Work and Energy
| サイト: | Young Education |
| コース: | Work Energy Power |
| ブック: | Work and Energy |
| 印刷者: | Guest user |
| 日付: | 2026年 11月 9日(月曜日) 16:41 |
1. Energy Stores
Learning outcomes
- I can identify and describe the main energy stores, including kinetic, gravitational potential, chemical, elastic, nuclear, electrostatic, and internal (thermal) energy stores.
- I can explain how energy can be transferred between stores through mechanical work, electrical currents, heating, and waves.
- I can distinguish between energy stores and energy transfer pathways.
- I can interpret and construct simple energy flow diagrams showing energy transfers within a system.
- I can apply the principle of conservation of energy to explain how energy is transformed and transferred between stores.
What Is Energy?
Energy is a quantity that can be stored and transferred.
When something happens—an object moves, a battery powers a device, water boils, or a ball falls—energy is being transferred between energy stores.
A useful way to think about energy is:
Energy starts in a store → is transferred → ends in another storeEnergy is measured in joules (J).
Importantly, energy does not disappear when it is used. It is transferred from one store to another.
The Main Energy Stores
There are several important ways in which energy can be stored within a system.
Kinetic Energy Store
A moving object has energy in its kinetic energy store.
The faster an object moves, the greater its kinetic energy.
Examples include:
- A moving car
- A running person
- A spinning fan
- A flying football
- Water flowing in a river
An object that is not moving has no kinetic energy.
Gravitational Potential Energy Store
An object can have energy in a gravitational potential energy store because of its position in a gravitational field.
Near Earth's surface, raising an object increases its gravitational potential energy.
For example, a book on a high shelf has more gravitational potential energy than the same book on the floor.
Examples include:
- Water stored behind a dam
- A person at the top of a diving board
- A roller coaster at the top of a hill
- A rock held above the ground
When these objects fall, energy can be transferred from the gravitational potential store to the kinetic store.
Chemical Energy Store
Energy can be stored chemically in substances.
Examples include:
- Food
- Fuels
- Batteries
- Wood
- Natural gas
When chemical reactions occur, energy can be transferred from the chemical energy store to other stores.
For example, when fuel burns in a car engine, energy from the chemical store of the fuel is transferred to other stores, including the kinetic store of the car and the internal energy stores of the engine and surroundings.
Elastic Energy Store
Energy can be stored when an elastic object is stretched or compressed.
This is called an elastic energy store.
Examples include:
- A stretched elastic band
- A compressed spring
- A drawn bow
- A stretched trampoline
When the object returns toward its original shape, energy can be transferred from its elastic store.
For example:
Nuclear Energy Store
Energy is stored within atomic nuclei.
This is called a nuclear energy store.
Energy can be transferred from nuclear stores during processes such as:
- Nuclear fission – splitting large nuclei
- Nuclear fusion – joining small nuclei
- Radioactive decay
The Sun transfers enormous amounts of energy from nuclear stores through nuclear fusion.
Electrostatic Energy Store
Energy can be stored when electric charges attract or repel each other.
This is called an electrostatic energy store.
For example, rubbing a balloon against hair can cause electric charge to build up. The charged balloon and nearby objects form an electrostatic system in which energy can be stored.
Electrostatic stores become especially important when studying electric fields and charged particles.
Internal (Thermal) Energy Store
Objects contain energy because of the movement and arrangement of their particles.
This is called the internal energy store.
When an object's temperature increases, its internal energy usually increases.
For example:
The hot water has a greater internal energy store.
The term thermal energy is often used informally, but in the energy-stores model we normally refer to an object's internal energy store.
Energy Stores vs Energy Transfers
This distinction is extremely important.
An energy store describes where energy is held.
An energy transfer pathway describes how energy moves from one store to another.
For example, heating is not an energy store. Heating is a way of transferring energy.
Similarly, light is not an energy store. Electromagnetic waves can transfer energy.
Think of it as:
STORE → TRANSFER PATHWAY → STOREThe Four Main Energy Transfer Pathways
Energy can be transferred between stores in four main ways.
| Transfer Pathway. | Description | Example |
|---|---|---|
| Mechanical work | A force moves an object | Lifting a box |
| Electrical work | Charges move through a circuit | Battery powering a motor |
| Heating | Energy transfers because of a temperature difference | Heating water |
| Waves | Energy is carried by waves | Sunlight reaching Earth |
Let's examine each one.
Mechanical Work
Energy is transferred mechanically when a force causes an object to move.
Imagine lifting a box from the floor.
Energy is transferred from the chemical store in your body to the gravitational potential store of the box.
Gravitational potential storeMechanical energy transfer also occurs when:
- A car accelerates.
- A person pushes a trolley.
- A falling object accelerates.
- A spring launches an object.
Electrical Work
Energy can be transferred when an electric current flows.
For example, consider a battery connected to a lamp.
Some energy is also transferred away from the lamp by electromagnetic waves, including visible light and infrared radiation.
Heating
Energy can be transferred because there is a temperature difference.
Energy transfers from hotter regions toward colder regions.
For example, when a metal spoon is placed in hot soup, energy is transferred from the soup to the spoon by heating.
The internal energy store of the spoon increases.
Waves
Waves can transfer energy from one place to another.
Examples include:
- Visible light
- Infrared radiation
- Sound waves
- Microwaves
For example, energy from the Sun reaches Earth mainly through electromagnetic waves.
A speaker transfers energy to its surroundings through sound waves.
Remember:
Waves are a transfer pathway, not an energy store.
Energy Flow Diagrams
An energy flow diagram shows where energy starts, how it is transferred, and where it ends up.
Consider a falling ball.
Initially, the ball has energy in its gravitational potential store.
As it falls:
Gravitational potential store → Kinetic storeThe gravitational potential energy decreases while the kinetic energy increases.
When the ball hits the ground, some of its kinetic energy is transferred to the internal energy stores of the ball and surroundings and some is transferred away by sound waves.
A more complete diagram might therefore be:
Dissipated energy usually spreads into the internal energy stores of the surroundings.
It has not been destroyed. It has simply become less useful for doing the task we wanted.
Conservation of Energy
One of the most important principles in physics is the principle of conservation of energy.
It states:
Energy cannot be created or destroyed.Energy can only be:
- Transferred between stores.
- Transferred between objects.
- Spread into the surroundings.
Therefore, in a closed system:
Total energy before = Total energy afterWorked Example: A Roller Coaster
Imagine a roller coaster at the top of a hill.
At the top
The roller coaster has a large amount of energy in its:
gravitational potential storeMoving downhill
As the roller coaster falls, energy is transferred:
The roller coaster speeds up.
At the bottom
Its gravitational potential energy has decreased while its kinetic energy has increased.
Some energy has also been transferred to the internal energy stores of the wheels, track, air, and surroundings because of friction and air resistance.
The total energy is still conserved.
Worked Example: A Bouncing Ball
A ball is held above the ground.
Initially:
Energy in gravitational potential storeThe ball is released:
The ball hits the ground and deforms:
The ball springs upward:
The ball does not normally return to its original height because some energy is transferred to the internal energy stores of the ball, floor, and surroundings and some is transferred by sound waves.
Energy has not disappeared.
Common Mistakes
Mistake 1: Calling heat an energy store
Instead of:
The object contains heat energy.
It is better to say:
The object has energy in its internal energy store.
Heating describes a transfer pathway.
Mistake 2: Saying energy is used up
Energy is not destroyed.
Instead of:
The car uses up energy.
Say:
Energy is transferred between stores and some is dissipated to the surroundings.
Mistake 3: Calling light an energy store
Light transfers energy through electromagnetic waves.
Therefore, waves are a transfer pathway, not an energy store.
Key Vocabulary
Energy store – A way in which energy is stored within a system.
Energy transfer – The movement of energy from one store or system to another.
Kinetic energy store – Energy associated with a moving object.
Gravitational potential energy store – Energy associated with an object's position in a gravitational field.
Chemical energy store – Energy associated with chemical substances.
Elastic energy store – Energy stored when an object is stretched or compressed.
Nuclear energy store – Energy associated with atomic nuclei.
Electrostatic energy store – Energy associated with the positions of electric charges.
Internal energy store – Energy associated with the particles within a substance.
Dissipation – The spreading of energy into the surroundings, usually into internal energy stores.
Conservation of energy – The principle that energy cannot be created or destroyed.
Key Takeaways
- Energy can be held in kinetic, gravitational potential, chemical, elastic, nuclear, electrostatic, and internal energy stores.
- An energy store describes where energy is held.
- A transfer pathway describes how energy moves between stores.
- The four main transfer pathways are mechanical work, electrical work, heating, and waves.
- Energy flow diagrams can show energy moving between different stores.
- Energy that spreads into the surroundings is dissipated, not destroyed.
- According to the principle of conservation of energy, energy cannot be created or destroyed.
- In a closed system: Total energy before = Total energy after
2. Work Done
Learning outcomes
- I can explain work done as the transfer of energy by a force acting through a displacement.
