功、能与功率

AP 物理 1 · 第 3 主题

训练

讲义词汇表

3.1

平动动能

大纲

Learning Objective	Essential Knowledge
3.1.A Describe the translational kinetic energy of an object in terms of the object's mass and velocity.	3.1.A.1 An object's translational kinetic energy is given by the equation Equation: $K = \dfrac{1}{2}mv^2$ 3.1.A.2 Translational kinetic energy is a scalar quantity. 3.1.A.3 Different observers may measure different values of the translational kinetic energy of an object, depending on the observer's frame of reference.

来源:美国大学理事会 AP 课程与考试说明

能量(energy)是做功的能力,以焦耳(joules)(J)测量。一个移动的物体有动能(kinetic energy):

$$K=\tfrac{1}{2}mv^2.$$

它取决于速率的平方,所以把速率加倍使动能变为四倍。动能是一个标量而从不为负。

Worked example. 一辆 $1500\ \text{kg}$ 的车以 $20\ \text{m/s}$ 行驶。它的动能是 $K=\tfrac12\times1500\times20^2=3.0\times10^{5}\ \text{J}=300\ \text{kJ}$。若它加速到 $40\ \text{m/s}$(两倍),动能变成 $4\times$ 大,$1200\ \text{kJ}$ ——这就是为什么停车距离随速率增长得如此快。

词汇表训练

英文	中文	拼音
Energy	能量	néng liàng
joules	焦耳	jiāo ěr
kinetic energy	动能	dòng néng
Work	功	gōng

练习卷

3.2

功

大纲

Learning Objective

Essential Knowledge

3.2.A
Describe the work done on an object or system by a given force or collection of forces.

3.2.A.1 Work is the amount of energy transferred into or out of a system by a force exerted on that system over a distance.
- 3.2.A.1.i The work done by a conservative force exerted on a system is path-independent and only depends on the initial and final configurations of that system.
- 3.2.A.1.ii The work done by a conservative force on a system—or the change in the potential energy of the system—will be zero if the system returns to its initial configuration.
- 3.2.A.1.iii Potential energies are associated only with conservative forces.
- 3.2.A.1.iv The work done by a nonconservative force is path-dependent.
- 3.2.A.1.v Examples of nonconservative forces are friction and air resistance.
3.2.A.2 Work is a scalar quantity that may be positive, negative, or zero.
3.2.A.3 The amount of work done on a system by a constant force is related to the components of that force and the displacement of the point at which that force is exerted.
- 3.2.A.3.i Only the component of the force exerted on a system that is parallel to the displacement of the point of application of the force will change the system's total energy.
  - Equation: $W = F_{\parallel}d = Fd\cos\theta$
- 3.2.A.3.ii The component of the force exerted on a system perpendicular to the direction of the displacement of the system's center of mass can change the direction of the system's motion without changing the system's kinetic energy.
3.2.A.4 The work-energy theorem states that the change in an object's kinetic energy is equal to the sum of the work (net work) being done by all forces exerted on the object.
- Equation: $\Delta K = \sum_{i} W_i = \sum_{i} F_{\parallel,i}\,d$
- 3.2.A.4.i An external force may change the configuration of a system. The component of the external force parallel to the displacement times the displacement of the point of application of the force gives the change in kinetic energy of the system.
- 3.2.A.4.ii If the system's center of mass and the point of application of the force move the same distance when a force is exerted on a system, then the system may be modeled as an object, and only the system's kinetic energy can change.
- 3.2.A.4.iii The energy dissipated by friction is typically equated to the force of friction times the length of the path over which the force is exerted
  - Equation: $\Delta E_{\text{mech}} = F_f\,d\cos\theta$
3.2.A.5 Work is equal to the area under the curve of a graph of $F_{\parallel}$ as a function of displacement.

Boundary statement: AP Physics 1 only expects students to analyze the transfer of mechanical energy (as defined in Unit 3, Topic 4: Conservation of Energy), although students should be aware that mechanical energy may be dissipated in the form of thermal energy or sound. In AP Physics 2, students will also study how thermal energy can be transferred between systems through heating or cooling.

