IB Physics: Derivation of the Force and Energy Conservation Law | IB物理：力与能量守恒定律的公式推导

📚 IB Physics: Derivation of the Force and Energy Conservation Law | IB物理：力与能量守恒定律的公式推导

In IB Physics, understanding the relationship between force and energy is fundamental. The principle of conservation of energy, together with Newton’s laws, allows us to derive powerful equations that describe motion without directly solving for forces. This article systematically derives the key formulas connecting work, kinetic energy, potential energy, and the conservation of mechanical energy, using rigorous yet accessible steps aligned with the IB syllabus.

在IB物理中，理解力与能量的关系是基础。能量守恒定律与牛顿定律相结合，使我们能够推导出描述运动的强大方程，而无需直接求解力。本文系统性地推导了功、动能、势能和机械能守恒的关键公式，采用与IB教学大纲一致且严谨易懂的步骤。

1. The Work-Energy Connection | 功与能的关系

Work is defined as the transfer of energy by a force acting over a displacement. It bridges the gap between force and energy, forming the basis for the work-energy theorem.

功定义为一个力在位移上所做的能量转移。它架起了力与能量之间的桥梁，构成了动能定理的基础。

In the SI system, work is measured in joules (J), where 1 J = 1 N·m. Understanding work is essential for deriving energy conservation laws.

在国际单位制中，功以焦耳 (J) 为单位，1 J = 1 N·m。理解功对于推导能量守恒定律至关重要。

2. Work Done by a Constant Force | 恒力做功

A constant force F acting on an object along a straight displacement d does work given by:

一个恒力F沿直线位移d对物体所做的功由下式给出：

W = F d cos θ

where θ is the angle between the force vector and the displacement vector. When force and displacement are in the same direction (θ = 0°), cos 0° = 1, so W = F d. When they are perpendicular (θ = 90°), cos 90° = 0, so no work is done.

其中θ是力矢量与位移矢量之间的夹角。当力与位移方向相同时 (θ = 0°)，cos 0° = 1，因此 W = F d。当它们垂直时 (θ = 90°)，cos 90° = 0，因而不做功。

Work can be positive, negative, or zero. Positive work adds energy to the system; negative work (e.g., friction) removes energy.

功可以为正、负或零。正功向系统输入能量；负功（如摩擦力）将能量从系统中移除。

3. Work Done by a Variable Force | 变力做功

When the force is not constant, we must break the displacement into infinitesimal segments. The total work is the sum of the work over each small segment Δx:

当力不是恒力时，必须将位移分割为无限小段。总功是每一小段Δx上所做功的总和：

W ≈ Σ F_x Δx

In the limit as Δx → 0, this becomes an integral: W = ∫_x₁^x₂ F_x dx. Graphically, work is the area under the force-position curve.

当Δx → 0时，这成为一个积分：W = ∫_x₁^x₂ F_x dx。在图像上，功就是力-位置曲线下的面积。

For a spring force obeying Hooke’s law (F = -k x), this integral yields the expression for elastic potential energy, derived later.

对于遵循胡克定律 (F = -k x) 的弹簧力，该积分可以得出弹性势能的表达式，稍后推导。

4. Kinetic Energy and the Work-Energy Theorem | 动能与动能定理

The net work done on a particle equals its change in kinetic energy. This is the work-energy theorem:

作用在质点上的净功等于其动能的变化量。这就是动能定理：

W_net = ΔK = K₂ – K₁

Kinetic energy K is defined as ½ m v². We can derive the theorem from Newton’s second law for a constant net force along the direction of motion:

动能K定义为½ m v²。我们可以从牛顿第二定律推导出该定理，假设恒定净力沿运动方向：

F = m a, and for constant acceleration, v₂² = v₁² + 2 a d. Solving for a d = (v₂² – v₁²) / 2, then W = F d = m a d = m (v₂² – v₁²) / 2 = ½ m v₂² – ½ m v₁² = ΔK.

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