📚 Mechanics and Materials | 力学与材料
Mechanics and materials is a core AQA A-level physics topic. It brings together the mathematical description of motion, the vector nature of forces, and the behaviour of solids under load. A strong exam technique means linking equations to physical situations, drawing clear force diagrams, and interpreting stress-strain graphs correctly.
力学与材料是 AQA A-level 物理的核心主题。它将运动的数学描述、力的矢量性质以及固体在载荷下的行为结合在一起。扎实的应试技巧意味着将方程与物理情境联系起来，画出清晰的受力图，并正确解读应力-应变图。
1. Scalars, Vectors and Motion | 标量、矢量与运动
Scalars have magnitude only, while vectors have both magnitude and direction. In AQA questions you are often asked to state whether a quantity is scalar or vector when defining it, so this distinction must be automatic.
标量只有大小，矢量既有大小又有方向。在 AQA 试题中，定义物理量时经常要求说明它是标量还是矢量，因此这一区别必须熟练掌握。
Typical scalars include distance, speed, mass, energy, time and temperature. Typical vectors include displacement, velocity, acceleration, force, momentum and weight.
常见标量包括路程、速率、质量、能量、时间和温度。常见矢量包括位移、速度、加速度、力、动量和重力。
| Scalar | 标量 | Vector | 矢量 |
| distance, speed, mass, energy, time | displacement, velocity, acceleration, force, momentum |
| 路程、速率、质量、能量、时间 | 位移、速度、加速度、力、动量 |
Average speed is total distance divided by total time, whereas average velocity is total displacement divided by total time. Displacement is the straight-line distance in a stated direction from a reference point.
平均速率等于总路程除以总时间，而平均速度等于总位移除以总时间。位移是从参考点到终点的有向直线距离。
average speed = total distance ÷ total time
average velocity = total displacement ÷ total time
In one-dimensional motion, choose a positive direction consistently. A negative velocity then means motion in the opposite direction, not a smaller speed.
在一维运动中，要始终选择正方向。速度为负表示向相反方向运动，而不是速率更小。
2. Equations of Motion | 运动方程
The four equations of motion, often called the suvat equations, only apply when acceleration is constant. They link initial velocity u, final velocity v, displacement s, acceleration a and time t.
四个运动方程，通常称为 suvat 方程，仅适用于加速度恒定的情况。它们联系初速度 u、末速度 v、位移 s、加速度 a 和时间 t。
v = u + at
This equation relates final velocity to initial velocity, acceleration and time.
该方程将末速度与初速度、加速度和时间联系起来。
s = ½(u + v)t
This form is useful when the average velocity is known or can be found quickly.
当已知或可以快速求出平均速度时，这个形式很有用。
s = ut + ½at²
This equation is used when displacement is needed and time is known, but final velocity is not.
当需要求位移且已知时间，但不知道末速度时，使用该方程。
v² = u² + 2as
This equation connects velocity and displacement without involving time directly.
该方程不直接涉及时间，将速度与位移联系起来。
tion under gravity, a = 9.81 m s⁻² vertically downwards when air resistance is negligible.
应用这些方程前，必须检查加速度是否恒定。在重力作用下的运动中，若空气阻力可忽略，a = 9.81 m s⁻²，方向竖直向下。
3. Newton's Laws and Forces | 牛顿定律与力
Newton's first law states that an object remains at rest or in uniform motion in a straight line unless acted on by a resultant force. This is the principle of inertia.
牛顿第一定律指出，除非受到合外力作用，物体保持静止或沿直线做匀速运动。这就是惯性原理。
Newton' t force as the rate of change of momentum. When mass is constant, it simplifies to F = ma.
牛顿第二定律将合外力定义为动量的变化率。当质量不变时，它简化为 F = ma。
F = ma
Force is measured in newtons, N, mass in kilograms and acceleration in m s⁻². One newton is the force needed to accelerate 1 kg at 1 m s⁻².
力的单位是牛顿 N，质量的单位是千克，加速度的单位是 m s⁻²。一牛顿是使 1 kg 物体产生 1 m s⁻² 加速度所需的力。
Newton's third law states that if body A exerts a force on body B, body B exerts an equal and opposite force on body A. The two forces act on different bodies, so they do not cancel each other.
牛顿第三定律指出，若物体 A 对物体 B 施加力，物体 B 对 A 施加大小相等、方向相反的力。这两个力作用在不同物体上，因此不能相互抵消。
ght W = mg, normal contact force, tension, friction, air resistance or drag, and upthrust.
力学中常见的力包括重力 W = mg、法向接触力、张力、摩擦力、空气阻力或曳力，以及浮力。
4. Free-body Diagrams and Equilibrium | 受力分析与平衡
A free-body diagram shows all the forces acting on one object. It is the starting point for resolving forces and deciding whether an object is in equilibrium.
受力分析图显示作用在单个物体上的所有力。它是分解力和判断物体是否平衡的起点。
For an object in translational equilibrium, the vector sum of all forces is zero. This means the horizontal components balance and the vertical components balance separately.
物体处于平动平衡时，所有力的矢量和为零。这意味着水平分量相互平衡，竖直分量也相互平衡。
ΣF = 0
To resolve a force F at an angle θ to a chosen direction, use F cos θ along that direction and F sin θ perpendicular to it. Always show these components clearly on a diagram.
要分解与选定方向成 θ 角的力 F，沿该方向使用 F cos θ，垂直于该方向使用 F sin θ。务必在图上清晰标出这些分量。
Friction acts parallel to the contact surface and opposes sliding or attempted sliding. Normal contact force acts perpendicular to the surface. These two forces are not an action-reaction pair.
