📚 Mastering Maths Mechanics: Forces, Motion and Moments | A-Level 数学力学核心考点精讲
Mechanics is the branch of mathematics that models how objects move under the action of forces. In A-Level Maths, mechanics questions combine algebraic manipulation, trigonometry and vector reasoning with physical principles such as Newton's laws, energy conservation and momentum.
力学是数学中研究物体在力作用下如何运动的分支。在 A-Level 数学中，力学题将代数运算、三角恒等式和向量推理与牛顿定律、能量守恒和动量等物理原理结合起来。
1. Kinematics and the SUVAT Equations | 运动学与 SUVAT 方程
SUVAT equations describe motion with constant acceleration in a straight line. The five quantities are displacement s, initial velocity u, final velocity v, acceleration a and time t.
SUVAT 方程描述匀加速直线运动。五个物理量分别是位移 s、初速度 u、末速度 v、加速度 a 和时间 t。
Memorise the five forms and identify which three quantities are known before choosing the most direct equation.
记住五种形式，并在选择最直接的方程之前，先确定已知哪三个量。
v = u + at
s = ut + ½at²
s = ½(u + v)t
v² = u² + 2as
s = vt − ½at²
Always choose one direction as positive. If a particle is slowing down, a has the opposite sign to u.
始终选择一个方向为正方向。如果物体正在减速，则 a 与 u 的符号相反。
2. Resolving Forces on an Inclined Plane | 斜面上的力分解
For a mass m on a slope at angle θ to the horizontal, the weight mg is resolved into components parallel and perpendicular to the plane.
对于质量为 m、斜面倾角为 θ 的物体，重力 mg 被分解为沿斜面和垂直于斜面的两个分力。
Perpendicular component: mg cos θ
Parallel component: mg sin θ
The normal reaction R balances the perpendicular component when there is no acceleration perpendicular to the plane.
当垂直于斜面方向没有加速度时，法向反力 R 与垂直分力平衡。
he particle is in equilibrium on a rough slope, friction acts up the slope and equals mg sin θ.
对于极限摩擦，使用 F = μR。若物体在粗糙斜面上处于平衡，摩擦力沿斜面向上，大小等于 mg sin θ。
3. Newton's Laws and Connected Particles | 牛顿定律与连接体
Newton's second law states that the resultant force on a particle equals its mass times its acceleration.
牛顿第二定律指出，作用在质点上的合力等于其质量与加速度的乘积。
ΣF = ma
For connected particles, apply the equation of motion to each mass or to the whole system to find the common acceleration.
对于连接体问题，可以对每个物体或对整个系统应用运动方程，求出共同的加速度。
For a particle hanging from a light inextensible string over a smooth pulley
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