-component / N |
| 6.0 N at 0° | +6.0 | 0 |
| 5.0 N at 90° | 0 | +5.0 |
| 4.0 N at 180° | −4.0 | 0 |
| Total | ΣFₓ = +2.0 | ΣFy = +5.0 |
Now the two totals are perpendicular, so Pythagoras and the tangent ratio apply.
现在这两个总量互相垂直，因此可以应用勾股定理和正切关系。
R = √2.0² + 5.0² = √29.0 = 5.39 N
θ = tan⁻¹(5.0 / 2.0) = 68.2°
The resultant is therefore 5.4 N acting at 68.2° above the east direction, that is, in the first quadrant between east and north.
因此合力为 5.4 N，方向在向东方向之上 68.2°，即位于东与北之间的第一象限。
7. Equilibrium: The Condition ΣF = 0 | 平衡条件：ΣF = 0
A body is in equilibrium when its acceleration is zero. By Newton's first law this happens when the body is at rest or moving with constant velocity in a straight line, and it requires the vector sum of all forces acting on the body to be zero.
当物体的加速度为零时，物体处于平衡状态。根据牛顿第一定律，这发生在物体静止或沿直线匀速运动时，并要求作用在物体上所有力的矢量和为零。
ΣF = 0, so ΣFₓ = 0 and ΣFy = 0
The two scalar equations are the working equations: the sum of the components in one direction equals the sum of the components in the opposite direction. This is why equilibrium questions almost always reduce to resolving forces and setting up simultaneous equations.
这两个标量方程就是解题方程：某一方向的分量之和等于相反方向的分量之和。这就是为什么平衡问题几乎总是归结为分解力并列出联立方程。
If a set of forces has a non-zero resultant, the single force that would cancel it exactly is called the equilibrant. It has the same magnitude as the resultant but acts in the opposite direction.
如果一组力的合力不为零，能够恰好抵消它的那个单独的力称为平衡力。它与合力大小相同，但方向相反。
Equilibrium does not require the body to be stationary; constant velocity is also equilibrium.
平衡并不要求物体静止；匀速运动同样是平衡状态。
Always resolve in two perpendicular directions, even if one of them is along a slope or along a string.
始终沿两个互相垂直的方向分解，即使其中一个方向是沿斜面或沿绳方向。
ntal surface with no vertical applied force, N = mg is a special case, not a general rule.
对于水平面上且无竖直外力的物体，N = mg 只是特例，并非普遍规律。
8. Three Forces in Equilibrium: Triangle of Forces and Lami's Theorem | 三力平衡：力三角形与拉密定理
If three non-parallel forces act on a body in equilibrium, their lines of action must be concurrent, and the three force vectors drawn head-to-tail form a closed triangle. This is the triangle of forces.
如果有三个互不平行的力作用在处于平衡的物体上，它们的作用线必须共点，而且这三个力矢量首尾相接可以构成一个闭合三角形。这就是力三角形。
Once the triangle is closed, the sine rule can be
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