摩擦力平行于接触面，阻碍滑动或滑动趋势。法向接触力垂直于接触面。这两个力不是一对作用力与反作用力。
5. Moments and Couples | 力矩与力偶
Moment of a force about a pivot is the product of the force and the perpendicular distance from the pivot to the line of action of the force.
力对支点的力矩等于力与支点到力作用线的垂直距离的乘积。
M = Fd
Moment is measured in newton metres, N m. The perpendicular distance d must be measured at right angles to the force line.
力矩的单位是牛·米，N m。垂直距离 d 必须沿垂直于力作用线的方向测量。
The principle of moments states that for an object in rotational equilibrium, the sum of clockwise moments about any pivot equals the sum of anticlockwise moments.
力矩原理指出，物体处于转动平衡时，对任意支点的顺时针力矩之和等于逆时针力矩之和。
Σ clockwise moments = Σ anticlockwise moments
A couple consists of two equal and opposite parallel forces whose lines of action do not coincide. The moment of a couple is one force times the perpendicular separation between the forces.
力偶由两个大小相等、方向相反、作用线不重合的平行力组成。力偶的力矩等于其中一个力乘以两力之间的垂直距离。
When solving equilibrium problems, draw all forces, choose a convenient pivot, and write one moment equation. Then use vertical and horizontal force balances to find unknown forces.
解决平衡问题时，画出所有力，选择合适的支点，并写出一个力矩方程。然后利用竖直和水平方向力的平衡求出未知力。
6. Work, Energy and Power | 功、能量与功率
Work done by a force is the product of the force and the displacement in the direction of the force. If the force acts at an angle to the displacement, include the cosine factor.
力所做的功等于力与沿力方向位移的乘积。如果力与位移成一定角度，则需要加入余弦因子。
W = Fs cos θ
Work is a scalar quantity measured in joules, J. One joule is the work done when a force of 1 N moves its point of application 1 m in the direction of the force.
功是标量，单位是焦耳 J。1 J 等于 1 N 的力使作用点沿力的方向移动 1 m 所做的功。
Kinetic energy is the energy of motion, while gravitational potential energy is the energy stored due to height in a gravitational field.
动能是物体由于运动而具有的能量，重力势能是物体由于在重力场中处于某一高度而储存的能量。
Eₖ = ½mv²
Eₚ = mgΔh
The work-energy principle states that the net work done on an object equals its change in kinetic energy. This is often a quicker route than using suvat equations.
功能原理指出，对物体做的净功等于其动能的变化量。这通常比使用 suvat 方程更快捷。
Power is the rate of doing work. For a constant force moving an object at constant velocity, power equals force times velocity.
功率是做功的速率。对于以恒定速度运动且力恒定的情况，功率等于力乘以速度。
P = W ÷ t = Fv
is the useful output power divided by total input power, often expressed as a percentage.
功率的单位是瓦特 W，1 W = 1 J s⁻¹。效率等于有用输出功率除以总输入功率，通常用百分数表示。
7. Momentum and Impulse | 动量与冲量
Linear momentum is the product of mass and velocity. It is a vector quantity with unit kg m s⁻¹ or N s.
线动量是质量与速度的乘积，是矢量，单位为 kg m s⁻¹ 或 N s。
p = mv
Momentum depends on both mass and velocity, so a small fast object can have the same momentum as a large slow object if the product mv is equal.
动量取决于质量和速度，因此小质量高速物体与大质量低速物体只要 mv 乘积相等，就可以有相同的动量。
Impulse is the change in momentum of an object. It equals the average resultant force multiplied by the time for which the force acts.
冲量是物体动量的变化量，等于平均合外力乘以力的作用时间。
Impulse = Δp = FΔt
The area under a force-time graph gives the impulse. This is useful in car safety problems when analysing crumple zones, airbags and seatbelts.
力-时间图下方的面积表示冲量。这在分析汽车安全问题时很有用，例如碰撞缓冲区、安全气囊和安全带。
8. Conservation of Momentum and Collisions | 动量守恒与碰撞
In a closed system with no external resultant force, total momentum before a collision is equal to total momentum after the collision. This is the principle of conservation of momentum.
在没有合外力的封闭系统中，碰撞前的总动量等于碰撞后的总动量。这就是动量守恒原理。
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Here m₁ and m₂ are the masses, u₁ and u₂ are the initial velocities, and v₁ and v₂ are the final velocities. All velocities must be measured in the same chosen positive direction.
式中 m₁ 和 m₂ 是质量，u₁ 和 u₂ 是初速度，v₁ 和 v₂ 是末速度。所有速度必须沿同一选定正方向测量。
Elastic collisions conserve both momentum and kinetic energy. Inelastic collisions conserve momentum but not kinetic energy, as some kinetic energy is transformed into heat, sound or permanent deformation.
弹性碰撞同时守恒动量和动能。非弹性碰撞守恒动量但不守恒动能，因为部分动能转化为热、声或永久形变。
Explosions are treated as the reverse of collisions: the total momentum before an explosion is zero if the system is initially at rest, so the fragments must carry equal and opposite total momentum afterwards.
爆炸可视为碰撞的逆过程：如果系统最初静止，爆炸前的总动量为零，因此爆炸后各碎片的合动量必须等大反向。
9. Hooke's Law and Springs | 胡克定律与弹簧
Hooke's law states that the extension of a spring is directly
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