来源:美国大学理事会 AP 课程与考试说明

功(work)是一个力作用在一个位移上转移的能量:

$$W=F\,d\cos\theta,$$

其中 $\theta$ 是力和位移之间的角。功在力有一个沿运动的分量时是正的(添加能量)、在它反对运动时是负的(移除能量),而在力垂直时是零。在一个力-位置图上,功是曲线下的面积。动能定理(work–energy theorem)陈述合功等于动能的变化:$W_{\text{net}}=\Delta K$。

只有沿位移的力分量做功

Worked example. 一个 $2.0\ \text{kg}$ 的木块在一个无摩擦地板上以 $3.0\ \text{m/s}$ 移动,被一个 $5.0\ \text{N}$ 的力沿运动方向推 $4.0\ \text{m}$。求它的最终速率。合功是 $W=Fd=5.0\times4.0=20\ \text{J}$,而由动能定理 $W=\tfrac12 m(v^2-v_0^2)$:

$$20=\tfrac12\times2.0\times(v^2-3.0^2)\;\Rightarrow\;v^2=29\;\Rightarrow\;v=5.4\ \text{m/s}.$$

词汇表训练

英文	中文	拼音
work–energy theorem	动能定理	dòng néng dìng lǐ

练习卷

3.3

势能

大纲

Learning Objective

Essential Knowledge

3.3.A
Describe the potential energy of a system.

3.3.A.1 A system composed of two or more objects has potential energy if the objects within that system only interact with each other through conservative forces.
3.3.A.2 Potential energy is a scalar quantity associated with the position of objects within a system.
3.3.A.3 The definition of zero potential energy for a given system is a decision made by the observer considering the situation to simplify or otherwise assist in analysis.
3.3.A.4 The potential energy of common physical systems can be described using the physical properties of that system.
- 3.3.A.4.i The elastic potential energy of an ideal spring is given by the following equation, where $\Delta x$ is the distance the spring has been stretched or compressed from its equilibrium length.
  - Equation: $U_s = \dfrac{1}{2}k(\Delta x)^2$
- 3.3.A.4.ii The general form for the gravitational potential energy of a system consisting of two approximately spherical distributions of mass (e.g., moons, planets or stars) is given by the equation
  - Equation: $U_g = -G\dfrac{m_1 m_2}{r}$
- 3.3.A.4.iii Because the gravitational field near the surface of a planet is nearly constant, the change in gravitational potential energy in a system consisting of an object with mass $m$ and a planet with gravitational field of magnitude $g$ when the object is near the surface of the planet may be approximated by the equation
  - Equation: $\Delta U_g = mg\Delta y$
3.3.A.5 The total potential energy of a system containing more than two objects is the sum of the potential energy of each pair of objects within the system.

来源:美国大学理事会 AP 课程与考试说明

势能(potential energy)是取决于位置或构型的储存的能量:

表面附近的重力势能(gravitational potential energy):$U_g=mgh$(一个参考水平之上的高度 $h$)。
一个弹簧里的弹性势能(elastic potential energy):$U_s=\tfrac{1}{2}kx^2$。

势能只对保守力(conservative forces)(重力、弹簧)定义,对它们储存的能量取决于位置,不是路径。只有势能的变化重要,所以你可以把零水平放在任何方便的地方。

$U_g=mgh$ 的形式只在 $g$ 大致恒定的表面附近成立。一般形式,对相距 $r$ 的两个球形质量,是

$$U_g=-\frac{G m_1 m_2}{r}.$$

它是负的,并被定义为在无穷远处为零,所以引力势能随质量分开而向零上升。$mgh$ 只是它在表面附近的近似。(这是你处理卫星和逃逸速度所需的形式。)

探索

Store elastic potential energy in a spring

Stretching a spring stores elastic potential energy $\tfrac12 kx^2$ — the area under the force-extension line. Release it and that energy becomes kinetic.

词汇表训练

英文	中文	拼音
Potential energy	势能	shì néng
Gravitational potential energy	重力势能	zhòng lì shì néng
Elastic potential energy	弹性势能	tán xìng shì néng
conservative forces	保守力	bǎo shǒu lì

练习卷

3.4

能量守恒

大纲

Learning Objective

Essential Knowledge

3.4.A
Describe the energies present in a system.

3.4.A.1 A system composed of only a single object can only have kinetic energy.
3.4.A.2 A system that contains objects that interact via conservative forces or that can change its shape reversibly may have both kinetic and potential energies.

3.4.B
Describe the behavior of a system using conservation of mechanical energy principles.

3.4.B.1 Mechanical energy is the sum of a system's kinetic and potential energies.
3.4.B.2 Any change to a type of energy within a system must be balanced by an equivalent change of other types of energies within the system or by a transfer of energy between the system and its surroundings.
3.4.B.3 A system may be selected so that the total energy of that system is constant.
3.4.B.4 If the total energy of a system changes, that change will be equivalent to the energy transferred into or out of the system.

3.4.C
Describe how the selection of a system determines whether the energy of that system changes.

