📚 Forces as Vectors | 力作为矢量
In AS Physics, understanding the vector nature of force is fundamental. A force is not just a push or a pull with a certain size; its direction is equally important. To analyse complex situations, you must treat forces as vectors, using methods such as vector addition, resolution, and equilibrium conditions.
在AS物理中，理解力的矢量性质是基础。力不仅仅是有一定大小的推或拉；它的方向同样重要。为了分析复杂情况，你必须将力作为矢量来处理，使用矢量加法、分解和平衡条件等方法。
1. Scalar vs Vector Quantities | 标量与矢量量
A scalar quantity has magnitude only, e.g., mass (kg), speed (m/s), energy (J). A vector quantity has both magnitude and direction, e.g., force (N), velocity (m/s), displacement (m). This distinction means that force calculations must follow vector rules.
标量只有大小，例如质量(kg)、速率(m/s)、能量(J)。矢量既有大小又有方向，例如力(N)、速度(m/s)、位移(m)。这一区别意味着力的计算必须遵循矢量法则。
2. Representing a Force Vector | 表示力矢量
Forces are drawn as arrows. The arrow's length is proportional to the magnitude, and the arrowhead points in the direction of the force. A scale is chosen, e.g., 1 cm represents 10 N. The straight line of the arrow indicates the line of action.
力用箭头表示。箭头的长度与力的大小成正比，箭头指向力的方向。选择一个比例尺，例如1厘米代表10牛。箭头所在的直线表示力的作用线。
In diagrams, a force is usually labelled, e.g., F = 50 N at 30° above the horizontal.
在图中，力通常会标注，例如 F = 50 N，与水平方向成30°角向上。
3. Adding Collinear Forces | 共线力的加法
If forces act along the same line, you can add them algebraically. Choose a positive direction. For example, 8 N to the right (+) and 3 N to the right (+) give a resultant of 11 N to the right. 8 N to the right and 3 N to the left (–) give 5 N to the right.
如果力沿同一直线作用，可以进行代数加法。选择一个正方向。例如，向右8 N (+) 和向右3 N (+) 得到的合力为向右11 N。向右8 N和向左3 N (–) 得到向右5 N。
Resultant force: R = F₁ + F₂ (signed). In equilibrium, R = 0.
合力：R = F₁ + F₂（带符号）。平衡时，R = 0。
4. Perpendicular Forces and Pythagoras | 垂直力与勾股定理
is found using the Pythagorean theorem. If F₁ = 3 N east and F₂ = 4 N north, resultant R = √3² + 4² = √25 = 5 N.
当两个力垂直时，合力的大小可用勾股定理求得。若 F₁ = 3 N 向东，F₂ = 4 N 向北，合力 R = √3² + 4² = √25 = 5 N。
ion is given by tanθ = opposite/adjacent = 4/3, so θ = tan⁻¹(1.333) ≈ 53.1° north of east.
合力相对于正东方向的角度 θ 由 tanθ = 对边/邻边 = 4/3 给出，θ = tan⁻¹(1.333) ≈ 53.1° 北偏东。
This method is essential for any two perpendicular force components.
此方法对任意两个垂直分力都必不可少。
5. Vector Addition Using Parallelogram and Triangle Methods | 使用平行四边形和三角形法则进行矢量加法
For forces at any angle, construct a parallelogram where the sides represent the two forces. The diagonal from the common origin gives the resultant in both magnitude and direction.
对于任意角度的力，构建一个平行四边形，其两边表示两个力。从共同起点出发的对角线给出合力的大小和方向。
Alternatively, the triangle method places the tail of the second force at the head of the first; the resultant is the vector from the start of the first to the head of the second.
另一种方法是三角形法则，将第二个力的尾部放在第一个力的头部；合力是从第一个力的起点指向第二个力头部的矢量。
These graphical methods help visualise, but for precise calculations, trigonometry or components are used.
这些图解法有助于直观理解，但精确计算则使用三角法或分解法。
6. Cosine and Sine Rules for Non-Perpendicular Forces | 非垂直力情形的余弦和正弦定理
When two forces F₁ and F₂ have an angle φ between them, the resultant magnitude R can be calculated using the cosine rule:
当两个力 F₁ 和 F₂ 之间夹角为 φ 时，合力大小 R 可用余弦定理计算：
R² = F₁² + F₂² + 2F₁F₂ cosφ
tors, not the angle to the horizontal. If φ = 90°, cos90° = 0, giving the Pythagoras case.
注意：φ 是两个力矢量之间的夹角，不是与水平方向的夹角。若 φ = 90°，cos90° = 0，即勾股定理情形。
lative to F₁ can be found via the sine rule: sinα / F₂ = sinφ / R, or from the components.
合力 R 相对于 F₁ 的方向 α 可通过正弦定理求得：sinα / F₂ = sinφ / R，或从分量得到。
These rules are directly applicable to any triangle formed by the forces.
这些定理可直接应用于力形成的任意三角形。
7. Resolving Forces into Rectangular Components | 将力分解为直角分量
al) can be resolved into two perpendicular components: Fₓ = F cosθ along the x-axis, and Fy = F sinθ along the y-axis.
任何与给定方向（通常水平）成 θ 角的力 F 都可分解为两个垂直分量：沿 x 轴的 Fₓ = F cosθ，沿 y 轴的 Fy = F sinθ。
This is extremely useful when several forces act at different angles. Resolve all forces, sum the x-components (ΣFₓ) and y-components (ΣFy) separately, then combine to find the resultant:
当多个力以不同角度作用时，这极其有用。将所有力分解，分别求 x 分量之和 (ΣFₓ) 和 y 分量之和 (ΣFy)，再合成得到合力：
R = √((ΣFₓ)² + (ΣFy)²), θ = tan⁻¹(|ΣFy / ΣFₓ|)
Always consider the quadrant when determining the final direction.
确定最终方向时始终要考虑象限。
8. Free-Body Diagrams (FBD) | 受力图（隔离体图）
A free-body diagram isolates a single object, showing all forces acting on that object as vectors. Forces include weight (mg down), normal reaction (perpendicular to surface), tension (along a string), friction (opposing motion or tendency), and applied forces.
受力图将单个物体隔离，标出作用在该物体上的所有力矢量。力包括重力 (mg 向下)、法向反力（垂直于表面）、张力（沿绳方向）、摩擦力（与运动或趋势相反）以及外加力。
Drawing an accurate FBD is the first step in solving any mechanics problem. Only forces on the body are included; forces exerted by the body are omitted.
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