Vector Applications in Mechanics | 向量在力学中的应用

📚 Vector Applications in Mechanics | 向量在力学中的应用

Vectors are essential mathematical tools for describing physical quantities that have both magnitude and direction. In mechanics, forces, velocities, accelerations, and displacements are all vector quantities, and their combined behaviour follows the rules of vector algebra.

向量是描述既有大小又有方向的物理量的基本数学工具。在力学中，力、速度、加速度和位移都是向量量，它们的合成与分解遵循向量代数的运算法则。

1. Scalar vs Vector Quantities | 标量与向量的区别

A scalar quantity has only magnitude, such as mass, time, and speed. A vector quantity has both magnitude and direction, such as force, velocity, and displacement. In written notation, a vector is often denoted by a bold letter v or an arrow ᵥ, and its magnitude is written as |v|.

标量只有大小，如质量、时间和速率。向量既有大小又有方向，如力、速度和位移。书写时常用粗体字母v或带箭头符号表示向量，其大小记作|v|。

When working in a coordinate plane, a vector can be expressed in component form. For example, a force F of magnitude 10 N acting at 30° above the horizontal has components Fₓ = 10 cos 30° = 5√3 N and Fᵧ = 10 sin 30° = 5 N.

在坐标平面中，向量可以用分量形式表示。例如，大小为10 N、方向与水平方向成30°角的力F，其分量为 Fₓ = 10 cos 30° = 5√3 N，Fᵧ = 10 sin 30° = 5 N。

2. Force as a Vector | 力的向量表示

Forces are vectors because they have both magnitude and direction. When several forces act on a body, the total effect is given by the resultant force, which is the vector sum of all individual forces.

力是向量，因为它既有大小也有方向。当多个力作用在同一物体上时，总效果由合力给出，合力就是所有力的向量和。

For two forces F₁ and F₂ acting at a point, the resultant R is found by placing the vectors tip-to-tail:

对于作用在同一点的两个力F₁和F₂，合力R通过将向量首尾相接来求得：

R = F₁ + F₂

In component form, if F₁ = (F₁ₓ, F₁ᵧ) and F₂ = (F₂ₓ, F₂ᵧ), then R = (F₁ₓ + F₂ₓ, F₁ᵧ + F₂ᵧ).

在分量形式中，若 F₁ = (F₁ₓ, F₁ᵧ)，F₂ = (F₂ₓ, F₂ᵧ)，则 R = (F₁ₓ + F₂ₓ, F₁ᵧ + F₂ᵧ)。

3. Resolution of Forces into Perpendicular Components | 力的正交分解

Resolving a force means splitting it into two perpendicular components, usually horizontal and vertical. If a force F makes an angle θ with the horizontal, the horizontal component is F cos θ and the vertical component is F sin θ.

力的分解是指将一个力拆分为两个互相垂直的分量，通常为水平和竖直方向。若力F与水平方向夹角为θ，则水平分量为 F cos θ，竖直分量为 F sin θ。

This technique simplifies equilibrium and motion problems because the components along each axis can be treated independently. For example, a weight on a smooth inclined plane has a component of its weight down the plane equal to mg sin θ.

这种分解技巧简化了平衡与运动问题，因为各坐标轴上的分量可以独立处理。例如，在光滑斜面上的物体，其重力沿斜面向下的分量等于 mg sin θ。

4. Resultant of Multiple Forces | 多个力的合力

To find the resultant of several forces, add their components in each direction separately. For forces F₁, F₂, …, Fₙ, the resultant R is given by:

要求多个力的合力，需要分别将每个方向的分量相加。对于力F₁, F₂, …, Fₙ，合力R为：

Rₓ = ΣFᵢₓ, Rᵧ = ΣFᵢᵧ

Then the magnitude of the resultant is |R| = √(Rₓ² + Rᵧ²), and its direction angle θ satisfies tan θ = Rᵧ / Rₓ.

合力的大小为 |R| = √(Rₓ² + Rᵧ²)，其方向角θ满足 tan θ = Rᵧ / Rₓ。

Consider a particle acted on by two forces: F₁ = (3, 4) N and F₂ = (−2, 7) N. Then R = (1, 11) N, so |R| = √122 N and θ = arctan(11) ≈ 84.8°.

考虑一个质点受到两个力：F₁ = (3, 4) N，F₂ = (−2, 7) N。则 R = (1, 11) N，所以 |R| = √122 N，θ = arctan(11) ≈ 84.8°。

5. Equilibrium of Forces | 力的平衡

A body is in equilibrium when the vector sum of all forces acting on it is zero. This means that the resultant force has no magnitude and no specific direction: ΣF = 0.

