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Energy stores, work done and energy calculations

Energy stores, work done and energy calculations

8.1.1 Energy stores and transfer diagrams

Systems and energy stores

Definition

System

A system is the object or group of objects chosen for study, with everything outside it treated as the surroundings.

Definition

Energy store

An energy store is a way in which energy is held in a system, such as a kinetic, gravitational, elastic, chemical, magnetic, electrostatic, nuclear or thermal store.

  1. A system is the object or group of objects you decide to study, and everything outside it is called the surroundings.
  2. Drawing the boundary of a system decides which energy changes you count and which ones are treated as happening outside it.
  3. Energy is measured in joules, J\text{J}J, whichever store is holding it.
  4. Every store is named after the reason the energy is there, so naming a store also means naming the object it belongs to.
  5. Eight energy stores are used at this level.
    1. A kinetic energy store holds energy because an object is moving.
    2. A gravitational potential energy store holds energy because two objects are separated in a gravitational field.
    3. An elastic potential energy store holds energy because an object has been stretched, squashed or twisted.
    4. A thermal energy store holds energy because of the random motion of the particles inside an object, and it grows as the temperature rises.
    5. A chemical energy store holds energy in the bonds inside fuels, food and the chemicals in a battery.
    6. An electrostatic energy store holds energy because electric charges have been pulled apart.
    7. A magnetic energy store holds energy because magnets or magnetic materials are held apart or pushed together.
    8. A nuclear energy store holds energy inside the nuclei of atoms.
  6. The gravitational potential energy store belongs to the object and the Earth together, because a separation needs two objects.
  7. A store is where energy sits, so words such as light, sound and electricity name ways of moving energy rather than stores.

Changes in the way energy is stored

  1. When a system changes, one store decreases while one or more other stores increase.
  2. Describing a change means naming the store that falls, naming the store that rises, and naming the object each store belongs to.
  3. As a ball falls, the gravitational potential energy store of the ball and Earth decreases while the ball's kinetic energy store increases, so the ball speeds up.
  4. When a stretched catapult is released, the elastic potential energy store of the rubber decreases and the kinetic energy store of the stone increases.
  5. When a battery drives a motor, the chemical energy store of the battery decreases and the kinetic energy store of the turning parts increases.
  6. When a mug of tea cools on a kitchen worktop, the thermal energy store of the tea decreases and the thermal energy store of the room increases.
  7. When a crane raises a steel beam, the gravitational potential energy store of the beam and Earth increases.
  8. When a firework burns, the chemical energy store of the powder decreases while the thermal and kinetic energy stores of the hot gases increase.

A ball held above the ground and then released, showing the gravitational potential energy store of the ball and Earth emptying into the ball's kinetic energy store as it falls.

Drawing energy transfer diagrams

  1. An energy transfer diagram shows the store the energy starts in, the store or stores it ends in, and an arrow for each transfer.
  2. Write both the store and the object it belongs to inside each box, such as chemical store of the battery.
  3. Draw every arrow from the store that decreases towards the store that increases, and put an arrowhead on it.
  4. One arrow means all of the energy ends in a single store, while several arrows mean the energy is shared between several stores.
  5. Label an arrow with the way the energy travels when the question asks for it, for example an arrow marked electrically for a battery driving a motor.
  6. For that battery and motor the diagram reads: chemical store of the battery ⟶\longrightarrow⟶ kinetic store of the motor.
  7. To interpret a diagram, follow each arrow from its tail to its head, then state that the tail store decreases and the head store increases.
  8. A Sankey diagram is an energy transfer diagram drawn with arrow widths in proportion to the number of joules, so the widest branch carries the most energy.
  9. The single arrow entering a Sankey diagram on the left stands for the total energy supplied, and the branches leaving it show how those joules are shared out.
  10. The widths of all the branches leaving a point add up to the width of the arrow entering it, which is a quick check that the diagram is drawn correctly.

general-sankey-diagram-27614283-genie.png

Energy in a closed system

Definition

Closed system

A closed system is a system across whose boundary no energy is transferred, so its total energy stays constant.

Definition

Conservation of energy

Conservation of energy means that energy cannot be created or destroyed, only transferred between stores.

