Work, energy and power · 功、能量与功率
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| work done/wɜːk dʌn/ | 功 | gōng |
| displacement/dɪˈspleɪsmənt/ | 位移 | wèiyí |
| conservation of energy/ˌkɒnsəˈveɪʃn ɒv ˈenədʒi/ | 能量守恒 | néng liàng shǒu héng |
| efficiency/ɪˈfɪʃənsi/ | 效率 | xiào lǜ |
| thermal energy/ˈθɜːml ˈenədʒi/ | 热能 | rè néng |
| power/ˈpaʊə/ | 功率 | gōnglǜ |
Holding a bag does no work
- Hold a heavy bag still and your arm aches — but in physics you do zero work.
- Work needs the force to move something along its own direction.
- No movement (or a sideways force) means no work done 功.
举着一个袋子不做功
- 静止举着一个重袋子,你的手臂会酸——但在物理里你做了 零功。
- 做功需要力 沿它自己的方向 移动某物。
- 没有移动(或力是横向的)就没有做功。
Work done by a force
- $W = F s \cos\theta$, where $\theta$ is the angle between force and displacement 位移.
- Only the part of the force along the motion does work. Unit: $\text{J} = \text{N}\cdot\text{m}$.
力做的功
- $W = F s \cos\theta$,其中 $\theta$ 是力与位移之间的夹角。
- 只有力中 沿 运动方向的部分做功。单位:$\text{J} = \text{N}\cdot\text{m}$。
Work, energy & power · 功、能量与功率
PE + KE = constant
Work transfers energy · 能量; as it falls, PE becomes KE with the total fixed. · 做功转移 能量;下落时势能变成动能,总量不变。
A force of $10\ \text{N}$ pushes a box $3.0\ \text{m}$ in the direction of the force. How much work is done? · 一个 $10\ \text{N}$ 的力沿力的方向把箱子推动 $3.0\ \text{m}$。做了多少功?
Force is along the motion ($\theta = 0$), so $W = Fs = 10 \times 3.0 = 30\ \text{J}$. · 力沿运动方向($\theta = 0$),所以 $W = Fs = 10 \times 3.0 = 30\ \text{J}$。
Positive, negative, zero
- Force along the motion → positive work (energy given to the object).
- Force opposite the motion → negative work (e.g. friction takes energy away).
- Force at right angles → zero work.
Only the component F cos(theta) along the displacement does work
正功、负功、零功
- 力 沿 运动方向 → 正功(把能量给了物体)。
- 力 反 运动方向 → 负功(例如摩擦带走能量)。
- 力 成直角 → 零功。
只有沿位移的分量 F cos(θ) 做功
A force acting at right angles to the motion does how much work? · 与运动方向成直角的力做多少功?
$W = Fs\cos 90^{\circ} = 0$. The normal contact force on a car on a flat road does no work. · $W = Fs\cos 90^{\circ} = 0$。平路上汽车所受的法向接触力不做功。
Friction on a sliding object does negative work. · 作用在滑动物体上的摩擦力做负功。
Friction points opposite to the motion ($\theta = 180^{\circ}$), so $W = -Fs$ — it takes energy away. · 摩擦力方向与运动相反($\theta = 180^{\circ}$),所以 $W = -Fs$——它带走能量。
Conservation of energy 能量守恒
- Energy is never made or destroyed — it only changes form or moves between objects.
- In a closed system the total energy stays constant.
Positive work (force along motion) and negative work (force opposite motion)
能量守恒
- 能量既不会被创造也不会被消灭——它只会 改变形式 或在物体之间转移。
- 在封闭系统中,总能量保持不变。
正功(力沿运动方向)和负功(力反运动方向)
Energy cannot be created or destroyed, only ____ from one form to another. · 能量不能被创造或消灭,只能从一种形式 ____ 到另一种形式。
That is conservation of energy — the total in a closed system stays the same. · 这就是能量守恒——封闭系统中的总量保持不变。
Match each term to the definition the examiner marks. · 把每个术语与评分认可的定义配对。
The work definition's last phrase is what makes a force at right angles to the motion do no work at all. · 功的定义里最后那半句,正是使垂直于运动方向的力完全不做功的原因。
Efficiency 效率
- $\text{efficiency} = \dfrac{\text{useful output}}{\text{total input}} \times 100\%$.
- It is always below $100\%$ — some energy ends up as "useless" thermal energy 热能.
The same work done in less time means more power
效率
- 效率 = 有用输出 ÷ 总输入 × 100%。
- 它总是低于 100%——一部分能量最终变成“无用的”热能。
Energy flow & efficiency · 能量流动与效率
The input energy divides into useful work and wasted energy; efficiency = useful ÷ input, and useful + wasted always equals the input. · 输入能量分成有用功和浪费的能量;效率 = 有用 ÷ 输入,而有用 + 浪费始终等于输入。
A machine takes in $200\ \text{J}$ and gives out $60\ \text{J}$ of useful energy. What is its efficiency (in %)? · 一台机器输入 $200\ \text{J}$,输出 $60\ \text{J}$ 的有用能量。它的效率是多少(用 %)?
$\dfrac{60}{200} \times 100\% = 30\%$. The other $140\ \text{J}$ is wasted, mostly as heat. · $\dfrac{60}{200} \times 100\% = 30\%$。另外的 $140\ \text{J}$ 被浪费,主要变成热。
Power 功率
- Power is the rate of doing work: $P = \dfrac{W}{t} = \dfrac{\Delta E}{\Delta t}$.