- I can describe the relationship between work done, energy transferred, and the principle of conservation of energy.
- I can determine when work is done on an object and identify situations in which no mechanical work is done.
- I can apply the equations and to calculate the work done by a constant force.
- I can analyze energy transfers in mechanical and electrical systems using the concept of work done.
What Is Work Done?
In physics, work is done when a force causes an object to move through a displacement.
Work is not simply about how hard something feels. A force must act through a distance for mechanical work to be done.
The simplest equation is:
\[ \boxed{W=Fd} \]
where:
- \(W\) = work done, measured in joules (J)
- \(F\) = force, measured in newtons (N)
- \(d\) = displacement in the direction of the force, measured in metres (m)
This equation applies when the force and displacement act in the same direction.
Work Is an Energy Transfer
Work done is closely connected to energy.
When a force does work on an object, energy is transferred.
For example, suppose you push a box across the floor.
Chemical energy stored in your body is transferred into:
- kinetic energy of the box
- thermal energy due to friction
- thermal energy in your muscles and surroundings
The amount of work done represents the amount of energy transferred by the force.
Therefore:
\[ \boxed{1\text{ J of work}=1\text{ J of energy transferred}} \]
This is one of the most important ideas in mechanics.
The Joule
The SI unit of work is the joule (J).
One joule of work is done when a force of 1 newton moves an object 1 metre in the direction of the force.
\[ \boxed{1\text{ J}=1\text{ N}\cdot\text{m}} \]
For example:
\[ W=Fd \]
If:
\[ F=1\text{ N} \]
and:
\[ d=1\text{ m} \]
then:
\[ W=1\times1 \]\[ \boxed{W=1\text{ J}} \]
Worked Example: Pushing a Box
A person pushes a box with a constant force of 50 N over a distance of 4 m.
Calculate the work done.
Use:
\[ W=Fd \]
Substitute:
\[ W=50\times4 \]\[ \boxed{W=200\text{ J}} \]
Therefore, 200 J of energy has been transferred by the force.
When Is Mechanical Work Done?
Mechanical work is done when:
- a force acts on an object
- the object moves through a displacement
- at least part of the force acts in the direction of the displacement
If any of these conditions is missing, mechanical work may be zero.
Example: Lifting an Object
Suppose you lift a 10 N object vertically upward by 2 m.
The force you apply is upward.
The displacement is also upward.
So work is done.
\[ W=Fd \]\[ W=10\times2 \]\[ \boxed{W=20\text{ J}} \]
The work done transfers energy to the object.
In this case, the object's gravitational potential energy increases.
Example: Pushing a Wall
Imagine pushing against a solid wall with a force of 200 N.
You may become tired, but the wall does not move.
Therefore:
\[ d=0 \]
Using:
\[ W=Fd \]\[ W=200\times0 \]\[ \boxed{W=0\text{ J}} \]
No mechanical work is done on the wall because there is no displacement.
Your body still transfers chemical energy internally, but that is different from mechanical work being done on the wall.
Example: Holding a Heavy Object
Suppose you hold a heavy box stationary for 30 seconds.
You exert an upward force on the box, but the box does not move.
Therefore:
\[ d=0 \]
and:
\[ \boxed{W=0} \]
No mechanical work is done on the box.
Even though your muscles become tired, the box has no displacement.
This is a common example showing that the everyday meaning of "work" is different from the physics definition.
Force at an Angle
The force does not always act in exactly the same direction as the displacement.
For example, imagine pulling a suitcase using a handle angled upward.
Only the component of the force parallel to the displacement does work.
The more general equation is:
\[ \boxed{W=Fd\cos\theta} \]
where:
- \(W\) = work done
- \(F\) = force
- \(d\) = displacement
- \(\theta\) = angle between the force and displacement
Understanding the Angle
The value of \(\theta\) is the angle between the force vector and the displacement vector.
This gives several important situations.
Force in the Same Direction
\[ \theta=0^\circ \]
Since:
\[ \cos0^\circ=1 \]
then:
\[ W=Fd \]
The force does the maximum possible positive work.
Force Perpendicular to the Motion
\[ \theta=90^\circ \]
Since:
\[ \cos90^\circ=0 \]
then:
\[ W=0 \]
The force does no work.
Force Opposite the Motion
\[ \theta=180^\circ \]
Since:
\[ \cos180^\circ=-1 \]
then:
\[ W=-Fd \]
The force does negative work.
Worked Example: Pulling at an Angle
A person pulls a suitcase with a force of 80 N at an angle of \(30^\circ\) above the horizontal.
The suitcase moves 5 m horizontally.
Calculate the work done by the pulling force.
Use:
\[ W=Fd\cos\theta \]
Substitute:
\[ W=(80)(5)\cos30^\circ \]\[ W=400(0.866) \]\[ \boxed{W\approx346\text{ J}} \]
Only the horizontal component of the pulling force contributes to the work done on the suitcase.
Positive Work
A force does positive work when it acts at least partly in the same direction as the object's displacement.
For example:
- pushing a trolley forward
- gravity pulling a falling object downward
- a motor accelerating a vehicle
Positive work generally transfers energy to the object or system.
For example, if a force increases an object's speed, its kinetic energy increases.
Negative Work
A force does negative work when it acts in the opposite direction to the displacement.
Friction commonly does negative work.
Imagine a sliding box.
The box moves to the right while friction acts to the left.
The angle between force and displacement is:
\[ 180^\circ \]
Therefore:
\[ W=Fd\cos180^\circ \]\[ W=-Fd \]
The negative sign indicates that mechanical energy is being transferred away from the object's kinetic energy store.
Much of this energy becomes thermal energy.
Zero Work
A force can act on an object without doing mechanical work.
There are two particularly important situations.
No Displacement
If:
\[ d=0 \]
then:
\[ W=0 \]
Example: pushing against a wall that does not move.
Force Perpendicular to Displacement
If:
\[ \theta=90^\circ \]
then:
\[ W=Fd\cos90^\circ \]\[ W=0 \]
Example: the normal force on an object moving horizontally across a level floor.
Work Done by the Normal Force
Consider a box sliding horizontally across a flat floor.
The box moves horizontally.
The normal force acts vertically upward.
The force and displacement are perpendicular.
Therefore:
\[ \theta=90^\circ \]
and:
\[ \boxed{W_N=0} \]
The normal force does no mechanical work on the box in this situation.
Circular Motion and Zero Work
An important example occurs in uniform circular motion.
The centripetal force always points toward the centre of the circle.
The instantaneous displacement is tangential to the circular path.
These directions are perpendicular.
Therefore:
\[ \theta=90^\circ \]
and the centripetal force does:
\[ \boxed{W=0} \]
The force changes the direction of the velocity without changing the object's speed.
Work and Kinetic Energy
When a net force does work on an object, the object's kinetic energy can change.
This relationship is described by the work-energy theorem:
\[ \boxed{W_{\text{net}}=\Delta E_k} \]
or:
\[ W_{\text{net}}=E_{k,\text{final}}-E_{k,\text{initial}} \]
This means:
- positive net work → kinetic energy increases
- negative net work → kinetic energy decreases
- zero net work → kinetic energy remains constant
Example: Accelerating a Car
Suppose the net force on a car does 30 000 J of positive work.
According to the work-energy theorem:
\[ \Delta E_k=30\,000\text{ J} \]
The car's kinetic energy increases by 30 000 J.
If its mass remains constant, this corresponds to an increase in speed.
Work Against Gravity
When an object is lifted, work is done against gravity.
For an object lifted vertically at constant speed:
\[ F=mg \]
and:
\[ W=Fd \]
Therefore:
\[ W=mgh \]
This work increases the object's gravitational potential energy:
\[ \boxed{\Delta E_p=mgh} \]
So:
\[ \boxed{W=\Delta E_p} \]
when no other significant energy transfers occur.
Worked Example: Lifting a Backpack
A student lifts a 6 kg backpack vertically through 1.5 m.
Use:
\[ g=9.8\text{ N/kg} \]
First find the weight:
\[ F=mg \]\[ F=6(9.8) \]\[ F=58.8\text{ N} \]
Now calculate work:
\[ W=Fd \]\[ W=58.8(1.5) \]\[ \boxed{W=88.2\text{ J}} \]
Approximately 88 J of energy is transferred into gravitational potential energy.
Conservation of Energy
The principle of conservation of energy states that:
Energy cannot be created or destroyed. It can only be transferred from one store to another or transformed from one form to another.
Work is one way that energy is transferred.
For example, when you push a box:
chemical energy in muscles
↓
work done by force
↓
kinetic energy + thermal energy
The energy has not disappeared.
It has simply changed form or location.
Energy Transfers with Friction
Suppose you push a box along a rough floor.
You do positive work on the box.
At the same time, friction does negative work.
Some of the energy transferred by your push increases the box's kinetic energy.
Some becomes thermal energy.