  1. Conservation of energy means the total number of joules in a closed system is the same before and after any change, so Etotal before=Etotal afterE_{\text{total before}} = E_{\text{total after}}Etotal before​=Etotal after​.
  2. Energy is never created and never destroyed, so a change only moves joules from one store to another.
  3. A closed system therefore shows no net change in its total energy, and any decrease in one store is matched by an equal total increase in the others.
  4. If one store of a closed system falls by 250 J250\ \text{J}250 J, the remaining stores must gain 250 J250\ \text{J}250 J between them.
  5. Whether a system counts as closed depends on where the boundary is drawn, so a swinging pendulum on its own is not closed while the pendulum together with the air and its support is much closer to closed.
  6. Widening the boundary to include the surroundings makes the total easier to account for, because joules that leave the object are still inside the system.
Example

Describing a falling ball

  • A ball is held above a lawn and then released, and air resistance is small enough to ignore.
  • The gravitational potential energy store of the ball and Earth decreases as the ball drops.
  • The ball's kinetic energy store increases by the same number of joules, so the ball speeds up as it falls.
  • Treating the ball and Earth as a closed system, the total energy does not change during the fall.
  • Just before the ball lands, almost all of the energy that left the gravitational store is held in the ball's kinetic store.
Example

Reading a Sankey diagram

  • A Sankey diagram for a lamp has one arrow of width 60 J60\ \text{J}60 J entering on the left.
  • Two branches leave it, one of width 6 J6\ \text{J}6 J carrying energy away by light and one of width 54 J54\ \text{J}54 J going to the thermal energy store of the surroundings.
  • The branch widths add to 6+54=60 J6 + 54 = 60\ \text{J}6+54=60 J, which matches the arrow entering, so energy is conserved.
  • The wide branch shows that most of the energy supplied ends in the thermal energy store of the surroundings.
Exam technique

Writing about energy stores

  • Name the store and the object together, so write kinetic energy store of the trolley rather than movement energy.
  • Use the words decreases and increases for the two stores in a change, because the direction of the transfer usually carries its own mark.
  • Quote conservation of energy whenever a question mentions a closed system, then state that the total energy does not change.
  • Label every box with a store and put an arrowhead on every arrow when a diagram is asked for, because an unlabelled box scores nothing.
  • Check that the branch widths add up before quoting any value read from a Sankey diagram.
Common Mistake
  • Do not write that energy is used up, lost or destroyed, because energy is always conserved.
  • Do not treat light, sound, electricity or heat as energy stores, because each of them describes a transfer.
  • Do not write heat energy or movement energy, and use thermal energy store and kinetic energy store instead.
  • Do not give the gravitational store to the raised object alone, because that store belongs to the object and the Earth together.
Self review
  • Define a system and state what its boundary decides.
  • Name the eight energy stores and give one object that holds energy in each.
  • Describe the changes in energy stores as a catapult launches a stone.
  • Explain what an arrow on an energy transfer diagram tells you.
  • State what conservation of energy says about the total energy of a closed system.

8.1.2 Ways of changing a system's energy

Changing the energy of a system

Definition

Energy transfer pathway

An energy transfer pathway is the route by which energy moves from one store to another, such as mechanically, electrically, by heating or by radiation.

  1. The energy of a system changes only when joules cross its boundary into or out of one of its stores.
  2. An energy transfer pathway is the route those joules take, so a pathway is how energy moves while a store is where it ends up.
  3. Three pathways change the energy of a system in this topic: work done by forces, electrical equipment and heating.
  4. Naming the pathway and naming the store that changes are separate marks, so a full answer needs both.
  5. No pathway makes or destroys joules, so the number leaving one store equals the number arriving in the others.

Work done by forces

  1. A force does work on an object when the object moves in the direction of that force, and the energy is then transferred mechanically.
  2. Lifting a crate off the floor transfers energy mechanically into the gravitational potential energy store of the crate and Earth.
  3. Pushing a supermarket trolley from rest transfers energy mechanically into the trolley's kinetic energy store.
  4. Drawing a bow transfers energy mechanically into the elastic potential energy store of the bow.
  5. Friction between brake pads and a wheel transfers energy mechanically out of the bicycle's kinetic energy store and into thermal energy stores.
  6. Holding a heavy bag still at one height transfers no energy mechanically, because the bag does not move.
  7. Carrying that bag horizontally at a steady speed adds nothing to its gravitational store either, because the upward force and the movement are at right angles.

