Work, Energy, and Power

3.1

Translational Kinetic Energy

Syllabus

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.

Source: College Board AP Course and Exam Description

Kinetic energy 动能 is the energy of motion, a scalar 标量 measured in joules 焦耳 (J):

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

It grows with the square of speed – doubling the speed quadruples $K$. One subtlety worth knowing: kinetic energy depends on the observer's reference frame 参考系. A passenger walking down a train has a small $K$ measured inside the train and a huge one measured from the ground – both observers are right, each in their own frame.

Vocabulary Train

English	Chinese	Pinyin
Kinetic energy	动能	dòng néng
scalar	标量	biāo liàng
joules	焦耳	jiāo ěr
reference frame	参考系	cān kǎo xì

Exercise sheet

3.2

Work

Syllabus

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 The most common 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 work done on an object by a variable force is calculated using Equation: $W = \displaystyle\int_a^b \vec{F}(r) \cdot d\vec{r}$, where the integral is taken over the path from point $a$ to point $b$. 3.2.A.3.i The dot product between two vectors, $\vec{A}$ and $\vec{B}$, results in a scalar quantity of magnitude $\vec{A} \cdot \vec{B} = AB\cos\theta$. 3.2.A.3.ii 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. 3.2.A.3.iii If the component of the force exerted on a system that is parallel to the displacement is constant, the work done on the system by the force is given by the derived equation $W = F_{\parallel}d = Fd\cos\theta$. 3.2.A.3.iv 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: $\Delta K = \displaystyle\sum W_i = \sum F_{\parallel,i}\, d_i$ 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 C: Mechanics only expects students to analyze the transfer of mechanical energy, although students should be aware that mechanical energy may be dissipated in the form of thermal energy or sound.

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 The most common 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 work done on an object by a variable force is calculated using
- Equation: $W = \displaystyle\int_a^b \vec{F}(r) \cdot d\vec{r}$, where the integral is taken over the path from point $a$ to point $b$.
- 3.2.A.3.i The dot product between two vectors, $\vec{A}$ and $\vec{B}$, results in a scalar quantity of magnitude $\vec{A} \cdot \vec{B} = AB\cos\theta$.
- 3.2.A.3.ii 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.
- 3.2.A.3.iii If the component of the force exerted on a system that is parallel to the displacement is constant, the work done on the system by the force is given by the derived equation $W = F_{\parallel}d = Fd\cos\theta$.
- 3.2.A.3.iv 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: $\Delta K = \displaystyle\sum W_i = \sum F_{\parallel,i}\, d_i$
- 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 C: Mechanics only expects students to analyze the transfer of mechanical energy, although students should be aware that mechanical energy may be dissipated in the form of thermal energy or sound.

Source: College Board AP Course and Exam Description

Work 功 is energy transferred into or out of a system by a force acting over a distance. For a variable force it is an integral 积分 along the path:

$$W=\int_a^b \vec{F}\cdot d\vec{r}=\int F\cos\theta\,dr.$$

For a constant force this reduces to $W=Fd\cos\theta$; on a force–position graph, work is the area under the curve. Work is a scalar with a sign, and the sign is physics, not bookkeeping:

Positive – force has a component along the motion (it speeds the object up).
Negative – force opposes the motion (friction 摩擦力 and air drag do negative work).
Zero – force perpendicular to the motion (the normal force 法向力 on a sliding block, gravity on a horizontal move, tension in a circular swing).

Only the force component along the displacement does work

The work–energy theorem 动能定理 collects every force's contribution: the net work equals the change in kinetic energy,

$$W_{\text{net}}=\sum W_i=\Delta K.$$

Worked example. A variable force $F(x)=3x^2\ \text{N}$ (along the motion) acts from $x=0$ to $x=2\ \text{m}$: $W=\displaystyle\int_0^2 3x^2\,dx=\big[x^3\big]_0^2=8\ \text{J}$. Acting alone on a body starting from rest, it would raise the kinetic energy to exactly $8\ \text{J}$.

