Work, Energy and Power功、能量与功率

The formula $W = Fd\cos\theta$ is the scalar (dot) product $\vec{F}\cdot\vec{d}$. For gravity ($\vec{F}_g = -mg\hat{j}$) on an object moving along an incline of angle $\alpha$, the displacement has a vertical component $d\sin\alpha$ downward. Gravity does work $W_g = mgd\sin\alpha$ (positive when descending, negative when ascending). This is why the work done against gravity on a ramp depends only on the vertical height gained $h = d\sin\alpha$, not on the length of the ramp — a result that underpins gravitational potential energy in §3.公式 $W = Fd\cos\theta$ 是标量（点）积 $\vec{F}\cdot\vec{d}$。重力（$\vec{F}_g = -mg\hat{j}$）作用于沿倾角 $\alpha$ 的斜面运动的物体时，位移有向下的竖直分量 $d\sin\alpha$。重力所做的功为 $W_g = mgd\sin\alpha$（下降时为正，上升时为负）。这就是为何在斜面上克服重力做的功只取决于上升的竖直高度 $h = d\sin\alpha$，而非斜面长度 — 这一结果是 §3 重力势能的基础。

Start with Newton's second law: $F_\text{net} = ma$. For constant force over displacement $d$: $W_\text{net} = F_\text{net}\,d$. From kinematics (recall from the Forces unit), $v_f^2 = v_i^2 + 2ad$, so $ad = \tfrac{v_f^2 - v_i^2}{2}$. Multiply both sides by $m$:从牛顿第二定律 $F_\text{net} = ma$ 出发。对位移 $d$ 上的恒力：$W_\text{net} = F_\text{net}\,d$。由运动学（回忆力学单元），$v_f^2 = v_i^2 + 2ad$，故 $ad = \tfrac{v_f^2 - v_i^2}{2}$。两边乘以 $m$：

This derivation shows the work-energy theorem is not a new law but a consequence of $F = ma$ plus kinematics. At AP Physics C level, the derivation generalises to variable forces via the integral $W = \int F\,dx$, where the result $W = \Delta KE$ still holds.这个推导表明，功能定理不是新定律，而是 $F = ma$ 加运动学的推论。在 AP Physics C 层面，推导通过积分 $W = \int F\,dx$ 推广至变力，结果 $W = \Delta KE$ 仍然成立。

A force is conservative if the work it does on an object moving between two points is independent of the path taken. For gravity: $W_\text{gravity} = -\Delta GPE = -(mgh_f - mgh_i)$ regardless of the route from $h_i$ to $h_f$. This path-independence is why we can assign a unique energy value to each position, define a potential energy function, and write a conservation law. Friction, by contrast, depends on path length (more distance $=$ more heat), so it is non-conservative and cannot be described by a potential energy function — that is why §6 treats friction separately.若一个力对物体在两点间做的功与路径无关，则该力是保守力。对重力：$W_\text{gravity} = -\Delta GPE = -(mgh_f - mgh_i)$，无论从 $h_i$ 到 $h_f$ 走哪条路。这种路径无关性使我们能为每个位置赋予唯一的能量值，定义势能函数，并写出守恒定律。相比之下，摩擦力与路径长度有关（走得越远产生越多热），故为非保守力，无法用势能函数描述 — 这正是 §6 单独处理摩擦力的原因。

Start with the work-energy theorem: $W_\text{net} = \Delta KE$. In an isolated system, the only work is done by gravity: $W_g = -\Delta GPE$ (work done by gravity equals the decrease in gravitational PE). Substituting: $-\Delta GPE = \Delta KE$, which rearranges to $\Delta KE + \Delta GPE = 0$, i.e. $\Delta(KE + GPE) = 0$. Since the total change is zero, $KE + GPE$ is constant — mechanical energy is conserved.从功能定理 $W_\text{net} = \Delta KE$ 出发。在孤立系统中，唯一做功的是重力：$W_g = -\Delta GPE$（重力所做的功等于重力势能的减少量）。代入：$-\Delta GPE = \Delta KE$，整理得 $\Delta KE + \Delta GPE = 0$，即 $\Delta(KE + GPE) = 0$。由于总变化量为零，$KE + GPE$ 是常数 — 机械能守恒。