Electrical equipment

  1. Energy is transferred electrically when charge is pushed through a component by a power supply.
  2. The store of the supply falls as this happens, so a torch battery's chemical energy store decreases while the torch is switched on.
  3. An electric motor transfers energy electrically and increases the kinetic energy store of its turning parts.
  4. A kettle element transfers energy electrically and increases the thermal energy store of the water.
  5. A filament lamp transfers energy electrically, then passes it on by light and by heating the surroundings.
  6. A loudspeaker transfers energy electrically and then carries it away by sound waves through the surrounding air.
  7. Electrical equipment never creates joules, so every joule delivered to the components has come out of the store of the supply.

Heating

Definition

Heating

Heating is the transfer of energy from a region at a higher temperature to a region at a lower temperature because of the temperature difference between them.

  1. Heating transfers energy whenever two regions are at different temperatures, and it always runs from the hotter region to the cooler one.
  2. The thermal energy store of the hotter region decreases while the thermal energy store of the cooler region increases.
  3. The transfer keeps going until the two temperatures are equal, and it then stops.
  4. A gas hob heats a pan, so the pan's thermal energy store increases and its temperature rises.
  5. A mug of coffee left on a desk cools down because energy is transferred by heating to the cooler air of the room.
  6. Energy transferred by heating can travel by conduction, by convection or by radiation.
  7. Burning gas in a boiler empties the chemical energy store of the fuel and then transfers energy by heating into the water in the radiators.
Example

Tracing the pathways in an electric kettle

  • A kettle is plugged in and switched on to boil water for tea.
  • Energy is first transferred electrically from the mains supply to the heating element.
  • Energy is then transferred by heating from the hot element to the water, so the water's thermal energy store increases.
  • Energy is also transferred by heating from the kettle body to the kitchen air, so the room's thermal energy store increases as well.
  • Two pathways are named in this change, electrically and by heating, and each one is matched to the store that increases.
Example

Comparing a straight lift with a ramp

  • A worker raises a 20 kg20\ \text{kg}20 kg box onto a shelf by lifting it straight up.
  • The upward force moves the box in the direction of that force, so work is done and energy is transferred mechanically into the gravitational store.
  • A second worker slides an identical box up a ramp onto the same shelf.
  • The push moves the box along the ramp, so work is done mechanically again, and friction means part of that work also raises the thermal energy stores of the ramp and box.
  • Both routes change the system's energy through work done by forces, so the pathway is the same even though the forces and distances differ.
Exam technique

Naming the pathway and the store

  • Read the question for the words that fix the pathway, such as a force, a circuit or a temperature difference.
  • Write the pathway in the board's own wording: mechanically, electrically or by heating.
  • Follow the pathway with the store that changes, for example energy is transferred electrically, so the thermal energy store of the water increases.
  • Give one pathway for each transfer the question asks about, because a two-mark question usually wants two separate transfers.
  • The mark for direction is lost unless you say which store increases and which decreases.
Common Mistake
  • Do not name a pathway when a store is asked for, because electrically and by heating are not stores.
  • Do not write that a device makes or produces energy, because equipment only moves joules between stores.
  • Do not say work is done when the object does not move, because a force on its own transfers no energy.
  • Do not use heat as a store, because heating is the pathway and the thermal energy store is the destination.
Self review
  • Name the three ways the energy of a system can be changed.
  • Explain the difference between an energy store and an energy transfer pathway.
  • State why holding a bag still at one height transfers no energy mechanically.
  • Describe the pathway and the store that changes when a kettle heats water.
  • Explain what decides the direction of an energy transfer by heating.

8.1.3 Work done by forces

Work done and energy transferred

Definition

Work done

Work done is the energy transferred when a force moves an object through a distance in the direction of the force.

Definition

Joule

The joule is the unit of energy and of work done, equal to the work done when a force of one newton moves an object one metre in the direction of the force.

