Vector Representation of Forces | 力的矢量表示

📚 Vector Representation of Forces | 力的矢量表示

Forces are not just numbers; they have both size and direction. This is why we call them vectors. The ability to represent a force as a vector allows us to predict its effect, combine forces, and solve mechanics problems systematically.

力不只是数字，它既有大小又有方向，因此我们称之为矢量。能够将力表示为矢量，使我们能够预测它的效果、合成多个力，并有条不紊地解决力学问题。

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1. Understanding Force as a Vector | 理解力作为矢量

Force is a push or pull that can change the motion or shape of an object. A scalar quantity, like mass, can be described by one number; a vector quantity, like force, requires both a magnitude and a direction.

力是改变物体运动状态或形状的推或拉。标量（如质量）只需一个数即可描述；矢量（如力）则需要大小和方向两个信息。

• Magnitude: measured in newtons (N).

  大小：以牛顿（N）为单位。

• Direction: described by an angle relative to a reference line or by coordinate axes.

  方向：用相对于参考线的角度或坐标轴来表示。

• Point of application: the location on the body where the force acts.

  作用点：力作用在物体上的位置。

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2. Scalars vs Vectors | 标量与矢量

Some quantities have only a value; others have both value and direction. The table below shows the difference.

有些量只有数值，有些量则同时具有数值和方向。下表展示了它们的区别。

Quantity Type / 类型	Examples / 例子	Description / 描述
Scalar / 标量	mass, temperature, speed, energy / 质量、温度、速率、能量	described by magnitude only / 仅用大小描述
Vector / 矢量	force, velocity, displacement, acceleration / 力、速度、位移、加速度	described by magnitude and direction / 需要用大小和方向描述

An object's speed is scalar, but its velocity is a vector: it includes direction. Similarly, knowing a force is 10 N is not enough; we also need to know which way it acts.

物体的速率是标量，速度却是矢量，因为速度包含方向。同样，只知道力为 10 N 还不够，还必须知道它朝哪个方向作用。

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3. Directed Line Segments | 有向线段

A force can be represented graphically by a directed line segment. The length of the arrow shows the magnitude to a chosen scale, and the arrowhead shows the direction.

力可以用有向线段来图示。箭头的长度按选定的比例尺表示力的大小，箭头方向表示力的方向。

For example, if 1 cm represents 5 N, then a 20 N force is drawn as an arrow 4 cm long.

例如，若 1 cm 表示 5 N，那么 20 N 的力应画成 4 cm 长的箭头。

• Tail: point of application.

  箭尾：作用点。

• Length: magnitude, proportional to the size of the force.

  长度：大小，与力的大小成正比。

• Arrowhead: direction in which the force acts.

  箭头：力的作用方向。

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4. Component Form | 分量形式

On a coordinate grid, any force F can be split into a horizontal component F_x and a vertical component F_y. This is called the component form.

在坐标网格中，任意力 F 可以分解为水平分量 F_x 和竖直分量 F_y。这种表示方法称为分量形式。

For a 2D force, we can write:

F = F_x i + F_y j

where i and j are unit vectors in the x and y directions.

其中 i 和 j 是沿 x 轴和 y 轴方向的单位矢量。

In three dimensions we add a third component: F = F_x i + F_y j + F_z k.

在三维空间中我们添加第三个分量：F = F_x i + F_y j + F_z k。

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5. Magnitude and Direction from Components | 由分量求大小与方向

If the components F_x and F_y are known, the magnitude can be found using Pythagoras' theorem.

如果已知分量 F_x 和 F_y，则可以用勾股定理求出合力的大小。

|F| = √F_x² + F_y²

The angle θ that the force makes with the x-axis is given by:

力与 x 轴之间的夹角 θ 由下式给出：

θ = tan⁻¹(F_y / F_x)

Remember to check which quadrant the force lies in before trusting the calculator's angle.

注意：使用计算器前，要先判断力所在的象限，不能直接照搬计算器给出的角度。

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6. Resolving a Force into Components | 将一个力分解为分量

If a force F acts at an angle θ to the positive x-axis, its components are:

若力 F 与 x 轴正方向成 θ 角，则它的分量为：

F_x = F cos θ

F_y = F sin θ

These are sometimes called the resolved parts of the force. If the angle is given to the vertical, swap sin and cos.

这两个分量有时称为力的正交分量。如果已知的是与竖直方向的夹角，则 sin 和 cos 要对调。

Resolving is one of the most important skills in A-Level mechanics.

分解是 A-Level 力学中最关键的技能之一。

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7. Adding Forces: Resultant | 力的合成：合力

To find the resultant of several forces, add their vector components. If F = F_x i + F_y j and G = G_x i + G_y j, then:

欲求多个力的合力，只需将它们各自的分量相加。若 F = F_x i + F_y j，G = G_x i + G_y j，则：

R = F + G = (F_x + G_x) i + (F_y + G_y) j

Graphically, we draw the forces head-to-tail; the resultant is the vector from the first tail to the last arrowhead.

在图形上，我们把力首尾相接；合力就是从第一个箭尾指向最后一个箭头的矢量。

Never add magnitudes of two non-parallel forces directly.

两个不平行力的大小绝不能直接相加。

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8. Equilibrium | 平衡

When several forces act on a body and the resultant is zero, the body is in equilibrium. This means:

当多个力作用在物体上且合力为零时，物体处于平衡状态。这意味着：

ΣF_x = 0 and ΣF_y = 0

Simultaneously, the vector sum of all forces is the zero vector.

与此同时，所有力的矢量和为零矢量。

Common exam situations involve three forces in equilibrium; they can be represented as a closed triangle of vectors.

常见的考题情境是三个力使物体平衡；这三个力可以构成一个闭合矢量三角形。

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9. Negative Vector and Subtracting Forces | 负矢量与力的减法

The negative of a force F, written −F, has the same magnitude but exactly the opposite direction.

力 F 的负矢量记作 −F，它的大小相同，但方向恰好相反。

Subtraction of vectors is the same as adding the negative vector:

矢量的减法等同于加上负矢量：

F − G = F + (−G)

This idea is useful when calculating relative forces or net forces in opposite directions.

这一概念在计算相反方向的力或净力时非常有用。

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10. Worked Example 1 | 例题 1

Two forces act on a particle: 6 N due east and 8 N due north. Find the magnitude and direction of the resultant force.

一质点上作用有两个力：6 N 向东，8 N 向北。求合力的大小和方向。

Take east as i and north as j.

取东为 i，北为 j。

F₁ = 6i, F₂ = 8j

R = 6i + 8j

Find the magnitude:

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