📚 Edexcel Further Mechanics: Complete Revision Guide | Edexcel 进阶力学：完整复习指南
This revision guide covers the core topics in Edexcel Further Mathematics Further Mechanics 1 and Further Mechanics 2. It focuses on vector methods, conservation laws, energy methods, and mathematical modelling.
本复习指南涵盖 Edexcel 进阶数学 Further Mechanics 1 和 Further Mechanics 2 的核心主题。重点包括向量方法、守恒定律、能量方法以及数学建模。
1. Momentum and Impulse in Two Dimensions | 二维动量与冲量
In Further Mechanics, linear momentum and impulse are vector quantities. For a particle of mass m moving with velocity v, momentum is p = mv. The impulse-momentum principle states that impulse I equals the change in momentum, so I = m(v − u), where u is the initial velocity and v is the final velocity.
在进阶力学中，线动量和冲量都是向量。质量为 m 的质点以速度 v 运动时，动量为 p = mv。冲量-动量原理指出冲量 I 等于动量的变化，即 I = m(v − u)，其中 u 为初速度，v 为末速度。
I = m(v − u)
For a collision between two particles, total momentum is conserved in every direction. In component form, the x-direction gives m₁u₁ₓ + m₂u₂ₓ = m₁v₁ₓ + m₂v₂ₓ, and the y-direction gives a similar equation.
对于两质点的碰撞，总动量在每个方向上守恒。用分量表示时，x 方向给出 m₁u₁ₓ + m₂u₂ₓ = m₁v₁ₓ + m₂v₂ₓ，y 方向给出类似方程。
2. Oblique Collisions and Newton’s Law of Restitution | 斜碰与牛顿恢复系数
When a smooth sphere hits a smooth plane obliquely, the velocity component parallel to the plane is unchanged because there is no friction. The component perpendicular to the plane is reversed and multiplied by the coefficient of restitution e.
光滑球体斜碰光滑平面时，由于没有摩擦，平行于平面的速度分量不变。垂直于平面的速度分量反向并乘以恢复系数 e。
vₙ = −e uₙ and vₜ = uₜ
The coefficient of restitution is defined by e = speed of separation / speed of approach along the line of impact. For two smooth spheres colliding obliquely, use conservation of momentum along the line of centres and the restitution equation, while the perpendicular components remain unchanged.
恢复系数定义为碰撞线方向上分离速度与接近速度之比：e = 分离速度 / 接近速度。两个光滑球体斜碰时，沿球心连线方向使用动量守恒和恢复系数方程，垂直方向分量保持不变。
- If e = 1, the collision is perfectly elastic.
- If e = 0, the collision is perfectly inelastic and the spheres do not separate along the impact line.
若 e = 1，碰撞为完全弹性碰撞。若 e = 0，碰撞为完全非弹性碰撞，两球在碰撞线方向不分离。
3. Centre of Mass of Plane Laminas | 平面薄片的质心
The centre of mass of a uniform plane lamina can be found by taking moments of area. For a composite lamina, treat each part as a point mass at its own centre of mass and use the weighted average formula.
均匀平面薄片的质心可以通过面积矩求得。对于复合薄片，可将每一部分视为位于其自身质心的质点，并使用加权平均公式。
x̄ = Σ Aᵢ xᵢ / Σ Aᵢ and ȳ = Σ Aᵢ yᵢ / Σ Aᵢ
Standard centres of mass are essential. A uniform triangular lamina has its centroid at the intersection of the medians, located 2/3 of the way from each vertex to the midpoint of the opposite side. A uniform semicircular lamina of radius r has its centre of mass at distance 4r / (3π) from the diameter.
标准质心位置非常重要。均匀三角形薄片的质心位于中线的交点，从每个顶点到对边中点距离的 2/3 处。半径为 r 的均匀半圆形薄片质心距直径为 4r / (3π)。
- Uniform rectangular lamina: centre of mass at the geometric centre.
- Uniform solid cone or pyramid: centre of mass lies on the axis, 1/4 of the height from the base.
均匀矩形薄片：质心在几何中心。均匀实心圆锥或棱锥：质心位于轴上，距底面高度的 1/4 处。
4. Equilibrium of Rigid Bodies | 刚体的平衡
A rigid body is in equilibrium when both the resultant force and the resultant moment about any point are zero. These two vector equations give up to three scalar equations in a plane.
刚体平衡的条件是合外力为零，且对任意点的合力矩为零。这两个向量方程在平面内最多给出三个标量方程。
ΣF = 0 and ΣM = 0
When a body rests on a rough surface, friction satisfies F ≤ μR, where μ is the coefficient of friction and R is the normal reaction. Sliding occurs when F = μR. Toppling occurs when the line of action of the weight passes beyond the edge of the base.
当刚体放在粗糙表面上时，摩擦力满足 F ≤ μR，其中 μ 为摩擦系数，R 为法向反力。当 F = μR 时开始滑动。当重力作用线超出底边时，刚体发生倾覆。
5. Work, Energy and Power | 功、能量与功率
Work is done when a force moves its point of application. For a constant force F acting at angle θ to displacement s, work done is W = Fs cos θ. Kinetic energy is ½ mv² and gravitational potential energy is mgh.
力使其作用点移动时做功。恒力 F 与位移 s 成 θ 角时，做功为 W = Fs cos θ。动能为 ½ mv²，重力势能为 mgh。
W = Fs cos θ and KE = ½ mv²
The work-energy principle states that the total work done by all external forces equals the change in kinetic energy plus the change in potential energy. Power is the rate of doing work, P = Fv for a force acting in the direction of motion.
功能原理指出，所有外力所做的总功等于动能变化量与势能变化量之和。功率是做功的速率，当力沿运动方向作用时，P = Fv。
6. Elastic Strings and Springs | 弹性绳与弹簧
An elastic string or spring obeys Hooke’s law within its elastic limit. The tension T is proportional to extension x, so T = kx. For a string of natural length L and modulus of elasticity λ, the stiffness is k = λ / L.
弹性绳或弹簧在弹性限度内服从胡克定律。张力 T 与伸长量 x 成正比，即 T = kx。自然长度为 L、弹性模量为 λ 的绳子，刚度 k = λ / L。
T = (λ / L) x and EPE = λ x² / (2L)
Elastic potential energy stored in an extension x is EPE = λ
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