Cambridge IGCSE IGCSE & A-Level Sheet preview

Work, Energy, and Power

AP Physics 1 · Topic 3

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Handout Vocabulary
3.1

Translational Kinetic Energy
Syllabus
Learning ObjectiveEssential 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

Energy 能量 is the capacity to do work, measured in joules 焦耳 (J). A moving object has kinetic energy 动能:

$$K=\tfrac{1}{2}mv^2.$$
It depends on the square of the speed, so doubling the speed quadruples the kinetic energy. Kinetic energy is a scalar and is never negative.

Worked example. A $1500\ \text{kg}$ car travels at $20\ \text{m/s}$. Its kinetic energy is $K=\tfrac12\times1500\times20^2=3.0\times10^{5}\ \text{J}=300\ \text{kJ}$. If it speeds up to $40\ \text{m/s}$ (double), the kinetic energy becomes $4\times$ larger, $1200\ \text{kJ}$ – which is why stopping distance grows so fast with speed.

Vocabulary Train
English Chinese Pinyin
Energy 能量 néng liàng
joules 焦耳 jiāo ěr
kinetic energy 动能 dòng néng
Exercise sheet
3.2

Work
Syllabus
Learning ObjectiveEssential 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.

Source: College Board AP Course and Exam Description

Work 功 is energy transferred by a force acting over a displacement:

$$W=F\,d\cos\theta,$$
where $\theta$ is the angle between the force and the displacement. Work is positive when the force has a component along the motion (adds energy), negative when it opposes the motion (removes energy), and zero when the force is perpendicular. On a force–position graph, work is the area under the curve. The work–energy theorem 动能定理 states that the net work equals the change in kinetic energy: $W_{\text{net}}=\Delta K$.

Only the force component along the displacement does work Only the force component along the displacement does work

Worked example. A $2.0\ \text{kg}$ block moving at $3.0\ \text{m/s}$ on a frictionless floor is pushed by a $5.0\ \text{N}$ force over $4.0\ \text{m}$ in the direction of motion. Find its final speed. The net work is $W=Fd=5.0\times4.0=20\ \text{J}$, and by the work–energy theorem $W=\tfrac12 m(v^2-v_0^2)$:

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

Vocabulary Train
English Chinese Pinyin
Work 功 gōng
work–energy theorem 动能定理 dòng néng dìng lǐ
Exercise sheet
3.3

Potential Energy
Syllabus
Learning ObjectiveEssential 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.

Source: College Board AP Course and Exam Description

Potential energy 势能 is stored energy that depends on position or configuration:

  • Gravitational potential energy 重力势能 near the surface: $U_g=mgh$ (height $h$ above a reference level).
  • Elastic potential energy 弹性势能 in a spring: $U_s=\tfrac{1}{2}kx^2$.

Potential energy is defined only for conservative forces 保守力 (gravity, springs), for which the stored energy depends on position, not path. Only changes in potential energy matter, so you may put the zero level wherever is convenient.

The $U_g=mgh$ form only holds near a surface where $g$ is roughly constant. The general form, for two spherical masses a distance $r$ apart, is

$$U_g=-\frac{G m_1 m_2}{r}.$$
It is negative and defined to be zero at infinite separation, so gravitational PE rises toward zero as the masses move apart. $mgh$ is simply its near-surface approximation. (This is the form you need for satellites and escape speed.)

Explore

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
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
Exercise sheet
3.4

Conservation of Energy
Syllabus
Learning ObjectiveEssential 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.

Source: College Board AP Course and Exam Description

Energy conservation: KE ⇄ PE

A roller coaster with tall loops of green and red track against the sky A roller coaster trades energy back and forth: it is highest (most potential energy) at the top and fastest (most kinetic energy) at the bottom

The total mechanical energy 机械能 is $E=K+U$. When only conservative forces do work, mechanical energy is conserved 守恒:

$$K_1+U_1=K_2+U_2.$$
When friction or other non-conservative forces act, they transfer mechanical energy to thermal energy 热能; then the general statement is that total energy (including thermal) is conserved. Energy bar charts are a good way to track where the energy goes.

A swinging pendulum trades gravitational potential energy for kinetic energy and back A swinging pendulum trades gravitational potential energy for kinetic energy and back

Worked example. A ball is released from rest at the top of a frictionless ramp $5.0\ \text{m}$ high. Find its speed at the bottom. All the gravitational potential energy becomes kinetic energy:

$$mgh=\tfrac12 mv^2\;\Rightarrow\;v=\sqrt{2gh}=\sqrt{2\times9.8\times5.0}=9.9\ \text{m/s}.$$
The mass cancels, so every object reaches the same speed – exactly the free-fall result, now got from energy. If instead $30\ \text{J}$ were lost to friction, you would subtract it: $mgh-30=\tfrac12 mv^2$.

Explore

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
total mechanical energy 机械能 jī xiè néng
conserved 守恒 shǒu héng
thermal energy 热能 rè néng
Exercise sheet
3.5

Power
Syllabus
Learning ObjectiveEssential 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: $P_{\text{inst}} = F_{\parallel}v = Fv\cos\theta$

Source: College Board AP Course and Exam Description

Power 功率 is the rate of doing work or transferring energy, measured in watts 瓦特 (W):

$$P=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t},\qquad\text{and instantaneously}\qquad P=Fv.$$
So the same job done faster requires more power. On an energy–time graph, power is the slope.

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

Worked example. A motor lifts a $50\ \text{kg}$ load at a steady $2.0\ \text{m/s}$. Because it moves at constant speed, the lifting force equals the weight, so

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

Real machines waste some energy, so we quote efficiency 效率 – useful output power divided by total input power. If this motor draws $1400\ \text{W}$ of electrical power to deliver $980\ \text{W}$ of useful lifting, its efficiency is $980/1400=0.70$, or $70\%$; the other $30\%$ becomes heat and sound.

Vocabulary Train
English Chinese Pinyin
Power 功率 gōng lǜ
watts 瓦特 wǎ tè
efficiency 效率 xiào lǜ
Exercise sheet
3.5

Exam tips
  • Use $W=Fd\cos\theta$: work is zero when the force is perpendicular to the motion, and negative when it opposes it.
  • Reach for the work–energy theorem ($W_{\text{net}}=\Delta K$) or energy conservation ($K_1+U_1=K_2+U_2$) instead of forces whenever the path is complicated.
  • When friction acts, mechanical energy is not conserved — subtract the energy lost to heat.
  • Remember $K\propto v^2$: doubling the speed quadruples the kinetic energy (and the stopping distance).
  • Use $P=Fv$ for power at a steady speed; at constant velocity the net force is zero but the power is not.

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