当物体所受全部力的向量和为零时，物体处于平衡状态。这意味着合力没有大小和确定方向：ΣF = 0。

For forces in a plane, this gives two scalar equations:

对于平面力系，这给出两个标量方程：

ΣFₓ = 0 and ΣFᵧ = 0

If three non-parallel forces keep a body in equilibrium, their vectors must form a closed triangle when drawn tip-to-tail. This is known as the triangle of forces or Lami’s theorem, which states that F₁ / sin α = F₂ / sin β = F₃ / sin γ, where α, β, γ are the angles between the other two forces.

若三个不平行力使物体平衡，则它们的向量首尾相接构成封闭三角形。这就是力的三角形法则或拉密定理，即 F₁ / sin α = F₂ / sin β = F₃ / sin γ，其中α、β、γ分别是其余两个力之间的夹角。

6. Position Vectors and Displacement | 位置向量与位移

The position vector r of a point P relative to an origin O describes the location of P. If P has coordinates (x, y), then r = x i + y j, where i and j are unit vectors along the x and y axes.

点P相对于原点O的位置向量r描述P的位置。若P的坐标为(x, y)，则 r = x i + y j，其中i和j分别表示沿x轴和y轴的单位向量。

Displacement is the change in position. If a body moves from point A with position vector rₐ to point B with position vector rᵦ, then the displacement vector is s = rᵦ − rₐ.

位移是位置的变化量。若物体从位置向量为rₐ的点A运动到位置向量为rᵦ的点B，则位移向量为 s = rᵦ − rₐ。

The distance between A and B is the magnitude of the displacement, |s| = √((xᵦ−xₐ)² + (yᵦ−yₐ)²). This is exactly the vector form of Pythagoras’ theorem.

A、B两点间的距离是位移的大小，|s| = √((xᵦ−xₐ)² + (yᵦ−yₐ)²)。这正是勾股定理的向量形式。

7. Velocity and Acceleration as Vectors | 速度与加速度的向量表示

Velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. Both are vector quantities. If a particle has position vector r(t) at time t, then its velocity is v = dr/dt and its acceleration is a = dv/dt = d²r/dt².

速度是位移的变化率，加速度是速度的变化率。两者都是向量量。若质点在时刻t的位置向量为r(t)，则其速度为 v = dr/dt，加速度为 a = dv/dt = d²r/dt²。

In component form, for motion in a plane, r = x(t) i + y(t) j, then v = (dx/dt) i + (dy/dt) j and a = (d²x/dt²) i + (d²y/dt²) j.

在平面运动的分量形式中，r = x(t) i + y(t) j，则 v = (dx/dt) i + (dy/dt) j，a = (d²x/dt²) i + (d²y/dt²) j。

Projectile motion is a classic example. Ignoring air resistance, the only acceleration is g downward, so the horizontal component of velocity remains constant while the vertical component changes uniformly.

抛体运动是典型例子。忽略空气阻力时，唯一加速度是向下的g，因此速度的水平分量保持不变，而竖直分量做匀变速变化。

8. Relative Velocity and Position | 相对速度与相对位置

Relative velocity describes the velocity of one body as observed from another moving body. If body A has velocity vₐ and body B has velocity vᵦ, then the velocity of B relative to A is vᵦₐ = vᵦ − vₐ.

相对速度描述一个物体相对于另一个运动物体的速度。若物体A的速度为vₐ，物体B的速度为vᵦ，则B相对于A的速度为 vᵦₐ = vᵦ − vₐ。

Similarly, the position of B relative to A is rᵦₐ = rᵦ − rₐ. The vector difference gives both the direction and distance from A to B.

类似地，B相对于A的位置为 rᵦₐ = rᵦ − rₐ。向量差同时给出从A到B的方向和距离。

For two ships moving in a plane, the relative velocity determines whether they will collide. If the vector from ship A to ship B is parallel to the relative velocity vᵦₐ, then their paths intersect at the same time, implying a possible collision.

对于在同一平面内运动的船只，相对速度决定它们是否会相撞。若从船A到船B的向量与相对速度vᵦₐ平行，则它们的路径在同一时刻相交，可能发生碰撞。

9. Newton’s Second Law in Vector Form | 牛顿第二定律的向量形式

Newton’s second law states that the resultant force on a body is equal to its mass times its acceleration. In vector notation this is written as:

牛顿第二定律指出，物体所受合力等于其质量乘以加速度。用向量记号写出：

F = m a

Since mass is a scalar, the acceleration vector is parallel to the resultant force vector. For example, a particle of mass 2 kg subject to F = (6, 8) N has acceleration a = (3, 4) m/s², with magnitude 5 m/s².