Starting from $P = W/t$ and $W = Fd$ (force parallel to displacement): $P = Fd/t = F \cdot (d/t) = Fv$. This assumes constant force and constant velocity. If force varies, instantaneous power is $P = F(t) \cdot v(t)$, and average power requires integrating work over time — the AP Physics C extension.从 $P = W/t$ 和 $W = Fd$（力平行于位移）出发：$P = Fd/t = F \cdot (d/t) = Fv$。这假设力与速度均为恒量。若力变化，瞬时功率为 $P = F(t) \cdot v(t)$，平均功率需对功关于时间积分 — 这是 AP Physics C 的延伸。

Efficiency: $\eta = W_\text{useful}/W_\text{input} = P_\text{useful}/P_\text{input}$. The "lost" energy goes to heat (friction, air resistance). NGSS HS-PS3-3 frames efficiency as the key design constraint for energy-converting devices.效率：$\eta = W_\text{有用}/W_\text{输入} = P_\text{有用}/P_\text{输入}$。"损失"的能量转化为热能（摩擦、空气阻力）。NGSS HS-PS3-3 把效率视为能量转换装置设计的关键约束。

When friction acts, the work-energy theorem still holds at the molecular scale: the surfaces' atoms gain kinetic energy (random thermal motion). At the macroscopic scale this shows up as a temperature rise in the sliding surfaces. The total energy of the universe (system + surroundings) is unchanged — the first law of thermodynamics. Only the mechanical energy of the sliding object decreases. This distinction (mechanical vs total) is exactly what NGSS HS-PS3-2 means by modelling energy as "a combination of energy associated with motion [KE] and energy associated with relative positions [PE]" at the macroscopic scale, with the remainder showing up as internal (thermal) energy at the microscopic scale.当摩擦力作用时，功能定理在分子尺度上仍然成立：两个表面的原子获得动能（随机热运动）。在宏观尺度上，这表现为滑动表面温度升高。宇宙（系统 + 环境）的总能量不变 — 热力学第一定律。只有滑动物体的机械能减少了。这种区分（机械能 vs 总能量）正是 NGSS HS-PS3-2 所说的"在宏观尺度上把能量建模为与运动相关的能量 [动能] 和与相对位置相关的能量 [势能] 的组合"，其余部分在微观尺度上表现为内能（热能）。

A simple pendulum of length $L$ is released from angle $\theta$ from the vertical. The height above the lowest point is $h = L(1 - \cos\theta)$. At the lowest point, all PE has converted to KE:一根长度为 $L$ 的单摆从与竖直方向成 $\theta$ 角处释放。高于最低点的高度为 $h = L(1 - \cos\theta)$。在最低点，所有势能已转化为动能：

At any intermediate angle $\phi$, the height is $h' = L(1 - \cos\phi)$, and the speed is $v = \sqrt{2g(h - h')} = \sqrt{2gL(\cos\phi - \cos\theta)}$. This tracks the continuous exchange between GPE and KE that NGSS HS-PS3-2 models — a pendulum is an ideal demonstration that mechanical energy is conserved (in the absence of air resistance and pivot friction).在任意中间角度 $\phi$ 处，高度为 $h' = L(1 - \cos\phi)$，速度为 $v = \sqrt{2g(h - h')} = \sqrt{2gL(\cos\phi - \cos\theta)}$。这追踪了 NGSS HS-PS3-2 所建模的重力势能与动能之间的持续交换 — 摆是机械能守恒（在无空气阻力和转轴摩擦的情况下）的理想演示。