  1. Work done is the energy a force transfers when it moves an object through a distance in the direction of that force.
  2. Work done and energy transferred are the same quantity, so the work done by a force equals the energy it transfers.
  3. Both are measured in joules, J\text{J}J, while force is measured in newtons, N\text{N}N, and distance in metres, m\text{m}m.
  4. One joule is the work done when a force of 1 N1\ \text{N}1 N moves an object 1 m1\ \text{m}1 m in the direction of the force, so 1 J=1 N m1\ \text{J} = 1\ \text{N}\ \text{m}1 J=1 N m.
  5. A force that acts without producing any movement does no work, so it transfers no energy.
  6. Only the part of the movement lying along the force counts, so a bag carried horizontally gains nothing from the upward force holding it.

The work done equation

  1. Work done is calculated from E=F×dE = F \times dE=F×d.
  2. In this equation EEE is the work done in joules, FFF is the force in newtons and ddd is the distance moved in the direction of the force in metres.
  3. Rearranging gives F=EdF = \dfrac{E}{d}F=dE​ when the force is wanted and d=EFd = \dfrac{E}{F}d=FE​ when the distance is wanted.
  4. Convert every distance into metres before substituting, so 60 cm=0.60 m60\ \text{cm} = 0.60\ \text{m}60 cm=0.60 m and 1.5 km=1500 m1.5\ \text{km} = 1500\ \text{m}1.5 km=1500 m.
  5. For a lift straight upwards at a steady speed the force needed equals the weight, so the work done is the weight in newtons multiplied by the vertical distance in metres.
  6. Doubling the force doubles the work done over the same distance, and doubling the distance doubles the work done for the same force.
  7. A force at right angles to the movement contributes no distance in its own direction, so it does no work.
Example

Dragging a crate across a floor

  • A stagehand drags a crate 4.0 m4.0\ \text{m}4.0 m along a flat floor using a horizontal force of 25 N25\ \text{N}25 N.
  • The equation is E=F×dE = F \times dE=F×d.
  • Substituting gives E=25×4.0E = 25 \times 4.0E=25×4.0.
  • The work done is E=100 JE = 100\ \text{J}E=100 J, so the pulling force transfers 100 J100\ \text{J}100 J of energy.
Example

Finding a force from the work done

  • A winch transfers 4500 J4500\ \text{J}4500 J while pulling a sledge 30 m30\ \text{m}30 m along level ground.
  • Rearranging E=F×dE = F \times dE=F×d gives F=EdF = \dfrac{E}{d}F=dE​.
  • Substituting gives F=450030F = \dfrac{4500}{30}F=304500​.
  • The force is F=150 NF = 150\ \text{N}F=150 N, acting along the direction of the movement.
Practical

Measuring the work done by a force

  • Aim: to measure the work done by a known force as it drags a wooden block along a bench.
  • Apparatus: wooden block, newton meter reading to 0.1 N0.1\ \text{N}0.1 N, metre rule, bench or runway, masking tape, slotted masses, balance and clamp.
  • Variables: the distance moved in the direction of the force is the independent variable, the work done is the dependent variable, and the block, the mass on top of it, the bench surface and the pulling direction are controlled.
  • Method, set-up:
    • Weigh the block, then tape a start line and a finish line to the bench and measure the distance between them with the metre rule.
    • Hook the newton meter to the block and hold it horizontally, in line with the direction of travel, so the reading is the pulling force.
    • Zero the newton meter while it is held in the pulling direction with nothing attached.
  • Method, measurements:
    • Pull the block from the start line to the finish line at a slow, steady speed so the reading stays as constant as possible.
    • Read the newton meter while the block is moving rather than before it starts, because a larger force is needed to break it free.
    • Record the steady force in newtons and the distance in metres.
    • Repeat each run three times and calculate the mean force.
    • Repeat for at least five distances by moving the finish line further along the bench.
  • Results: the work done rises in proportion to the distance moved while the pulling force is kept the same.
  • Maths: calculate the work done for each distance from E=F×dE = F \times dE=F×d, then plot EEE on the vertical axis against ddd on the horizontal axis; the graph is a straight line through the origin and its gradient equals the pulling force FFF in newtons.
  • Watch out: a newton meter held at an angle reads more than the horizontal pulling force, a block that accelerates is not being pulled by a steady force, and a dusty or worn patch of bench changes the friction part way along.
  • Safety: clamp the runway or keep the block well away from the bench edge, keep fingers clear of falling masses, and use modest masses so the block stays under control.