由于质量是标量，加速度向量与合力向量方向相同。例如，质量为2 kg的质点受F = (6, 8) N作用，其加速度为 a = (3, 4) m/s²，大小为5 m/s²。

In component form, Fₓ = m aₓ and Fᵧ = m aᵧ. This allows us to solve problems involving forces in two directions independently, such as a particle on a rough inclined plane where friction acts parallel to the plane.

在分量形式中，Fₓ = m aₓ，Fᵧ = m aᵧ。这样我们可以独立求解两个方向上的受力问题，例如粗糙斜面上的物体，摩擦力沿斜面方向作用。

10. Impulse and Momentum as Vector Quantities | 冲量与动量的向量性

Momentum is defined as the product of mass and velocity, p = m v. Impulse is the product of force and time, J = F Δt. Both are vectors.

动量定义为质量与速度的乘积，p = m v。冲量是力与时间的乘积，J = F Δt。两者均为向量。

The impulse-momentum principle states that the impulse on a body equals the change in momentum:

冲量-动量定理指出，物体所受冲量等于其动量变化量：

F Δt = m v₂ − m v₁

This vector equation can be applied separately in the x and y directions. For example, in a two-dimensional collision, the total momentum before impact equals the total momentum after impact:

该向量方程可分别在x和y方向独立应用。例如，在二维碰撞中，碰撞前总动量等于碰撞后总动量：

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂

where u and v are initial and final velocity vectors respectively.

其中u和v分别为初始和末速度向量。

11. Work Done and the Dot Product | 做功与点积

When a constant force F acts on a body while it undergoes displacement s, the work done by the force is given by the scalar product (dot product) of the force and displacement:

当恒力F作用在物体上并产生位移s时，力所做的功等于力与位移的点积（标量积）：

W = F ⋅ s = |F| |s| cos θ

where θ is the angle between F and s. If F and s are perpendicular, cos θ = 0, so no work is done. For example, when a particle moves in a circular path under a centripetal force, the force is always perpendicular to the displacement and does no work.

其中θ是F与s之间的夹角。若F与s垂直，则cos θ = 0，因此做功为零。例如，质点做圆周运动时向心力始终垂直于位移，所以不做功。

In component form, if F = (Fₓ, Fᵧ) and s = (sₓ, sᵧ), then W = Fₓ sₓ + Fᵧ sᵧ. The dot product of two vectors always yields a scalar, which is why work has magnitude but no direction.

在分量形式中，若F = (Fₓ, Fᵧ)，s = (sₓ, sᵧ)，则 W = Fₓ sₓ + Fᵧ sᵧ。两个向量的点积总是产生标量，因此功只有大小没有方向。

12. Three-Dimensional Vector Mechanics | 三维向量力学

In three dimensions, vectors are expressed with three components, often using the unit vectors i, j, k along the x, y, and z axes. A position vector becomes r = x i + y j + z k, and its magnitude is |r| = √(x² + y² + z²).

在三维空间中，向量用三个分量表示，常用沿x、y、z轴的单位向量i、j、k。位置向量变为 r = x i + y j + z k，其大小为 |r| = √(x² + y² + z²)。

Equilibrium in three dimensions requires that ΣFₓ = 0, ΣFᵧ = 0, and ΣF_z = 0 simultaneously. A typical example is a particle suspended by three strings in space, each string exerting a tension vector that must add to zero with the particle’s weight.

三维平衡要求同时满足 ΣFₓ = 0、ΣFᵧ = 0 和 ΣF_z = 0。典型例子是空间中被三根绳子悬挂的质点，每根绳子的张力向量必须与质点的重力相加为零。

The dot product in 3D is F ⋅ s = Fₓ sₓ + Fᵧ sᵧ + F_z s_z, and the cross product can be used to calculate moments of forces. The moment of a force about a point is given by τ = r × F, where r is the position vector from the point to the line of action of the force.

三维点积为 F ⋅ s = Fₓ sₓ + Fᵧ sᵧ + F_z s_z，叉积可用于计算力矩。力对某点的矩为 τ = r × F，其中r是从该点到力的作用线的位置向量。

Vector methods therefore extend naturally from one to three dimensions, making them indispensable for analysing real-world mechanical systems such as rigid bodies, structural frameworks, and robotic motion.

因此，向量方法可以自然地从一个维度扩展到三个维度，使其成为分析现实机械系统（如刚体、结构框架和机器人运动）不可或缺的工具。

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Vector Applications in Mechanics | 向量在力学中的应用