If the box moves at constant speed, its kinetic energy does not change.
In that case, the positive work done by the applied force is balanced by the negative work done by friction.
Worked Example: Work Against Friction
A box is pushed 8 m across a floor at constant speed.
The friction force is 25 N.
Calculate the work done against friction.
\[ W=Fd \]\[ W=25(8) \]\[ \boxed{W=200\text{ J}} \]
Approximately 200 J of mechanical energy is transferred into thermal energy.
If the box moves at constant speed, the applied force must do approximately +200 J of work while friction does −200 J of work.
The net work is therefore:
\[ W_{\text{net}}=0 \]
and the kinetic energy remains constant.
Work in Electrical Systems
The concept of work also applies to electricity.
When electric charge moves through a potential difference, energy is transferred.
Electrical work can be calculated using:
\[ \boxed{W=VQ} \]
where:
- \(W\) = energy transferred or work done, in joules
- \(V\) = potential difference, in volts
- \(Q\) = charge transferred, in coulombs
Recall:
\[ 1\text{ V}=1\text{ J/C} \]
So a potential difference of 6 V transfers 6 J of energy for every coulomb of charge.
Worked Example: Electrical Work
A charge of 4 C moves through a potential difference of 12 V.
Calculate the electrical energy transferred.
\[ W=VQ \]\[ W=12(4) \]\[ \boxed{W=48\text{ J}} \]
Therefore, 48 J of electrical energy is transferred.
Mechanical and Electrical Work
Although mechanical and electrical systems may look very different, the underlying idea is similar.
Mechanical system
A force moves an object:
\[ W=Fd \]
Energy is transferred mechanically.
Electrical system
Charge moves through a potential difference:
\[ W=VQ \]
Energy is transferred electrically.
In both cases:
\[ \boxed{\text{Work done = energy transferred}} \]
Example: Electric Motor
An electric motor provides an excellent connection between electrical and mechanical work.
Electrical energy enters the motor.
The motor produces a force that causes something to move.
For example:
electrical energy
↓
motor
↓
mechanical work
↓
kinetic energy
Some energy is also transferred into:
- thermal energy
- sound
Because energy is conserved:
\[ E_{\text{input}}=E_{\text{useful}}+E_{\text{wasted}} \]
Example: Electric Lift
Imagine an electric motor lifting an elevator.
Electrical energy is supplied to the motor.
The motor does mechanical work against gravity.
Energy is transferred into the gravitational potential energy of the elevator.
So the chain is:
electrical energy
↓
work done by motor
↓
gravitational potential energy
This is a good example of mechanical and electrical work being connected within the same system.
Work and Power
Work tells us how much energy is transferred.
Power tells us how quickly the energy is transferred.
The relationship is:
\[ \boxed{P=\frac{W}{t}} \]
where:
- \(P\) = power in watts
- \(W\) = work done in joules
- \(t\) = time in seconds
Two machines can do the same amount of work but have different powers if one completes the work more quickly.
Worked Example: Same Work, Different Power
Machine A does 600 J of work in 3 s.
\[ P=\frac{600}{3} \]\[ P=200\text{ W} \]
Machine B does the same 600 J of work in 12 s.
\[ P=\frac{600}{12} \]\[ P=50\text{ W} \]
Both machines perform the same amount of work, but Machine A transfers energy four times faster.
Real-World Connections
Work and energy transfer appear throughout everyday life and technology.
Climbing Stairs
Your muscles do work against gravity, increasing your gravitational potential energy.
Braking a Bicycle
Friction does negative work on the bicycle, reducing its kinetic energy and increasing thermal energy.
Cranes
A crane performs work when it lifts heavy loads against gravity.
Cars
Engines and motors perform mechanical work to increase a vehicle's kinetic energy and overcome friction and drag.
Electric Motors
Electrical work is converted into mechanical work.
Generators
Mechanical work is used to produce electrical energy.
Household Appliances
Electrical energy is transferred into thermal, kinetic, light, or sound energy depending on the appliance.
Common Mistakes
Mistake 1: A force automatically means work is done
A force alone is not enough.
There must also be a displacement.
Mistake 2: Being tired means mechanical work was done
You can hold a heavy object and become tired while doing zero mechanical work on the object.
If:
\[ d=0 \]
then:
\[ W=0 \]
Mistake 3: Using the full force when it acts at an angle
Only the component parallel to the displacement contributes to work.
Use:
\[ W=Fd\cos\theta \]
Mistake 4: Forgetting the angle is between force and displacement
The angle \(\theta\) must be measured between the force direction and displacement direction.
Mistake 5: Assuming work is always positive
Work can be:
- positive
- negative
- zero
depending on the directions of force and displacement.
Mistake 6: Saying energy is "used up"
Energy is conserved.
It may become less useful because it spreads into thermal energy, but it has not disappeared.
Mistake 7: Confusing work with power
Work measures energy transferred.
Power measures the rate of energy transfer.
Check Your Understanding
1. Define work done in physics.
2. State the SI unit of work.
3. Explain what 1 joule of work means.
4. Calculate the work done when a 30 N force moves an object 5 m in the same direction.
5. A student pushes against a wall with a force of 150 N, but the wall does not move. Calculate the mechanical work done on the wall.
6. Explain why holding a heavy object stationary involves no mechanical work on the object.
7. A 60 N force acts on an object moving 4 m in the same direction. Calculate the work done.
8. A 100 N force acts at \(60^\circ\) to the direction of motion. The object moves 3 m. Calculate the work done.
9. What is the work done by a force acting at \(90^\circ\) to the displacement?
10. Explain why the normal force does no work on a box sliding horizontally across a level floor.
11. What does negative work mean?
12. Explain why friction often does negative work.
13. A 5 kg object is lifted vertically through 2 m. Calculate the work done against gravity using \(g=9.8\text{ N/kg}\).
14. Explain how this work is related to gravitational potential energy.
15. A 12 V battery moves 5 C of charge through a circuit. Calculate the electrical work done.
16. Explain how an electric motor connects electrical work and mechanical work.
17. A box moves across a rough floor at constant speed. Explain why the net work done on the box is zero.
18. Challenge: A person pulls a trolley with a 120 N force at \(35^\circ\) above the horizontal for 8 m. Calculate the work done by the pulling force.
Key Terms
- Work done – energy transferred when a force acts through a displacement.
- Force – a push or pull that can change an object's motion.
- Displacement – the change in position of an object in a particular direction.
- Joule (J) – the SI unit of work and energy.
- Positive work – work done by a force with a component in the direction of displacement.
- Negative work – work done by a force opposing the displacement.
- Zero work – occurs when there is no displacement or when force is perpendicular to displacement.
- Energy transfer – movement of energy from one store or system to another.
- Conservation of energy – energy cannot be created or destroyed.
- Kinetic energy – energy associated with motion.
- Gravitational potential energy – energy associated with position in a gravitational field.
- Electrical work – energy transferred when charge moves through a potential difference.
- Power – the rate at which work is done or energy is transferred.
Key Takeaways
- Work is done when a force acts through a displacement.
- Work represents an energy transfer.
- Work is measured in joules (J).
- For a force acting parallel to displacement:
\[ \boxed{W=Fd} \]
- For a force acting at an angle:
\[ \boxed{W=Fd\cos\theta} \]
- If there is no displacement, no mechanical work is done on the object.
- A force perpendicular to displacement does zero work.
- A force acting in the same direction as displacement does positive work.
- A force opposing displacement does negative work.
- The net work done on an object is related to its change in kinetic energy:
\[ \boxed{W_{\text{net}}=\Delta E_k} \]
- Work done against gravity can increase gravitational potential energy.
- Energy is conserved; work transfers energy rather than creating or destroying it.
- Electrical work can be calculated using:
\[ \boxed{W=VQ} \]
- Mechanical and electrical systems both involve energy being transferred through work.
3. Energy Transfer Pathways
Learning outcomes
- I can identify the four main energy transfer pathways: mechanical work, electrical work, heating, and waves.
- I can explain how energy is transferred through each pathway and describe real-world examples of each.
- I can distinguish between energy stores and energy transfer pathways in physical systems.
- I can analyze energy transfers in everyday situations and identify the stores and pathways involved.
- I can construct and interpret energy flow diagrams that show how energy is transferred between stores.
Energy Transfer Pathways
Energy is involved whenever something changes: an object speeds up, a lamp switches on, water gets warmer, or sunlight reaches Earth. However, energy does not simply disappear from one place and appear in another. It is transferred between energy stores.
A useful model is:
ENERGY STORE → TRANSFER PATHWAY → ENERGY STORE
There are four main energy transfer pathways:
- Mechanical work
- Electrical work
- Heating
- Waves
Learning Targets
- I can identify the four main energy transfer pathways: mechanical work, electrical work, heating, and waves.
- I can explain how energy is transferred through each pathway and describe real-world examples of each.
- I can distinguish between energy stores and energy transfer pathways in physical systems.
