Mechanics and Materials | 力学与材料

📚 Mechanics and Materials | 力学与材料

Mechanics and materials form the foundation of classical physics, bridging the behaviour of moving objects with the internal responses of substances to external forces. This chapter consolidates the AQA A-Level Physics syllabus for the ‘Mechanics and Materials’ section, guiding you through every core principle from vector resolution to Young modulus.

力学与材料是经典物理学的基石，它将物体的运动行为与物质对外力的内部响应联系起来。本章系统整合 AQA A-Level 物理大纲中”力学与材料”部分的内容，带你从矢量分解到杨氏模量逐一掌握所有核心原理。

1. Scalars and Vectors | 标量与矢量

A scalar quantity has magnitude only, such as mass, temperature, speed and energy. A vector quantity has both magnitude and direction, including displacement, velocity, acceleration, force and momentum.

标量仅具有大小，例如质量、温度、速率和能量。矢量同时具有大小和方向，包括位移、速度、加速度、力和动量。

When two vectors act at a point, the resultant can be found by the parallelogram law or by resolving each vector into perpendicular components. For two perpendicular components, the magnitude of the resultant R is given by:

当两个矢量作用于同一点时，可以通过平行四边形法则或将每个矢量分解为垂直分量来求合力。对于两个垂直分量，合力 R 的大小由下式给出：

R = √(Fx² + Fy²)

and its direction relative to the x-axis is found using θ = tan⁻¹(Fy/Fx).

其相对于 x 轴的方向由 θ = tan⁻¹(Fy/Fx) 求得。

When adding vectors, always consider both magnitude and direction; a common error is treating them as scalars.
Resolving a vector F at angle θ to a reference axis gives components Fcosθ and Fsinθ along the two perpendicular axes.

矢量相加时必须同时考虑大小和方向，常见错误是将它们当作标量处理。
将矢量 F 分解为与参考轴成 θ 角时，沿两个垂直轴的分量分别为 Fcosθ 和 Fsinθ。

2. Moments and Equilibrium | 力矩与平衡

The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force. Its unit is the newton metre (N m).

力对某点的力矩等于力的大小与该点到力的作用线的垂直距离的乘积，单位是牛顿米（N m）。

Moment = F × d

A couple is a pair of equal and opposite forces whose lines of action do not coincide. The torque of a couple is the product of one force and the perpendicular separation between the forces.

力偶是一对大小相等、方向相反且作用线不重合的力。力偶的矩等于其中一个力与两力之间垂直距离的乘积。

For a body to be in equilibrium, two conditions must be satisfied simultaneously:

物体处于平衡状态需同时满足两个条件：

The resultant force in any direction is zero: ΣF = 0.
The resultant moment about any point is zero: ΣM = 0.

任意方向上的合力为零：ΣF = 0。
对任意一点的合力矩为零：ΣM = 0。

When solving equilibrium problems, choose a pivot that eliminates unknown forces where possible, and take clockwise moments as positive and anticlockwise as negative (or vice versa, consistently).

在求解平衡问题时，应选择能消去未知力的转轴，并规定顺时针力矩为正、逆时针力矩为负（或相反，保持一致即可）。

3. Linear Motion and SUVAT Equations | 直线运动与运动学公式

Linear motion describes the movement of an object along a straight line. The key quantities are displacement s, initial velocity u, final velocity v, acceleration a and time t. For uniform acceleration, the five SUVAT equations apply:

直线运动描述物体沿直线的运动，关键物理量为位移 s、初速度 u、末速度 v、加速度 a 和时间 t。在匀加速条件下，五个运动学公式适用：

v = u + at

s = ut + ½at²

s = ½(u + v)t

v² = u² + 2as

s = vt − ½at²

For vertical motion under gravity, the acceleration is g = 9.81 m s⁻², acting downward. A useful substitution is to take upward as positive, making a = −9.81 m s⁻² for objects moving freely under gravity.

在重力作用下的竖直运动中，加速度为 g = 9.81 m s⁻²，方向向下。通常取向上为正方向，则自由落体运动中 a = −9.81 m s⁻²。

Displacement-time graphs and velocity-time graphs provide quick insights into motion: the gradient of an s-t graph gives velocity, while the gradient of a v-t graph gives acceleration and the area under it gives displacement.

位移-时间图和速度-时间图能快速反映运动特征：s-t 图的斜率表示速度，v-t 图的斜率表示加速度，v-t 图下的面积表示位移。

4. Projectile Motion | 抛体运动

Projectile motion is two-dimensional motion under the influence of gravity alone, after an initial launch. The horizontal and vertical components of motion are independent of each other.