Energy changes when work is done

  1. Work done by a force always changes a store, so every E=F×dE = F \times dE=F×d calculation can also be described as an energy transfer.
  2. Lifting an object at a steady speed puts the work done into the gravitational potential energy store of the object and Earth.
  3. A resultant force that speeds an object up puts the work done into the object's kinetic energy store.
  4. Stretching a spring puts the work done into the elastic potential energy store of the spring.
  5. Work done against friction or air resistance puts the energy into the thermal energy stores of the surfaces and the surroundings, so their temperature rises.
  6. A car braking from 30 m/s30\ \text{m/s}30 m/s has the work done by the braking force taken out of its kinetic energy store and put into the thermal energy stores of the brakes, tyres and road.
  7. To describe any of these changes, name the force, state that the object moves in the direction of that force, state that work is done, then name the store that increases and the store that decreases.
Example

Work done against friction while braking

  • A cyclist and bicycle of total mass 85 kg85\ \text{kg}85 kg are brought to rest by a braking force of 340 N340\ \text{N}340 N acting over 12 m12\ \text{m}12 m.
  • The work done by the braking force is E=F×d=340×12E = F \times d = 340 \times 12E=F×d=340×12.
  • This gives E=4080 JE = 4080\ \text{J}E=4080 J.
  • The cyclist's kinetic energy store therefore falls by 4080 J4080\ \text{J}4080 J as the bicycle stops.
  • Those joules are transferred into the thermal energy stores of the brake blocks, the wheel rims and the surrounding air, which is why the rims feel warm.
Exam technique

Setting out a work done answer

  • Write E=F×dE = F \times dE=F×d before any numbers, because the equation itself often carries a mark.
  • Check that the distance used is measured along the direction of the force and not the whole path length.
  • Convert centimetres and kilometres into metres, and never substitute a mass where a force is needed.
  • Give the answer in joules, and use kJ\text{kJ}kJ only when the question asks for it.
  • For a describe question, join the ideas with so, such as the force moves the crate, so work is done and energy is transferred to the thermal store of the floor.
Common Mistake
  • Do not give a force in joules, because joules measure work done and energy while newtons measure force.
  • Do not multiply by the whole path length when only part of the movement lies along the force.
  • Do not claim work is done while a bag is simply held still, because there is no movement.
  • Do not confuse mass in kilograms with weight in newtons when the work done in a lift is calculated.
Self review
  • Define work done and state its unit.
  • Write the equation linking work done, force and distance.
  • Explain why 1 J1\ \text{J}1 J is the same as 1 N m1\ \text{N}\ \text{m}1 N m.
  • Describe how to measure the work done by a force dragging a block along a bench.
  • Name the store that increases when work is done against friction.

8.2.1 Gravitational potential energy

The gravitational potential energy store

Definition

Gravitational potential energy

Gravitational potential energy is energy stored by an object because of its position in a gravitational field.

Definition

Gravitational field strength

Gravitational field strength is the force per unit mass acting on an object placed in a gravitational field, measured in newtons per kilogram (N/kg).

  1. Gravitational potential energy is stored whenever an object is raised in a gravitational field, and the store belongs to the object and the Earth together.
  2. The change in this store depends on the object's mass, the gravitational field strength and the change in vertical height.
  3. A heavier object has a larger weight, so more energy has to be transferred to raise it through the same height.
  4. Raising the same object twice as high transfers twice as much energy into the store.
  5. Gravitational field strength at the Earth's surface is taken as g=10 N/kgg = 10\ \text{N/kg}g=10 N/kg unless a question gives a different value.
  6. The value of ggg on the Moon is much smaller, so the same lift through the same height stores far fewer joules there.
  7. Only changes in this store are calculated, because the height counted as zero is chosen rather than fixed by nature.