- I can analyze energy transfers in everyday situations and identify the stores and pathways involved.
- I can construct and interpret energy flow diagrams that show how energy is transferred between stores.
1. Energy Stores
An energy store describes where energy is held within a system.
The main energy stores are:
- Kinetic store – associated with moving objects
- Gravitational potential store – associated with an object's position in a gravitational field
- Elastic potential store – associated with stretched or compressed objects
- Chemical store – found in fuels, food and batteries
- Thermal store – associated with the internal energy and temperature of an object
- Magnetic store – associated with interacting magnetic fields
- Electrostatic store – associated with interacting electric charges
- Nuclear store – associated with atomic nuclei
For example, a moving bicycle has energy in its kinetic store, while a battery has energy in its chemical store.
Energy stores tell us where energy is.
Transfer pathways tell us how energy moves between stores.
2. Stores Are Not Pathways
This distinction is extremely important.
Suppose a battery powers a lamp.
The battery contains a chemical energy store.
Energy is transferred from the battery electrically.
The lamp becomes hot and produces light.
We could represent part of this process as:
Chemical store → electrical work → thermal store
Notice that electrical describes the transfer, not the store.
A useful rule is:
STORE = where energy is
PATHWAY = how energy gets somewhere else
3. The Four Transfer Pathways
| Pathway | How energy is transferred | Example |
|---|---|---|
| Mechanical work | A force causes movement | Lifting a box |
| Electrical work | Moving charges transfer energy | Battery powering a motor |
| Heating | Energy moves because of a temperature difference | Hot water warming a spoon |
| Waves | Waves carry energy | Sunlight reaching Earth |
These four pathways allow us to describe a huge range of physical systems.
4. Mechanical Work
Energy is transferred mechanically when a force acts and causes an object to move through a distance.
Examples include:
- Lifting a backpack
- Pushing a shopping cart
- Kicking a football
- Stretching a spring
- A falling object accelerating
- A crane lifting a load
- A cyclist pedalling uphill
Example: Lifting a Book
Imagine lifting a book from the floor onto a shelf.
Your muscles contain a chemical energy store.
Your muscles exert a force on the book and move it upwards.
Energy is transferred mechanically:
Chemical store in body → mechanical work → gravitational potential store of book
The book now has a larger gravitational potential energy store because it has been raised.
5. Mechanical Work and Moving Objects
Mechanical work does not always increase gravitational potential energy.
Suppose you push a stationary shopping cart.
Your muscles exert a force and the cart begins moving.
Chemical store → mechanical work → kinetic store
Some energy will also be transferred to thermal stores because of friction.
Therefore, a more realistic description is:
Chemical store → mechanical work → kinetic store + thermal stores
Energy has not been destroyed. It has simply been distributed among different stores.
6. Electrical Work
Energy is transferred electrically when electric charges move through a potential difference.
This commonly happens when an electric current flows through a circuit.
Examples include:
- A battery powering a lamp
- Charging a phone
- An electric fan operating
- A television operating
- An electric heater warming a room
- A motor turning
Example: Battery and Lamp
A battery begins with energy in its chemical store.
When the circuit is completed, energy is transferred electrically to the lamp.
The lamp becomes hot and produces light.
A simplified description is:
Chemical store → electrical work → thermal store of lamp
Energy then leaves the lamp through heating and electromagnetic waves.
7. Example: Electric Fan
An electric fan receives energy through an electrical pathway.
The motor causes the blades to rotate.
The blades therefore gain energy in their kinetic store.
We can represent this as:
Electrical work → motor → mechanical work → kinetic store of blades
Not all of the transferred energy becomes useful kinetic energy.
Some increases the thermal stores of:
- the motor
- the wires
- the surrounding air
Some energy is also carried away by sound waves.
That is why an electric fan produces both heat and sound as well as moving air.
8. Heating
Energy is transferred by heating when there is a temperature difference.
Energy moves from a region at a higher temperature toward a region at a lower temperature.
Imagine putting a cold metal spoon into hot soup.
The soup is hotter than the spoon.
Energy is transferred:
Thermal store of soup → heating → thermal store of spoon
As a result:
- The soup cools slightly.
- The spoon becomes warmer.
- Their temperatures move toward thermal equilibrium.
Heating between materials can occur through processes such as conduction and convection.
9. Examples of Heating
Heating occurs constantly in everyday life.
Examples include:
- A radiator warming a room
- Hot tea warming a cup
- Your hand warming an ice cube
- A frying pan heating food
- A hot engine warming the surrounding air
- Warm water melting an ice cube
In each case, energy is transferred because there is a temperature difference.
For example:
Thermal store of hot pan → heating → thermal store of food
Eventually, if no other processes interfere, the temperatures become closer together.
10. Transfer by Waves
Energy can also be transferred by waves.
Electromagnetic waves are particularly important because they can transfer energy through empty space.
Examples include:
- Visible light
- Infrared radiation
- Microwaves
- Radio waves
- Ultraviolet radiation
- X-rays
Mechanical waves such as sound waves can also carry energy through matter.
A major example is energy arriving at Earth from the Sun.
Space between the Sun and Earth is mostly empty, so conduction and convection cannot carry energy across it.
Instead, energy is transferred by electromagnetic waves.
11. Example: Sun to Earth
The Sun contains an enormous nuclear energy store.
Nuclear reactions in the Sun allow energy to be transferred outward.
Eventually, electromagnetic waves carry energy through space.
A simplified model is:
Nuclear store in Sun → electromagnetic waves → thermal and other stores on Earth
When sunlight is absorbed by the ground, oceans or buildings, their thermal stores can increase.
This is an important example because it demonstrates that energy can be transferred without matter travelling from the source to the receiver.
12. Example: Sound
Imagine a loudspeaker playing music.
Electrical energy is supplied to the speaker.
The speaker cone vibrates.
Those vibrations produce sound waves in the surrounding air.
We can describe the sequence as:
Electrical work → kinetic store of speaker cone → sound waves → surroundings
The sound waves carry energy away from the speaker.
13. Analyzing an Energy Transfer
When you encounter an energy-transfer problem, ask three questions:
Question 1: Where is the energy initially stored?
Identify the starting store.
Question 2: How is the energy transferred?
Identify the pathway.
Question 3: Where does the energy end up?
Identify the final store or stores.
This gives us:
Starting store → pathway → final store
This store–pathway–store approach is a useful way of analysing physical systems.
14. Worked Example: Falling Ball
Imagine holding a ball above the ground.
Initially, the ball has energy in its:
Gravitational potential store
Release the ball.
Gravity exerts a force on the ball while the ball moves downward.
Therefore, energy is transferred mechanically.
The ball speeds up, increasing its kinetic energy store.
Gravitational potential store → mechanical work → kinetic store
When the ball hits the ground, energy is transferred again.
Some increases thermal stores, while some is carried away by sound waves.
15. Worked Example: Phone Charging
Consider a phone connected to a charger.
Energy is transferred electrically through the charger and into the phone.
Much of this energy increases the battery's chemical energy store.
Electrical work → chemical store of battery
However, the phone may also become slightly warm.
Therefore:
Electrical transfer
↓
Phone
↙ ↘
Chemical Thermal
store storeThis reminds us that energy can be transferred into more than one store.
16. Worked Example: Cyclist Going Uphill
A cyclist obtains energy from the chemical store associated with food.
The cyclist's muscles exert forces while pedalling.
Energy is transferred mechanically.
As the cyclist climbs the hill, the cyclist-bike system gains gravitational potential energy.
Chemical store → mechanical work → gravitational potential store
If the cyclist also speeds up:
Chemical store → mechanical work → kinetic store
Some energy is also transferred to thermal stores because of friction and air resistance.
17. Energy Flow Diagrams
An energy flow diagram represents how energy is transferred between stores.
A simple diagram might look like this:
Chemical store
↓
mechanical work
↓
Kinetic storeOr:
Chemical store in battery
↓
electrical work
↓
Thermal store
of lampThe boxes or labels represent stores.
The arrows represent energy transfers.
18. More Complex Energy Flow Diagrams
Real systems often have several outputs.
Consider an electric motor:
Chemical store
in battery
↓
electrical work
↓
Motor
↙ ↘
mechanical heating
work
↓ ↓
Kinetic store Thermal stores
of shaft of surroundingsThe diagram shows that energy does not necessarily follow only one pathway.
19. Worked Example: Roller Coaster
At the top of a hill, a roller coaster has a large gravitational potential energy store.
As it travels downhill, gravity does mechanical work.
Gravitational potential store → mechanical work → kinetic store
The roller coaster speeds up.
Friction and air resistance also transfer energy into thermal stores.
Therefore:
Gravitational potential store
↓
mechanical work
↙ ↘
kinetic store thermal storesThis explains why a roller coaster cannot return indefinitely to exactly the same height without an external energy input.
20. Worked Example: Hydroelectric Power
Hydroelectric power provides an excellent real-world example of several energy transfers.