抛体运动是物体在初始发射后仅在重力影响下的二维运动。运动的水平分量和竖直分量相互独立。

For a projectile launched with speed u at angle θ above the horizontal:

对于以速度 u、与水平方向成 θ 角发射的抛体：

Horizontal component: uₓ = ucosθ, with zero horizontal acceleration (air resistance neglected).
Vertical component: uᵧ = usinθ, with acceleration a = −g.

水平分量：uₓ = ucosθ，水平方向加速度为零（忽略空气阻力）。
竖直分量：uᵧ = usinθ，加速度 a = −g。

The time of flight, maximum height H and range R are given by:

飞行时间、最大高度 H 和射程 R 分别由以下公式给出：

T = 2usinθ/g, H = u²sin²θ/(2g), R = u²sin2θ/g

For a fixed launch speed, the maximum range occurs at θ = 45°, and complementary angles (e.g., 30° and 60°) give the same range.

在发射速率固定时，最大射程出现在 θ = 45°，而互补角（如 30° 和 60°）给出相同的射程。

5. Newton’s Laws of Motion | 牛顿运动定律

Newton’s three laws are the cornerstones of classical mechanics:

牛顿三定律是经典力学的基石：

First law: A body remains at rest or in uniform motion in a straight line unless acted upon by a resultant external force.
Second law: The rate of change of momentum of a body is directly proportional to the resultant force and takes place in the direction of that force. In equation form: F = ma, or F = Δp/Δt.
Third law: When two bodies interact, the forces they exert on each other are equal in magnitude, opposite in direction and act along the same line.

第一定律：物体在不受合外力作用时，保持静止或匀速直线运动状态。
第二定律：物体动量的变化率与所受合外力成正比，且方向与合外力方向一致，即 F = ma 或 F = Δp/Δt。
第三定律：两个物体相互作用时，彼此施加的力大小相等、方向相反且作用在同一直线上。

Weight is the gravitational force on an object: W = mg. Normal reaction force and tension are examples of contact and tensile forces that arise in response to applied loads.

重力是物体所受的引力：W = mg。支持力和张力分别是接触力和拉伸力的典型例子，它们响应于外加载荷而出现。

In applying F = ma, ensure the resultant force is used, not merely an individual force. Draw free-body diagrams to identify all forces acting on the object.

应用 F = ma 时必须使用合力，而非单独某一分力。建议绘制受力分析图，明确作用在物体上的所有力。

6. Momentum and Collisions | 动量与碰撞

Linear momentum is defined as the product of mass and velocity: p = mv, measured in kg m s⁻¹. The principle of conservation of momentum states that the total momentum of a closed system remains constant, provided no external resultant force acts on it.

线性动量定义为质量与速度的乘积：p = mv，单位是 kg m s⁻¹。动量守恒定律指出：在没有合外力作用的封闭系统中，总动量保持不变。

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Collisions are classified according to kinetic energy conservation:

碰撞根据动能是否守恒进行分类：

Elastic collision: Both momentum and kinetic energy are conserved.
Inelastic collision: Momentum is conserved but kinetic energy is not; some is transformed into heat, sound or deformation energy.
Perfectly inelastic collision: The two bodies stick together and move with a common velocity after impact.

弹性碰撞：动量和动能均守恒。
非弹性碰撞：动量守恒但动能不守恒，部分动能转化为热能、声能或形变能。
完全非弹性碰撞：两物体碰撞后粘合在一起，以共同速度运动。

Impulse is the product of force and time of impact: I = FΔt = Δp. This relationship is particularly useful when forces act over very short time intervals, such as in sports collisions or vehicle safety design. Crumple zones and airbags increase impact time, thereby reducing the average force experienced by occupants.

冲量是力与作用时间的乘积：I = FΔt = Δp。这个关系在力作用时间极短的情况下尤其有用，例如运动碰撞或车辆安全设计。溃缩区和安全气囊通过延长冲击时间，从而减小乘员受到的平均作用力。

7. Work, Energy and Power | 功、能量与功率

Work is done when a force moves its point of application. The work done is given by:

当力使力的作用点发生位移时，力就做了功。功的定义式为：

W = Fs cosθ

where θ is the angle between the force and the displacement direction. In the SI system, work is measured in joules (J), where 1 J = 1 N m.

式中 θ 为力与位移方向之间的夹角。国际单位制中功的单位为焦耳（J），且 1 J = 1 N m。

Energy exists in many forms: kinetic energy, gravitational potential energy, elastic potential energy, thermal energy and chemical energy. The principle of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another.