3.4.C.1 Energy is conserved in all interactions.
3.4.C.2 If the work done on a selected system is zero and there are no nonconservative interactions within the system, the total mechanical energy of the system is constant.
3.4.C.3 If the work done on a selected system is nonzero, energy is transferred between the system and the environment.

Boundary statement: AP Physics 1 expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces.

来源:美国大学理事会 AP 课程与考试说明

能量守恒：动能↔势能

一座过山车,绿色和红色的高大轨道环立在天空下 — 过山车来回转换能量:它在最高点时势能最大,在最低点时速度最快(动能最大)

机械能(total mechanical energy)是 $E=K+U$。当只有保守力做功时,机械能守恒(conserved):

$$K_1+U_1=K_2+U_2.$$

当摩擦或其他非保守力作用时,它们把机械能转移到热能(thermal energy);那么一般的陈述是总能量(包括热能)守恒。能量条形图是追踪能量去哪里的一个好方式。

一个摆动的摆把重力势能换成动能并换回

Worked example. 一个球从一个 $5.0\ \text{m}$ 高的无摩擦坡道顶部从静止释放。求它在底部的速率。所有的重力势能变成动能:

$$mgh=\tfrac12 mv^2\;\Rightarrow\;v=\sqrt{2gh}=\sqrt{2\times9.8\times5.0}=9.9\ \text{m/s}.$$

质量约去,所以每个物体达到相同的速率——恰好是自由落体的结果,现在从能量得到。若相反 $30\ \text{J}$ 损失给摩擦,你会减去它:$mgh-30=\tfrac12 mv^2$。

探索

Watch energy convert as an object falls

With no friction, mechanical energy is conserved: as an object falls, gravitational potential energy turns into kinetic energy while the total stays fixed.

词汇表训练

英文	中文	拼音
total mechanical energy	机械能	jī xiè néng
conserved	守恒	shǒu héng
thermal energy	热能	rè néng

练习卷

3.5

功率

大纲

Learning Objective

Essential Knowledge

3.5.A
Describe the transfer of energy into, out of, or within a system in terms of power.

3.5.A.1 Power is the rate at which energy changes with respect to time, either by transfer into or out of a system or by conversion from one type to another within a system.
3.5.A.2 Average power is the amount of energy being transferred or converted, divided by the time it took for that transfer or conversion to occur.
- Equation: $P_{\text{avg}} = \dfrac{\Delta E}{\Delta t}$
3.5.A.3 Because work is the change in energy of an object or system due to a force, average power is the total work done, divided by the time during which that work was done.
- Equation: $P_{\text{avg}} = \dfrac{W}{\Delta t}$
3.5.A.4 The instantaneous power delivered to an object by the component of a constant force parallel to the object's velocity can be described with the derived equation.
- Equation: $P_{\text{inst}} = F_{\parallel}v = Fv\cos\theta$

来源:美国大学理事会 AP 课程与考试说明

功率(power)是做功或转移能量的速率,以瓦特(watts)(W)测量:

$$P=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t},\qquad\text{and instantaneously}\qquad P=Fv.$$

所以同样的工作更快地完成需要更多功率。在一个能量-时间图上,功率是斜率。

功率是功-时间图的斜率:同样的功在更少时间里意味着更多功率

Worked example. 一个马达以稳定的 $2.0\ \text{m/s}$ 提升一个 $50\ \text{kg}$ 的负载。因为它以恒定速率移动,提升力等于重量,所以

$$P=Fv=mgv=50\times9.8\times2.0=980\ \text{W}.$$

真实的机器浪费一些能量,所以我们引用效率(efficiency)——有用的输出功率除以总输入功率。若这个马达抽取 $1400\ \text{W}$ 的电功率来传递 $980\ \text{W}$ 的有用提升,它的效率是 $980/1400=0.70$,或 $70\%$;另外的 $30\%$ 变成热和声音。

A waterfall: power is how quickly energy is transferred — the same drop in less time is more power

词汇表训练

英文	中文	拼音
Power	功率	gōng lǜ
watts	瓦特	wǎ tè
efficiency	效率	xiào lǜ

练习卷

3.5

考试技巧

用 $W=Fd\cos\theta$:功在力垂直于运动时是零,而在它反对时是负的。
每当路径复杂时,伸手去拿动能定理($W_{\text{net}}=\Delta K$)或能量守恒($K_1+U_1=K_2+U_2$)而不是力。
当摩擦作用时,机械能不守恒——减去损失给热的能量。
记住 $K\propto v^2$:把速率加倍使动能(和停车距离)变为四倍。
对稳定速率的功率用 $P=Fv$;在恒定速度合力是零但功率不是。

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