Water stored behind a dam has a large gravitational potential energy store.
When released, the water falls and accelerates.
Its kinetic store increases.
The moving water turns a turbine, transferring energy mechanically.
The turbine drives a generator, which transfers energy electrically.
A simplified sequence is:
Gravitational potential store → kinetic store → mechanical work → electrical work
This demonstrates how several stores and pathways can be involved in a single system.
21. Other Real-World Examples
Wind Turbine
Moving air has a kinetic energy store.
The air exerts forces on the turbine blades.
Kinetic store of wind → mechanical work → kinetic store of turbine → electrical work
Electric Kettle
Energy is transferred electrically to the heating element.
The element becomes hot and transfers energy to the water.
Electrical work → thermal store of element → heating → thermal store of water
Car
Fuel has a chemical energy store.
Combustion allows energy ultimately to be transferred mechanically to the vehicle.
Chemical store → mechanical work → kinetic store
Some energy is also transferred to thermal stores and through sound waves.
22. Conservation of Energy
Energy can move between stores, but the total amount of energy is conserved.
Energy cannot be created or destroyed.
Therefore:
Total energy before = total energy after
Suppose a system initially has 500 J of energy.
After a transfer:
- 350 J enters a useful store.
- 100 J enters thermal stores.
- 50 J is transferred elsewhere.
Then:
All of the original energy can still be accounted for.
23. Useful and Dissipated Energy
Not every energy transfer produces the result we want.
Consider a motor.
We want energy transferred to the kinetic store of the moving parts.
But friction also increases the thermal stores of the motor and surroundings.
This energy becomes more spread out and less useful for doing further work. We say that energy has been dissipated.
It is important not to say that this energy has been "lost."
It still exists.
It has simply been transferred to stores where it is less useful.
24. Common Mistakes
Mistake 1: Treating a pathway as a store
Incorrect:
Electrical energy store
Better:
Energy is transferred electrically.
Electrical work describes a pathway.
Mistake 2: Saying energy is used up
Energy is not destroyed.
Instead say:
Energy is transferred to another store.
Mistake 3: Saying energy is lost
Energy may become dissipated into thermal stores in the surroundings, but it is not destroyed.
Mistake 4: Forgetting the pathway
Do not simply write:
Chemical → kinetic
A better description is:
Chemical store → mechanical work → kinetic store
Mistake 5: Confusing heating with a thermal store
Thermal describes an energy store.
Heating describes a transfer pathway.
25. Key Terms
Energy store – a way energy is held within a system.
Energy transfer pathway – a process through which energy moves between stores.
Mechanical work – energy transfer caused by a force acting through a distance.
Electrical work – energy transfer associated with moving electric charges.
Heating – energy transfer resulting from a temperature difference.
Waves – disturbances that can carry energy from one place to another.
Kinetic store – energy associated with movement.
Chemical store – energy associated with chemical substances such as fuels, foods and batteries.
Thermal store – internal energy associated with the particles in an object.
Gravitational potential store – energy associated with position in a gravitational field.
Dissipation – energy becoming spread into less useful stores, commonly thermal stores of the surroundings.
26. Check Your Understanding
1. Name the four energy transfer pathways.
2. Is a chemical energy store a store or a pathway?
3. What pathway transfers energy when you lift a backpack?
4. What pathway transfers energy from a battery through a circuit?
5. A hot cup of tea cools on a table. Which store decreases, and where does the energy go?
6. Explain why sunlight reaching Earth is an example of energy transfer by waves.
7. Describe the energy stores and transfer pathway when a ball falls.
8. Draw an energy flow diagram for a battery-powered toy car.
9. Draw an energy flow diagram for an electric kettle heating water.
10. A student says, "The energy disappeared because the motor got hot." Explain why this statement is incorrect.
Key Takeaways
- Energy is held in stores and transferred between stores.
- The four main transfer pathways are mechanical work, electrical work, heating, and waves.
- Mechanical transfer occurs when a force acts through a distance.
- Electrical transfer occurs through moving electric charges.
- Heating transfers energy because of a temperature difference.
- Waves can carry energy from one place to another.
- Stores and pathways are different concepts.
- Energy-flow diagrams help us track energy from its initial store through transfer pathways to final stores.
- Real systems usually involve several energy transfers at the same time.
- Energy may become dissipated, but it is never created or destroyed.
Did You Know?
Almost all of the energy used in ecosystems ultimately arrives from the Sun by electromagnetic waves. Plants transfer some of this energy into chemical stores through photosynthesis, and those chemical stores then support most food chains on Earth.
4. Energy Flow Diagrams
Learning outcomes
- I can construct energy flow diagrams that show how energy is transferred between stores within a system.
- I can identify the energy stores and transfer pathways involved in a variety of physical situations.
- I can distinguish between useful energy transfers and energy dissipated to the surroundings.
- I can interpret and construct Sankey diagrams to represent energy transfers and transformations.
- I can use energy flow diagrams and Sankey diagrams to demonstrate the conservation of energy in real-world systems.
Energy Flow Diagrams
Energy flow diagrams help us follow energy as it moves through a physical system. They show where energy begins, how it is transferred, where useful energy ends up, and where energy is dissipated to the surroundings.
A more quantitative type of energy flow diagram is called a Sankey diagram. In a Sankey diagram, the widths of the arrows represent the amounts of energy being transferred.
Learning Targets
- I can construct energy flow diagrams that show how energy is transferred between stores within a system.
- I can identify the energy stores and transfer pathways involved in a variety of physical situations.
- I can distinguish between useful energy transfers and energy dissipated to the surroundings.
- I can interpret and construct Sankey diagrams to represent energy transfers and transformations.
- I can use energy flow diagrams and Sankey diagrams to demonstrate the conservation of energy in real-world systems.
1. Following Energy Through a System
An energy flow diagram is a visual way of showing what happens to energy in a system.
A simple energy flow can be represented as:
Initial energy store → transfer pathway → final energy store
For example, when a person lifts a box:
Chemical store in body → mechanical work → gravitational potential store of box
This tells us three important things:
- Where the energy started
- How the energy was transferred
- Where the energy ended up
Energy is measured in joules (J).
2. Energy Stores
An energy store describes where energy is held in a system.
Important stores include:
- Kinetic
- Thermal
- Chemical
- Gravitational potential
- Elastic potential
- Magnetic
- Electrostatic
- Nuclear
For example, consider a stretched elastic band.
Before release:
Elastic potential store
After release:
Kinetic store
The energy is transferred mechanically:
Elastic potential store → mechanical work → kinetic store
3. Transfer Pathways
Stores should not be confused with transfer pathways.
The four main energy transfer pathways are:
- Mechanical work – a force acts through a distance.
- Electrical work – energy is transferred by moving electric charges.
- Heating – energy is transferred because of a temperature difference.
- Waves – energy is transferred by electromagnetic or mechanical waves.
A battery, for example, contains a chemical store.
When the battery powers a motor, energy is transferred electrically.
Therefore:
Chemical store → electrical work → kinetic store
4. Constructing an Energy Flow Diagram
Suppose a battery powers a small electric motor.
Start by identifying the initial store:
Chemical store in battery
Next, identify the pathway:
Electrical work
Then identify the useful final store:
Kinetic store of motor
A simple diagram is:
Chemical store
in battery
↓
Electrical work
↓
Kinetic store
of motorHowever, this is not the whole story.
The motor also becomes warm and produces sound.
A more realistic diagram is:
┌──→ Kinetic store
Chemical store │ of motor
in battery ─────────┤
│
└──→ Thermal stores
of surroundingsThe diagram now shows that the energy has been distributed among several stores.
5. Useful Energy Transfers
The useful energy transfer is the transfer that produces the effect we want from a device.
Consider an electric fan.
We want the fan blades to move.
Therefore, the increase in the kinetic store of the fan and moving air is useful.
Similarly:
| Device | Intended useful transfer |
|---|---|
| Lamp | Energy transferred by visible light |
| Electric heater | Increase in thermal store of room |
| Motor | Increase in kinetic store |
| Speaker | Energy transferred by sound waves |
| Crane | Increase in gravitational potential store |
| Phone charger | Increase in chemical store of battery |
What counts as "useful" depends on the purpose of the system.
For example, heating is useful in an electric heater but usually unwanted in an electric motor.
6. Dissipated Energy
Real devices do not transfer all their input energy into the desired store or pathway.
Some energy becomes dissipated to the surroundings.
Common unwanted transfers include:
- Heating caused by friction
- Heating of electrical components
- Sound produced by machines
- Vibrations
- Unwanted light
For example, an electric motor might receive 100 J of energy.
Suppose:
70 J → useful kinetic transfer
30 J → thermal stores of motor and surroundings
The 30 J has not disappeared. It has been dissipated into the surroundings.
7. Energy Is Not "Lost"
You will often hear the phrase wasted energy.
This does not mean the energy has been destroyed.
It means that energy has been transferred somewhere that is not useful for the intended purpose.