- Unit: $\text{W} = \dfrac{\text{J}}{\text{s}}$.
功率
- 功率(power) 是做功的快慢:$P = \dfrac{W}{t} = \dfrac{\Delta E}{\Delta t}$。
- 单位:$\text{W} = \dfrac{\text{J}}{\text{s}}$。
Power, force and velocity
- For a force $F$ along the motion at speed $v$: $P = Fv$.
- A car at steady speed needs $P = F_{\text{resist}} \times v$; lifting a weight at speed $v$ needs $P = mgv$.
功率、力和速度
- 对于沿运动方向、以速率 $v$ 作用的力 $F$:$P = Fv$。
- 匀速行驶的汽车需要 $P = F_{\text{resist}} \times v$;以速率 $v$ 提升一个重物需要 $P = mgv$。
A force $F$ acts on an object moving at speed $v$ in the direction of the force. The power delivered is: · 一个力 $F$ 作用在以速率 $v$ 沿力的方向运动的物体上。输出的功率是:
In time $\Delta t$ the displacement is $v\Delta t$ and the work is $Fv\Delta t$; dividing by $\Delta t$ gives $P = Fv$. · 在时间 $\Delta t$ 内位移是 $v\Delta t$,做的功是 $Fv\Delta t$;除以 $\Delta t$ 得到 $P = Fv$。
Worked example: a car at steady speed
- A car travels at a steady $25\ \text{m/s}$ against a total resistive force of $600\ \text{N}$. Find the useful output power of its engine.
- At steady speed the resultant force is zero, so the driving force equals the resistive force: $F = 600\ \text{N}$.
- $P = Fv = 600 \times 25 = 1.5\times10^{4}\ \text{W} = 15\ \text{kW}$.
- The step that carries the mark is stating why the driving force equals $600\ \text{N}$: constant velocity means zero resultant force.
- If the car were accelerating, the driving force would be larger than $600\ \text{N}$ and you could not use the resistive force in $P = Fv$ at all.
例题:匀速行驶的汽车
- 一辆汽车在总阻力 $600\ \text{N}$ 下以恒定的 $25\ \text{m/s}$ 行驶。求发动机的有用输出功率。
- 匀速时合力为零,所以牵引力等于阻力:$F = 600\ \text{N}$。
- $P = Fv = 600 \times 25 = 1.5\times10^{4}\ \text{W} = 15\ \text{kW}$。
- 带分的那一步是说出牵引力为什么等于 $600\ \text{N}$:匀速意味着合力为零。
- 如果车在加速,牵引力就会大于 $600\ \text{N}$,那时根本不能把阻力代进 $P = Fv$。
Marks that slip away
- Efficiency is useful output divided by total input. The input is always the larger number, so an answer above $100\%$ means the two were swapped.
- In $P = Fv$, $F$ is the driving force. It equals the resistive force only at constant speed.
- Use the vertical height in $mg\Delta h$, never the distance along a slope.
- Work done by a force is the force times the distance moved in the direction of the force. A force at right angles to the motion does no work.
- On a force-extension graph the area is the energy stored. The gradient is the stiffness, and reading one for the other is a whole question lost.
容易丢掉的分
- **效率是有用输出除以总输入。**输入总是较大的那个数,所以答案超过 $100\%$ 就说明两者弄反了。
- $P = Fv$ 中的 $F$ 是牵引力。只有在匀速时它才等于阻力。
- $mg\Delta h$ 中用的是竖直高度,绝不是沿斜面的距离。
- 力做的功是力乘以沿力的方向移动的距离。垂直于运动方向的力不做功。
- 力-伸长图上,面积是储存的能量。斜率是劲度,把两者读反就整道题都丢了。
A car travels at a steady 25 m/s against a total resistive force of 600 N. What is the useful output power, in kW? · 一辆汽车在总阻力 600 N 下以恒定的 25 m/s 行驶。有用输出功率是多少 kW?
Steady speed means zero resultant force, so the driving force is 600 N and P = Fv = 15 kW. Saying WHY the driving force equals the resistance is what carries the mark. · 匀速意味着合力为零,所以牵引力是 600 N,P = Fv = 15 kW。说出牵引力为什么等于阻力,才是带分的那一步。
In which cases does a force do no work? Select all · 所有 that apply. · 哪些情形下力不做功?选出所有适用的。
Braking does plenty of work, just negative work. The orbit is the neat case: gravity is always perpendicular to the motion, so the satellite's speed never changes. · 刹车做了很多功,只不过是负功。轨道那一条最漂亮:引力始终垂直于运动方向,所以卫星的速率从不改变。
Put an efficiency calculation in order. · 把效率的计算按顺序排列。
The input is always the larger number. An efficiency above 100% means the two were swapped, and dividing by the wasted energy is the other common wrong route. · 输入总是较大的那个数。效率超过 100% 说明两者弄反了,而除以浪费掉的能量是另一条常见的错路。
You've got it
- work $W = Fs\cos\theta$ — only the part of $F$ along the motion counts
- energy is conserved; efficiency = useful ÷ total (always < 100%)
- power $P = \dfrac{W}{t} = Fv$
你掌握了
- 功 $W = Fs\cos\theta$——只有 $F$ 沿运动方向的部分算数
- 能量 守恒;效率 = 有用 ÷ 总(总是 < 100%)
- 功率 $P = \dfrac{W}{t} = Fv$