Vocabulary Train

English	Chinese	Pinyin
Work	功	gōng
integral	积分	jī fēn
friction	摩擦力	mó cā lì
normal force	法向力	fǎ xiàng lì
work–energy theorem	动能定理	dòng néng dìng lǐ

Exercise sheet

3.3

Potential Energy

Syllabus

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 relationship between conservative forces exerted on a system and the system's potential energy is Equation: $\Delta U = -\displaystyle\int_a^b \vec{F}_{cf}(r) \cdot d\vec{r}$ 3.3.A.5 The conservative forces exerted on a system in a single dimension can be determined using the slope of the system's potential energy with respect to position in that dimension; these forces point in the direction of decreasing potential energy. Equation: $F_x = -\dfrac{dU(x)}{dx}$ 3.3.A.6 Graphs of a system's potential energy as a function of its position can be useful in determining physical properties of that system. 3.3.A.6.i Stable equilibrium is a location at which a small displacement in an object's position results in a force exerted on the object opposite to the direction of the small displacement, accelerating the object back toward the equilibrium position. 3.3.A.6.ii Unstable equilibrium is a location at which a small displacement in an object's position results in a force exerted on the object in the same direction as the small displacement, accelerating the object away from the equilibrium position. 3.3.A.6.iii In a given dimension, stable equilibrium positions exist at locations where the potential energy as a function of position in that dimension has a local minimum. 3.3.A.6.iv In a given dimension, unstable equilibrium positions occur at locations where the potential energy as a function of position in that dimension has a local maximum. 3.3.A.7 The potential energy of common physical systems can be described using the physical properties of that system. 3.3.A.7.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.7.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.7.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.8 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.

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 relationship between conservative forces exerted on a system and the system's potential energy is
- Equation: $\Delta U = -\displaystyle\int_a^b \vec{F}_{cf}(r) \cdot d\vec{r}$
3.3.A.5 The conservative forces exerted on a system in a single dimension can be determined using the slope of the system's potential energy with respect to position in that dimension; these forces point in the direction of decreasing potential energy.
- Equation: $F_x = -\dfrac{dU(x)}{dx}$
3.3.A.6 Graphs of a system's potential energy as a function of its position can be useful in determining physical properties of that system.
- 3.3.A.6.i Stable equilibrium is a location at which a small displacement in an object's position results in a force exerted on the object opposite to the direction of the small displacement, accelerating the object back toward the equilibrium position.
- 3.3.A.6.ii Unstable equilibrium is a location at which a small displacement in an object's position results in a force exerted on the object in the same direction as the small displacement, accelerating the object away from the equilibrium position.
- 3.3.A.6.iii In a given dimension, stable equilibrium positions exist at locations where the potential energy as a function of position in that dimension has a local minimum.
- 3.3.A.6.iv In a given dimension, unstable equilibrium positions occur at locations where the potential energy as a function of position in that dimension has a local maximum.
3.3.A.7 The potential energy of common physical systems can be described using the physical properties of that system.
- 3.3.A.7.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.7.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.7.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.8 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.

Source: College Board AP Course and Exam Description

Potential energy 势能 is energy a system stores by the positions of its parts – it exists only for conservative forces 保守力, whose work is path independent. It is defined through work:

$$\Delta U=-\int_a^b\vec{F}\cdot d\vec{r},$$

and you are free to choose where $U=0$ – only changes in $U$ matter, so pick the zero that makes the problem simplest. The standard results:

Gravity near a surface: $\Delta U_g=mg\,\Delta y$.
Gravity in general: $U_g=-\dfrac{Gm_1m_2}{r}$ (zero at infinite separation).
Spring: $U_s=\tfrac12k(\Delta x)^2$, with $\Delta x$ measured from natural length.

For systems of several objects, add the potential energy of each pair. Turning the definition around, a conservative force is minus the derivative 导数 of its potential energy:

$$F_x=-\frac{dU}{dx}.$$

The force points "downhill" on the $U(x)$ curve. Equilibrium sits where the slope is zero: a minimum is a stable equilibrium 稳定平衡 (displaced, the force pushes back), a maximum is an unstable equilibrium 不稳定平衡 (displaced, the force pushes away).

On a potential-energy curve the force points downhill and E = U marks the turning points

Worked example. Given $U(x)=2x^3-6x$ (joules), the force is $F=-\dfrac{dU}{dx}=6-6x^2$. Equilibria sit at $F=0$: $x=\pm1$. Since $\dfrac{d^2U}{dx^2}=12x$ is positive at $x=+1$ (a minimum – stable) and negative at $x=-1$ (a maximum – unstable), the two points behave oppositely.

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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.

Vocabulary Train

English	Chinese	Pinyin
Potential energy	势能	shì néng
conservative forces	保守力	bǎo shǒu lì
derivative	导数	dǎo shù
stable equilibrium	稳定平衡	wěn dìng píng héng
unstable equilibrium	不稳定平衡	bù wěn dìng píng héng

Exercise sheet

3.4

Conservation of Energy

Syllabus

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 C: Mechanics expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces.

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 C: Mechanics expects students to know that mechanical energy can be dissipated as thermal energy or sound by nonconservative forces.