For example, friction in a motor causes components to become warm.
Energy has been transferred into the thermal stores of:
- The motor
- The surrounding air
- Other nearby materials
Once energy has spread through the surroundings, it becomes more difficult to use for further useful work. This is called energy dissipation.
A better scientific statement than "energy was lost" is:
Energy was dissipated to the surroundings.
8. Worked Example — Electric Kettle
Consider an electric kettle.
Energy is transferred electrically to the heating element.
The heating element transfers energy to the water.
The intended result is an increase in the thermal store of the water.
Some energy also increases the thermal stores of:
- The kettle
- The surrounding air
- The work surface
A simplified flow diagram is:
Electrical transfer
↓
Kettle
↙ ↘
Useful Dissipated
heating heating
↓ ↓
Thermal Thermal stores
store of of kettle and
water surroundings9. Introducing Sankey Diagrams
A Sankey diagram is a special energy flow diagram.
The key feature is:
The width of each arrow represents the amount of energy being transferred.
A large energy transfer has a wide arrow.
A small energy transfer has a narrow arrow.
Usually:
- Input energy enters from the left.
- Useful output continues forwards.
- Dissipated energy branches away, commonly downward.
10. Reading a Sankey Diagram
Suppose a lamp receives:
100 J input energy
It transfers:
40 J by useful light
and:
60 J by heating to the surroundings
The Sankey diagram would have an input arrow representing 100 J.
That arrow divides into:
100 J INPUT ━━━━━━━━━━┳━━━━→ 40 J useful
┃
┃
┃
↓
60 J
dissipatedThe 60 J arrow must be wider than the 40 J arrow because more energy is being transferred by heating.
The exact arrow widths should be proportional to the energy quantities.
11. Sankey Diagrams and Conservation of Energy
Sankey diagrams provide a visual demonstration of the law of conservation of energy.
Energy cannot be created or destroyed.
Therefore:
If there are useful and dissipated outputs:
This relationship is reflected in the widths of the arrows: the widths of all output branches together correspond to the width of the input arrow.
12. Worked Example — Lamp
A lamp receives 200 J of energy.
It transfers:
- 50 J usefully by light
- The remainder by heating
First calculate the dissipated energy:
Check conservation:
Therefore:
Input = 200 J
Useful = 50 J
Dissipated = 150 J
The dissipated branch of the Sankey diagram should be three times as wide as the useful branch.
13. Arrow Width Matters
Consider two machines that each receive 100 J.
Machine A
- Useful = 80 J
- Dissipated = 20 J
Machine B
- Useful = 30 J
- Dissipated = 70 J
Machine A's Sankey diagram would have a large forward useful arrow and a relatively narrow dissipated branch.
Machine B would have a much smaller useful arrow and a much wider dissipated branch.
This allows us to compare devices visually without immediately performing calculations.
14. Drawing a Sankey Diagram to Scale
Suppose a device receives 100 J and produces:
- 70 J useful energy
- 30 J dissipated energy
We could choose the scale:
1 mm = 10 J
The input arrow would therefore be:
The useful arrow would be:
The dissipated arrow would be:
Notice:
The arrow widths therefore demonstrate conservation of energy.
15. Worked Example — Hair Dryer
A hair dryer receives 500 J of energy.
Suppose:
- 350 J is transferred usefully to heating and moving the air.
- 150 J is dissipated through unwanted heating and sound.
Check:
A Sankey diagram would show:
500 J INPUT
━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━→ 350 J useful
┃
┃
┗━━━━━━↓
150 J dissipatedBecause 70% of the input is useful, the useful arrow should represent 70% of the original width.
16. Worked Example — Electric Motor
An electric motor receives 100 J.
Suppose:
- 70 J increases the useful kinetic store.
- 30 J is dissipated into thermal stores.
The energy balance is:
The Sankey diagram clearly shows that the useful output is larger than the dissipated output.
This diagram also connects directly to efficiency:
Therefore:
or:
17. Worked Example — Car
Fuel contains a chemical energy store.
When the engine operates, some energy is transferred mechanically and increases the car's kinetic store.
However, energy is also dissipated through:
- Heating of the engine
- Heating caused by friction
- Air resistance
- Sound
- Heating of tyres and road
A simplified flow is:
Chemical store
in fuel
↓
Mechanical transfers
↙ ↘
Kinetic Thermal stores
store of car + sound
USEFUL DISSIPATEDA Sankey diagram could then represent the relative amounts of these transfers.
18. Worked Example — Falling Ball
A ball held above the ground has energy in its gravitational potential store.
As it falls:
Gravitational potential store → mechanical transfer → kinetic store
Immediately before impact, much of the transferred energy is associated with the ball's kinetic store.
When the ball strikes the ground, energy is transferred again.
Some may enter:
- Elastic stores as the ball deforms
- Thermal stores of the ball and floor
- Sound waves
- Kinetic store if the ball rebounds
An energy flow diagram can therefore become more complex as a system changes over time.
19. Real-World Example — Hydroelectric Power
Water stored behind a dam has a gravitational potential energy store.
When released:
Gravitational potential store → kinetic store of water
The moving water turns a turbine:
Kinetic store → mechanical transfer → kinetic store of turbine
The turbine drives a generator:
Mechanical transfer → electrical transfer
Some energy is dissipated through:
- Friction
- Heating
- Sound
- Turbulence
A Sankey diagram can show how much of the original energy ultimately becomes useful electrical output.
20. Flow Diagrams vs Sankey Diagrams
The two diagrams are related, but they serve slightly different purposes.
| Energy Flow Diagram | Sankey Diagram |
|---|---|
| Shows the sequence of energy transfers | Shows energy transfers quantitatively |
| Identifies stores | Shows amounts of energy |
| Identifies pathways | Arrow width represents energy |
| Can show several stages | Clearly separates useful and dissipated outputs |
| Useful for explaining a system | Useful for analyzing conservation and efficiency |
In many situations, using both diagrams together gives the clearest picture.
21. Using Sankey Diagrams to Find Missing Energy
Sankey diagrams can also be used to solve problems.
Suppose:
Input = 800 J
Useful output = 600 J
The dissipated energy must be:
Check:
The energy balance confirms conservation of energy.
22. Conservation in Complex Systems
There may be more than two outputs.
Suppose an electric machine receives 1000 J and transfers:
- 650 J into useful kinetic output
- 200 J into thermal stores
- 100 J through sound
- 50 J through unwanted vibration
Then:
Therefore:
Even though the energy has been divided among several pathways and stores, none has disappeared.
This is one of the major advantages of Sankey diagrams: they make conservation of energy visually obvious.
23. Comparing Efficient and Inefficient Devices
Consider two lamps that each receive 100 J.
Lamp A
- 75 J useful
- 25 J dissipated
Lamp B
- 10 J useful
- 90 J dissipated
Their Sankey diagrams would look dramatically different because the relative widths of the useful and dissipated arrows would be different. This makes Sankey diagrams particularly useful for comparing devices.
24. How to Construct a Sankey Diagram
When drawing your own Sankey diagram:
Step 1: Identify the total input energy.
Step 2: Identify all useful outputs.
Step 3: Identify all dissipated outputs.
Step 4: Check conservation:
Step 5: Choose a suitable scale.
For example:
Step 6: Draw the input arrow to scale.
Step 7: Split the arrow into useful and dissipated branches.
Step 8: Label every transfer and include units.
25. Common Mistakes
Mistake 1 — Making all arrows the same width
In a Sankey diagram, arrow width represents energy.
More energy → wider arrow.
Less energy → narrower arrow.
Mistake 2 — Saying dissipated energy disappears
Energy cannot disappear.
It has been transferred to other stores, often thermal stores in the surroundings.
Mistake 3 — Confusing stores and pathways
Thermal store = where energy is stored.
Heating = how energy is transferred.
Mistake 4 — Forgetting conservation
If:
something is missing from your diagram or calculation.
Mistake 5 — Using arrow length instead of width
It is the width, not the length, of a Sankey arrow that represents the amount of energy.
Mistake 6 — Assuming heating is always wasted
Heating is unwanted in a motor but useful in a heater or kettle.
"Useful" depends on what the device is intended to do.
26. Key Terms
Energy flow diagram – a diagram showing energy stores and transfers within a system.
Sankey diagram – an energy-transfer diagram in which arrow widths represent quantities of energy.
Input energy – energy supplied to a system.
Useful energy – energy transferred in the way intended.
Dissipated energy – energy transferred into less useful stores, often thermal stores of the surroundings.
Energy store – a way energy is held in a system.
Transfer pathway – a process through which energy moves between stores.
Conservation of energy – the principle that energy cannot be created or destroyed.
Efficiency – the proportion of input energy transferred usefully.