能量以多种形式存在：动能、重力势能、弹性势能、热能和化学能。能量守恒定律指出能量既不能凭空产生也不能凭空消失，只能从一种形式转化为另一种形式。

For a body of mass m moving at speed v:

对于质量为 m、速度为 v 的物体：

Kinetic energy Eₖ = ½mv²

Gravitational potential energy Eₚ = mgh

动能 Eₖ = ½mv²

重力势能 Eₚ = mgh

Power is the rate of doing work or transferring energy:

功率是做功或能量转化的速率：

P = W/t = Fv

where P is measured in watts (W). The equation P = Fv is particularly useful for problems involving vehicles moving at constant speed against resistive forces.

功率 P 的单位为瓦特（W）。公式 P = Fv 对于涉及车辆克服阻力匀速行驶的问题尤为实用。

Efficiency is the ratio of useful output energy to total input energy, usually expressed as a percentage. In real systems, energy is dissipated as thermal energy due to friction and air resistance, so efficiency is always less than 100%.

效率是有用输出能量与总输入能量之比，通常用百分数表示。实际系统中，能量会因摩擦和空气阻力而以热能形式耗散，因此效率总是低于 100%。

8. Density and Pressure | 密度与压强

Density is defined as mass per unit volume:

密度的定义为单位体积的质量：

ρ = m/V

with the SI unit kg m⁻³. Density is a characteristic property of a material that can be used to identify substances and is temperature-dependent for most materials.

密度的 SI 单位为 kg m⁻³。密度是材料的特性物理量，可用于鉴别物质，并且对大多数材料而言随温度变化。

Pressure is defined as the normal force per unit area:

压强的定义为每单位面积所受的法向力：

p = F/A

measured in pascals (Pa), where 1 Pa = 1 N m⁻². Pressure in a static fluid increases linearly with depth:

压强单位为帕斯卡（Pa），1 Pa = 1 N m⁻²。静止流体中的压强随深度线性增加：

p = p₀ + ρgh

where p₀ is the pressure at the surface, ρ is the fluid density, g is gravitational field strength and h is the depth.

其中 p₀ 为液面处压强，ρ 为流体密度，g 为重力场强度，h 为深度。

9. Hooke’s Law and Spring Behaviour | 胡克定律与弹簧行为

When a material is stretched or compressed by a force, it deforms. Hooke’s law states that, for a limited range, the extension of a spring is directly proportional to the applied force:

材料在拉伸或压缩力作用下会发生形变。胡克定律表明，在弹性限度内，弹簧的伸长量与所受外力成正比：

F = kx

where k is the spring constant (N m⁻¹) and x is the extension from the natural length. The spring constant is a measure of the stiffness of the spring.

其中 k 为劲度系数（N m⁻¹），x 为相对于自然长度的伸长量。劲度系数表征弹簧的刚性程度。

For springs in series and parallel, the equivalent spring constants are:

对于串联和并联弹簧，等效劲度系数分别为：

Series: 1/k_total = 1/k₁ + 1/k₂ + …
Parallel: k_total = k₁ + k₂ + …

串联：1/k_总 = 1/k₁ + 1/k₂ + …
并联：k_总 = k₁ + k₂ + …

Elastic potential energy stored in a stretched spring is the area under the force-extension graph:

拉伸弹簧中储存的弹性势能等于力-伸长量图像下的面积：

Eₑ = ½kx²

Beyond the elastic limit, the material undergoes plastic deformation and will not return to its original shape when the load is removed.

超过弹性极限后，材料发生塑性形变，卸载后无法恢复原来形状。

10. Stress, Strain and Young Modulus | 应力、应变与杨氏模量

Stress and strain describe the internal response of a material to external loading, independent of the material’s dimensions.

应力与应变描述材料对外部加载的内部响应，与材料的具体尺寸无关。

Stress σ = F/A (unit: Pa or N m⁻²)

Strain ε = ΔL/L (no unit, dimensionless)

应力 σ = F/A（单位：Pa 或 N m⁻²）

应变 ε = ΔL/L（无量纲）

The Young modulus is the ratio of tensile stress to tensile strain, measuring the stiffness of a material:

杨氏模量是拉伸应力与拉伸应变的比值，用于衡量材料的刚性：

E = σ/ε = FL/(AΔL)

In the elastic region, stress is proportional to strain, obeying Hooke’s law, and the Young modulus is constant. The Young modulus is determined experimentally by measuring the extension of a wire under increasing loads, using a micrometer to measure diameter and a travelling microscope for extension.