Source: College Board AP Course and Exam Description

Energy conservation: KE ⇄ PE

Energy is conserved in all interactions – the question is only where it goes. Mechanical energy 机械能 is the sum $E=K+U$. If the external work on a system is zero and nothing inside it acts through nonconservative forces 非保守力, then

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

When friction or drag act, they convert mechanical energy into thermal energy 热能 ($\Delta E_{\text{mech}}=-F_fd$), and the balance must include that term. If external work is done, the system's total energy changes by exactly the energy transferred: $W_{\text{ext}}=\Delta E_{\text{sys}}$. Choosing the system is choosing the bookkeeping – a single object can only have kinetic energy; include the Earth or the spring and the system can store potential energy too.

A pendulum trades gravitational potential energy for kinetic energy and back

A potential-energy graph is a complete motion map: the horizontal line at height $E$ is the total energy, the gap $E-U(x)$ is the kinetic energy at each $x$, and the crossings $E=U$ are the turning points 转折点 where the object momentarily stops and reverses.

Worked example. A $2.0\ \text{kg}$ block slides down a ramp from rest at height $1.5\ \text{m}$, arriving at the bottom at $4.0\ \text{m/s}$. Energy accounting: $mgh=29.4\ \text{J}$ available; $\tfrac12mv^2=16\ \text{J}$ arrives as kinetic energy; so friction converted $29.4-16=13\ \text{J}$ into thermal energy along the way.

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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.

Vocabulary Train

English	Chinese	Pinyin
Mechanical energy	机械能	jī xiè néng
nonconservative forces	非保守力	fēi bǎo shǒu lì
thermal energy	热能	rè néng
turning points	转折点	zhuǎn zhé diǎn

Exercise sheet

3.5

Power

Syllabus

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 a force is given by the equation Equation: $P_{\text{inst}} = \dfrac{dW}{dt}$ 3.5.A.5 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: $P_{\text{inst}} = F_{\parallel}v = Fv\cos\theta$

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 a force is given by the equation
- Equation: $P_{\text{inst}} = \dfrac{dW}{dt}$
3.5.A.5 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: $P_{\text{inst}} = F_{\parallel}v = Fv\cos\theta$

Source: College Board AP Course and Exam Description

Power 功率 is the rate of energy transfer, in watts 瓦特 (W):

$$P_{\text{avg}}=\frac{\Delta E}{\Delta t}=\frac{W}{\Delta t},\qquad P_{\text{inst}}=\frac{dW}{dt}=\vec{F}\cdot\vec{v}.$$

It measures how fast work is done, not how much. The dot product matters: only the force component along the velocity delivers power.

Power is the slope of the work-time graph: the same work in less time means more power

Worked example. A block released from rest at the top of a frictionless $3.0\ \text{m}$-high ramp reaches the bottom at $v=\sqrt{2gh}=7.7\ \text{m/s}$ (from $mgh=\tfrac12mv^2$). A motor that then drives it at a steady $7.7\ \text{m/s}$ against a $20\ \text{N}$ resistance delivers $P=Fv=20(7.7)\approx150\ \text{W}$.

Worked example. A $1200\ \text{kg}$ car climbs a hill that rises $1.0\ \text{m}$ for every $20\ \text{m}$ of road, at a steady $15\ \text{m/s}$. The engine must supply gravity's power drain: $P=mg\,v\sin\theta=1200(9.8)(15)\big(\tfrac{1}{20}\big)\approx8.8\ \text{kW}$ – before adding air resistance.

Exam skill. Energy FRQs reward the accounting sentence: name your system, state which forces do work on it, and write the balance ($W_{\text{ext}}=\Delta K+\Delta U+\Delta E_{\text{thermal}}$) before plugging in numbers. "Friction is present, so mechanical energy is not conserved" is a scored statement.

Vocabulary Train

English	Chinese	Pinyin
Power	功率	gōng lǜ
watts	瓦特	wǎ tè

Exercise sheet

3.5

Exam tips

Use the work–energy theorem $W_{net}=\Delta KE$ and compute work as $W=\int \vec F\cdot d\vec r$ for a variable force.
Read work off a force–position graph as the area under the curve.
Power is $P=\tfrac{dW}{dt}=\vec F\cdot\vec v$; watch instantaneous vs average.
Split forces into conservative (define a potential energy) and non-conservative (dissipate energy).
Choose energy methods over kinematics when the force varies or the path is complex.

Translational Kinetic Energy

Work

Potential Energy

Store elastic potential energy in a spring

Conservation of Energy

Watch energy convert as an object falls

Power

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