27. Check Your Understanding
1. What information does an energy flow diagram show?
2. Name the four main energy transfer pathways.
3. A battery powers a motor. Identify the initial store, transfer pathway and useful final store.
4. What is meant by dissipated energy?
5. Why should we avoid saying that energy has been "lost"?
6. What does the width of an arrow represent in a Sankey diagram?
7. A machine receives 500 J and transfers 350 J usefully. How much energy is dissipated?
8. A lamp receives 100 J and transfers 20 J usefully by light. Sketch an appropriate Sankey diagram.
9. Explain how a Sankey diagram demonstrates conservation of energy.
10. A motor receives 600 J. It transfers 420 J to useful kinetic output, 150 J to thermal stores and 30 J through sound. Show that energy has been conserved.
Key Takeaways
- Energy flow diagrams show where energy starts, how it is transferred and where it ends up.
- Energy transfers may be useful or dissipated, depending on the intended purpose of the system.
- Dissipated energy has not disappeared; it has been transferred to other stores.
- Sankey diagrams represent energy transfers using arrows of proportional width.
- Wider arrows represent larger energy transfers.
- Sankey diagrams can show useful and dissipated energy at the same time.
- The total width of the output branches corresponds to the original input because energy is conserved.
- Therefore:
- Energy flow diagrams and Sankey diagrams allow us to analyze real machines, appliances, vehicles and energy-generation systems.
Did You Know?
Sankey diagrams are named after Matthew Henry Phineas Riall Sankey, who used this style of diagram in the late 19th century to illustrate energy flows and the efficiency of a steam engine. Today, Sankey diagrams are used far beyond school physics: engineers and energy analysts use them to visualize flows of electricity, fuel, heat, materials and even resources through entire countries and industries.
5. Conservation of Energy
Learning outcomes
- I can define mechanical energy as the energy associated with the motion and position of objects.
- I can identify situations in which energy is stored as kinetic, gravitational potential, or elastic potential energy.
- I can explain how mechanical energy can be transferred between different energy stores within a system.
- I can describe real-world examples of mechanical energy transformations, such as roller coasters, pendulums, and springs.
- I can predict qualitative changes in mechanical energy as objects move, speed up, slow down, rise, or fall.
Conservation of Energy
Mechanical energy is the energy associated with the motion and position of objects. A roller coaster racing down a hill, a swinging pendulum, a falling ball, and a compressed spring all involve mechanical energy.
One of the most important ideas in physics is that energy can move between different stores, but energy cannot be created or destroyed. This is the law of conservation of energy.
Learning Targets
- I can define mechanical energy as the energy associated with the motion and position of objects.
- I can identify situations in which energy is stored as kinetic, gravitational potential, or elastic potential energy.
- I can explain how mechanical energy can be transferred between different energy stores within a system.
- I can describe real-world examples of mechanical energy transformations, such as roller coasters, pendulums, and springs.
- I can predict qualitative changes in mechanical energy as objects move, speed up, slow down, rise, or fall.
1. What Is Mechanical Energy?
Mechanical energy is energy associated with an object's motion or position.
Three important mechanical energy stores are:
- Kinetic energy – associated with motion
- Gravitational potential energy – associated with position in a gravitational field
- Elastic potential energy – associated with stretching or compressing an elastic object
A moving car has a kinetic energy store.
A book sitting on a high shelf has a gravitational potential energy store.
A compressed spring has an elastic potential energy store.
Mechanical energy can be transferred between these stores as objects move.
2. Kinetic Energy
Any moving object has energy in its kinetic store.
Examples include:
- A moving car
- A rolling football
- A running person
- A falling rock
- A spinning turbine
- A roller coaster moving along a track
The amount of kinetic energy depends on the object's mass and speed.
The faster an object moves, the greater its kinetic energy.
The greater its mass, the greater its kinetic energy at the same speed.
Kinetic energy = ½ × mass × speed²
or:
KE = ½mv²
Kinetic energy is measured in joules (J).
Example
Two identical bicycles are travelling at different speeds.
Bicycle A travels at 5 m/s.
Bicycle B travels at 10 m/s.
Bicycle B has more kinetic energy because it is moving faster.
Notice that speed is squared in the kinetic energy equation. Doubling the speed therefore increases the kinetic energy by a factor of four.
3. Gravitational Potential Energy
An object can store energy because of its position in a gravitational field.
This is called gravitational potential energy (GPE).
Examples include:
- A book on a shelf
- A roller coaster at the top of a hill
- Water stored behind a dam
- A person standing at the top of stairs
- A rock held above the ground
The higher an object is raised, the greater its gravitational potential energy relative to the chosen reference level.
Gravitational potential energy also depends on mass and gravitational field strength.
Gravitational potential energy = mass × gravitational field strength × height
or:
GPE = mgh
The unit is the joule (J).
Example
If you lift a backpack from the floor onto a table, its gravitational potential energy increases.
If the backpack falls from the table, its gravitational potential energy decreases while its kinetic energy increases.
4. Elastic Potential Energy
Energy can also be stored when an elastic object is stretched or compressed.
This is called elastic potential energy.
Examples include:
- A stretched elastic band
- A compressed spring
- A stretched bow
- A trampoline pushed downward
- A diving board bending
The further a spring is stretched or compressed within its elastic limit, the more energy is stored in its elastic potential store.
When released, this energy can be transferred to a kinetic store.
For example:
Elastic potential store of spring → mechanical transfer → kinetic store of object
5. The Conservation of Energy
The law of conservation of energy states:
Energy cannot be created or destroyed. It can only be transferred between stores.
Imagine a falling ball.
At the beginning, the ball has a large gravitational potential energy store.
As the ball falls:
- Gravitational potential energy decreases.
- Kinetic energy increases.
- The ball speeds up.
Ignoring air resistance, the decrease in gravitational potential energy corresponds to the increase in kinetic energy.
We can describe this using ordinary copyable text:
Gravitational potential energy → kinetic energy
The energy has not disappeared. It has moved from one store to another.
6. Conservation of Mechanical Energy
In an ideal system where there is no friction or air resistance, the total mechanical energy remains constant.
For a simple system involving kinetic and gravitational potential energy:
Total mechanical energy = Kinetic energy + Gravitational potential energy
Suppose an object has:
GPE = 800 J
KE = 200 J
Its total mechanical energy is:
800 J + 200 J = 1000 J
Later, it might have:
GPE = 300 J
KE = 700 J
The total is still:
300 J + 700 J = 1000 J
The amounts stored as kinetic and gravitational potential energy have changed, but the total mechanical energy has remained constant.
7. Falling Objects
A falling object provides one of the simplest examples of mechanical energy transfer.
Imagine dropping a ball.
At the top
The ball is high above the ground.
GPE: high
KE: low
While falling
The ball loses height and speeds up.
GPE: decreasing
KE: increasing
Just before hitting the ground
The ball is moving fastest.
GPE: low
KE: high
Therefore:
GPE decreases → KE increases
If air resistance is negligible, the total mechanical energy remains approximately constant.
8. Objects Moving Upward
Now imagine throwing a ball upward.
The process reverses.
As the ball rises:
- Its speed decreases.
- Its kinetic energy decreases.
- Its height increases.
- Its gravitational potential energy increases.
Therefore:
KE decreases → GPE increases
At the highest point, the ball is momentarily stationary.
Its kinetic energy is at a minimum, while its gravitational potential energy is at a maximum relative to the release point.
Then it begins falling again.
9. Roller Coasters
Roller coasters provide an excellent example of repeated transfers between gravitational potential and kinetic energy.
At the top of the first hill, the coaster has a large gravitational potential energy store.
As it moves downhill:
GPE decreases
KE increases
Speed increases
Near the bottom:
GPE is relatively low
KE is relatively high
Speed is high
As the coaster climbs another hill:
KE decreases
GPE increases
Speed decreases
This repeated transfer between gravitational potential and kinetic energy occurs throughout the ride.
10. Predicting Roller-Coaster Motion
The relationship between energy and motion allows us to make predictions.
Going downhill
Height decreases → GPE decreases → KE increases → speed increases.
Going uphill
Height increases → GPE increases → KE decreases → speed decreases.
At the bottom of a hill
GPE is relatively low and KE is relatively high.
At the top of a hill
GPE is relatively high and KE is relatively low.
This means we can often predict how the speed of a roller coaster changes simply by looking at the shape of the track.
11. Why Is the First Roller-Coaster Hill Usually the Highest?
A roller coaster usually receives a major energy input as it is pulled up its first large hill.
At the top, the coaster has a large gravitational potential energy store.
As it travels around the track, energy is repeatedly transferred between gravitational potential and kinetic stores.
However, friction and air resistance transfer some mechanical energy into thermal stores.
Without another significant energy input, the coaster therefore cannot naturally climb a later hill higher than the initial hill.
This is why roller-coaster designers must carefully consider the heights of hills throughout the ride.
12. Pendulums
A swinging pendulum provides another clear example of conservation of mechanical energy.
Consider a pendulum released from one side.
Position A — Highest Point
The pendulum is momentarily stationary.
GPE: maximum
KE: minimum
Position B — Moving Down
The pendulum loses height and speeds up.