在弹性区域内，应力与应变成正比，遵循胡克定律，杨氏模量为常数。杨氏模量通过实验测定：在逐渐增加载荷的条件下测量金属丝的伸长量，用千分尺测量直径，用移测显微镜测量伸长量。

The gradient of the stress-strain graph in the elastic region gives the Young modulus. The area under the stress-strain curve represents the energy stored per unit volume of the material.

应力-应变图在弹性区域的斜率即为杨氏模量。应力-应变曲线下的面积表示单位体积材料所储存的能量。

11. Material Properties and Deformation | 材料性质与形变

Different materials exhibit distinct behaviours when subjected to stress. The stress-strain graph reveals key properties:

不同材料在受力时表现出不同的行为特征。应力-应变图揭示了关键的材料性能：

Brittle materials (glass, ceramics): fracture with little plastic deformation; no yield point is observed.
Ductile materials (copper, mild steel): undergo significant plastic deformation before fracture, with a clear yield point and plastic region.
Elastic materials (rubber): return to original shape after load removal, exhibiting a non-linear force-extension relationship.
Polymeric materials: exhibit time-dependent behaviour such as creep and stress relaxation.

脆性材料（玻璃、陶瓷）：几乎没有塑性形变即发生断裂，不出现明显屈服点。
延性材料（铜、低碳钢）：断裂前经历显著塑性形变，具有清晰的屈服点和塑性区。
弹性材料（橡胶）：卸载后恢复原状，力-伸长关系呈非线性。
聚合物材料：表现出与时间相关的行为，如蠕变和应力松弛。

Key points on the stress-strain graph include the limit of proportionality (where Hooke’s law ceases to hold), the elastic limit (beyond which permanent deformation occurs) and the yield point (where the material deforms without a significant increase in load). For ductile materials, the ultimate tensile strength is the maximum stress the material can withstand before necking leads to fracture.

应力-应变图上的关键点包括：比例极限（胡克定律不再成立处）、弹性极限（超过后产生永久形变）和屈服点（材料在载荷无明显增加的情况下发生显著形变）。对于延性材料，抗拉强度极限是材料在发生颈缩断裂前能承受的最大应力。

The toughness of a material is measured by the total energy absorbed per unit volume before fracture — the area under the entire stress-strain curve. The stiffness is given by the Young modulus, while Tensile strength and Breaking stress both describe maximum load capacity but differ: tensile strength refers to the ultimate stress, while breaking stress is the stress at the point of fracture.

材料的韧性由断裂前单位体积吸收的总能量来衡量，即整个应力-应变曲线下的面积。刚性由杨氏模量决定，而抗拉强度和断裂应力都描述最大承载能力，但二者存在区别：抗拉强度指极限应力，而断裂应力指断裂时刻的应力。

12. Experimental Methods in Mechanics | 力学实验方法

Accurate measurement is central to mechanics. For determining the Young modulus of a metal wire, the following procedure is recommended:

精确测量是力学的核心。测定金属丝杨氏模量的推荐步骤如下：

Measure the wire length L with a metre ruler and the diameter d with a micrometer screw gauge at several positions along the wire; calculate the mean cross-sectional area A = πd²/4.
Attach known masses to the wire and measure the corresponding extensions using a travelling microscope or a sensitive marker on a scale.
Plot stress against strain and calculate the gradient in the elastic region gives the Young modulus.

用米尺测量金属丝长度 L，用千分尺在金属丝多个位置测量直径 d，计算平均横截面积 A = πd²/4。
逐次挂上已知质量的砝码，用移测显微镜或精密刻度标尺测量对应的伸长量。
绘制应力-应变图，弹性区域内直线部分的斜率即为杨氏模量。

For verifying the principle of conservation of momentum, a linear air track with gliders fitted with light gates provides near-frictionless conditions. Velocities are calculated from the time taken to pass through the light gates, and masses are measured directly. The total momentum before and after collision is compared to verify conservation.

为验证动量守恒定律，使用带气垫导轨和光电门的滑块系统可以提供近似无摩擦的条件。速度由滑块通过光电门的时间计算，质量直接测量。比较碰撞前后总动量即可验证动量守恒。

When determining the acceleration of free fall, a ball is dropped between two light gates a known distance apart, or an electromagnetic release and a timer record the fall time. The value is obtained from s = ½gt² using the measured drop distance and time, with small systematic errors corrected through careful calibration.

测定重力加速度时，可采用小球在两光电门之间下落，或用电磁释放装置配合计时器记录下落时间。根据测量的下落距离和时间，利用 s = ½gt² 求得 g 值，并通过仔细校准修正系统误差。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mechanics and Materials | 力学与材料