GPE: decreasing
KE: increasing
Position C — Lowest Point
The pendulum is moving fastest.
GPE: minimum
KE: maximum
Position D — Moving Up
The pendulum gains height and slows down.
GPE: increasing
KE: decreasing
Position E — Highest Point
The pendulum again becomes momentarily stationary.
GPE: maximum
KE: minimum
In an ideal system:
GPE → KE → GPE → KE → GPE
The total mechanical energy remains constant.
13. Why Does a Real Pendulum Eventually Stop?
A real pendulum does not swing forever.
This does not mean energy has been destroyed.
Air resistance and friction at the pivot transfer some mechanical energy into thermal stores in the surroundings.
Some energy may also be transferred by sound waves.
As a result, the pendulum's mechanical energy gradually decreases.
The total energy of the larger system is still conserved.
This is an important distinction:
Total energy is always conserved.
But:
Mechanical energy may not remain constant if energy is transferred to non-mechanical stores.
14. Springs
Springs demonstrate transfers between elastic potential energy and kinetic energy.
Imagine compressing a spring and placing a small cart against it.
Before release:
Elastic potential energy: high
Kinetic energy: zero
Release the spring.
As the spring expands:
Elastic potential energy decreases
Kinetic energy of cart increases
The cart speeds up.
The transfer can be represented as:
Elastic potential energy → kinetic energy
15. Elastic Bands and Bows
The same principle applies to an elastic band or bow.
When you stretch a bow:
Chemical store in muscles → mechanical transfer → elastic potential store of bow
When the bowstring is released:
Elastic potential store → mechanical transfer → kinetic store of arrow
The arrow accelerates forward.
A stretched elastic band works in the same way.
The more it is stretched within its safe elastic range, the more elastic potential energy can be stored.
16. Bouncing Ball
A bouncing ball involves several mechanical energy stores.
As the ball falls:
GPE → KE
When the ball strikes the ground, it deforms.
Some kinetic energy is temporarily transferred into an:
Elastic potential store
As the ball returns to its original shape:
Elastic potential energy → kinetic energy
The ball rises again:
KE → GPE
So the sequence is approximately:
GPE → KE → Elastic PE → KE → GPE
In reality, some energy is also dissipated through heating and sound.
Therefore, each bounce is usually lower than the previous one.
17. Skateboarder on a Ramp
A skateboarder moving along a U-shaped ramp behaves similarly to a pendulum.
At the top:
GPE high → KE low
Moving downward:
GPE decreases → KE increases
At the bottom:
GPE low → KE high
Moving upward:
KE decreases → GPE increases
This means the skateboarder is generally moving fastest near the bottom of the ramp.
18. Mechanical Energy in Real Systems
In ideal physics problems, we sometimes ignore friction and air resistance.
In reality, these forces are almost always present.
Consider a bicycle coasting along a flat road.
Initially, the bicycle has kinetic energy.
Friction and air resistance cause it to slow down.
Its kinetic energy decreases.
Where did the energy go?
Much of it is transferred into thermal stores of:
- The tyres
- The road
- The bicycle
- The surrounding air
Some energy is also transferred through sound waves.
The energy has not disappeared.
It has been transferred into other stores and pathways.
19. Predicting Changes in Mechanical Energy
You can often predict energy changes without performing calculations.
| Situation | Kinetic Energy | Gravitational PE |
|---|---|---|
| Object speeds up | Increases | Depends on height |
| Object slows down | Decreases | Depends on height |
| Object rises | Often decreases | Increases |
| Object falls | Increases | Decreases |
| Object stationary at height | Zero or low | High |
| Object moving rapidly near ground | High | Low |
For elastic systems:
| Situation | Elastic PE |
|---|---|
| Spring stretched further | Increases |
| Spring compressed further | Increases |
| Spring returns toward equilibrium | Decreases |
| Unstretched spring | Minimum |
20. Worked Example — Ball Rolling Down a Hill
A ball begins at rest at the top of a hill.
At the top
Speed: zero
KE: zero
GPE: maximum
Halfway down
The ball has lost height and gained speed.
KE: increasing
GPE: decreasing
At the bottom
The ball is moving fastest.
KE: maximum
GPE: minimum
Ignoring friction:
Total mechanical energy remains constant.
The energy has simply shifted between stores.
21. Worked Example — Throwing a Ball Upward
A ball leaves a person's hand travelling upward.
Immediately after release, it has substantial kinetic energy.
As it rises:
Speed decreases
KE decreases
GPE increases
At maximum height:
Speed = 0 m/s momentarily
KE = minimum
GPE = maximum
As the ball falls:
GPE decreases
KE increases
The process reverses.
22. Conservation of Energy and Friction
Suppose a roller coaster begins with 10,000 J of mechanical energy.
Later, it has:
6,000 J gravitational potential energy
3,000 J kinetic energy
That accounts for:
6,000 J + 3,000 J = 9,000 J mechanical energy
The remaining:
10,000 J − 9,000 J = 1,000 J
may have been transferred to thermal stores and sound because of friction and air resistance.
Total energy is still:
10,000 J
Therefore:
Mechanical energy decreased, but total energy was conserved.
23. Mechanical Energy vs Total Energy
These two ideas should not be confused.
Conservation of total energy
Energy cannot be created or destroyed.
This is always true.
Conservation of mechanical energy
Mechanical energy remains constant only when energy is not being transferred significantly into or out of the system through processes such as friction or air resistance.
In an ideal roller coaster:
GPE + KE = constant
In a real roller coaster:
Some mechanical energy is gradually transferred into thermal stores and through sound.
The total energy still remains conserved.
24. Common Mistakes
Mistake 1 — Saying energy disappears
Incorrect:
"The roller coaster loses energy."
Better:
"Energy is transferred from mechanical stores into thermal stores because of friction."
Mistake 2 — Thinking a stationary object has no energy
A stationary object has no kinetic energy, but it may have gravitational or elastic potential energy.
A roller coaster stationary at the top of a hill can have a large gravitational potential energy store.
Mistake 3 — Thinking faster means more GPE
Speed affects kinetic energy.
Height affects gravitational potential energy.
Mistake 4 — Assuming mechanical energy is always conserved
Mechanical energy is conserved only under suitable conditions.
Friction and air resistance can transfer energy from mechanical stores into thermal stores.
Mistake 5 — Saying potential energy "turns into motion"
A more precise statement is:
Energy is transferred from a potential energy store to a kinetic energy store.
25. Key Terms
Mechanical energy – energy associated with the motion and position of objects.
Kinetic energy – energy stored by a moving object.
Gravitational potential energy – energy associated with an object's position in a gravitational field.
Elastic potential energy – energy stored when an elastic object is stretched or compressed.
Conservation of energy – the principle that energy cannot be created or destroyed.
Mechanical energy conservation – mechanical energy remains constant when there are no significant transfers to other stores.
Energy transfer – movement of energy from one store to another.
Dissipation – energy spreading into less useful stores, often thermal stores of the surroundings.
Friction – a force that opposes motion and can transfer mechanical energy into thermal stores.
26. Check Your Understanding
1. What is mechanical energy?
2. Name three mechanical energy stores discussed in these notes.
3. What happens to the kinetic energy of an object when its speed increases?
4. What happens to the gravitational potential energy of an object as it rises?
5. Describe the energy changes as a ball falls.
6. At what point in a pendulum's swing is its kinetic energy greatest?
7. At what point is its gravitational potential energy greatest?
8. Describe the energy transfer when a compressed spring launches a toy car.
9. Why does a bouncing ball eventually stop bouncing?
10. A roller coaster is travelling downhill. Predict what happens to its GPE, KE and speed.
11. Explain why a real roller coaster cannot rely on mechanical energy alone to climb a hill higher than its original starting hill.
12. A student says, "The pendulum eventually stops, so its energy has been destroyed." Explain why this statement is incorrect.
Key Takeaways
- Mechanical energy is associated with the motion and position of objects.
- Important mechanical energy stores include kinetic, gravitational potential and elastic potential energy.
- Moving faster increases kinetic energy.
- Moving higher increases gravitational potential energy.
- Stretching or compressing an elastic object increases its elastic potential energy.
- Falling objects transfer energy from gravitational potential stores to kinetic stores.
- Rising objects transfer energy from kinetic stores to gravitational potential stores.
- Springs can transfer energy between elastic potential and kinetic stores.
- Roller coasters and pendulums repeatedly transfer energy between gravitational potential and kinetic stores.
- In an ideal system without friction or air resistance, mechanical energy remains constant.
- In real systems, some mechanical energy is transferred into thermal stores and through sound.
- Total energy is always conserved, even when mechanical energy decreases.
Did You Know?
The same principles can be explored with PhET's Energy Skate Park: Basics. The simulation lets learners watch the kinetic and gravitational potential energy change as a skater moves along different tracks, making conservation of mechanical energy visible in real time. OpenStax specifically uses this simulation when teaching mechanical energy and conservation.