The change in GPE equation

  1. The change in the store is calculated from ΔGPE=m×g×Δh\Delta GPE = m \times g \times \Delta hΔGPE=m×g×Δh.
  2. Here ΔGPE\Delta GPEΔGPE is the change in gravitational potential energy in joules, mmm is the mass in kilograms, ggg is the gravitational field strength in N/kg\text{N/kg}N/kg and Δh\Delta hΔh is the change in vertical height in metres.
  3. The symbol Δ\DeltaΔ means change in, so Δh\Delta hΔh is the final height minus the starting height.
  4. Rearranging gives m=ΔGPEg×Δhm = \dfrac{\Delta GPE}{g \times \Delta h}m=g×ΔhΔGPE​ for the mass and Δh=ΔGPEm×g\Delta h = \dfrac{\Delta GPE}{m \times g}Δh=m×gΔGPE​ for the height.
  5. Convert grams into kilograms and centimetres into metres first, so 250 g=0.250 kg250\ \text{g} = 0.250\ \text{kg}250 g=0.250 kg and 40 cm=0.40 m40\ \text{cm} = 0.40\ \text{m}40 cm=0.40 m.
  6. The height Δh\Delta hΔh is measured straight upwards, so an object pushed up a ramp gains the store set by the vertical rise and not by the length of the slope.
  7. The store increases while the object is being raised and decreases while it falls, so a falling object is emptying this store.

Linking the store to work done

  1. The weight of an object is its mass multiplied by the gravitational field strength, measured in newtons, and this is the force a steady lift has to balance.
  2. The work done in that lift is E=F×d=m×g×ΔhE = F \times d = m \times g \times \Delta hE=F×d=m×g×Δh, which is the same expression as the change in the store.
  3. Lifting a mass therefore transfers the work done straight into the gravitational potential energy store, which is why the two equations agree.
  4. Work done against friction on a ramp is extra to m×g×Δhm \times g \times \Delta hm×g×Δh, so a ramp needs more energy in total than a straight lift to the same height.
Example

Lifting a box onto a shelf

  • A shelf stacker raises a box of mass 25 kg25\ \text{kg}25 kg through a vertical height of 3.0 m3.0\ \text{m}3.0 m, where g=10 N/kgg = 10\ \text{N/kg}g=10 N/kg.
  • The equation is ΔGPE=m×g×Δh\Delta GPE = m \times g \times \Delta hΔGPE=m×g×Δh.
  • Substituting gives ΔGPE=25×10×3.0\Delta GPE = 25 \times 10 \times 3.0ΔGPE=25×10×3.0.
  • The change in the store is ΔGPE=750 J\Delta GPE = 750\ \text{J}ΔGPE=750 J, and the store increases because the box has been raised.
Example

Pushing a wheelbarrow up a ramp

  • A wheelbarrow of mass 40 kg40\ \text{kg}40 kg is pushed 5.0 m5.0\ \text{m}5.0 m up a ramp that rises 1.2 m1.2\ \text{m}1.2 m vertically, with g=10 N/kgg = 10\ \text{N/kg}g=10 N/kg.
  • The vertical rise is the value needed, so Δh=1.2 m\Delta h = 1.2\ \text{m}Δh=1.2 m and the 5.0 m5.0\ \text{m}5.0 m measured along the slope is not used.
  • Substituting gives ΔGPE=40×10×1.2\Delta GPE = 40 \times 10 \times 1.2ΔGPE=40×10×1.2.
  • The gravitational potential energy store increases by ΔGPE=480 J\Delta GPE = 480\ \text{J}ΔGPE=480 J.
Example

Finding a height from the energy stored

  • A crane transfers 36 000 J36\,000\ \text{J}36000 J into the gravitational store of a 300 kg300\ \text{kg}300 kg girder, with g=10 N/kgg = 10\ \text{N/kg}g=10 N/kg.
  • Rearranging gives Δh=ΔGPEm×g\Delta h = \dfrac{\Delta GPE}{m \times g}Δh=m×gΔGPE​.
  • Substituting gives Δh=36 000300×10\Delta h = \dfrac{36\,000}{300 \times 10}Δh=300×1036000​.
  • The girder has been raised through Δh=12 m\Delta h = 12\ \text{m}Δh=12 m.
Exam technique

Getting the GPE values right

  • Write ΔGPE=m×g×Δh\Delta GPE = m \times g \times \Delta hΔGPE=m×g×Δh first, then list the three values you are about to substitute.
  • Take the value of ggg from the question, and use 10 N/kg10\ \text{N/kg}10 N/kg only when no value is given.
  • Read the diagram carefully for the vertical height, because ramp lengths and slope distances are printed there to catch you out.
  • State whether the store increases or decreases when a description is wanted as well as a number.
  • Answer in joules, and convert to kilojoules only when the question asks for them.
Common Mistake
  • Do not put a weight in newtons into the mmm position, because that position needs a mass in kilograms.
  • Do not use the distance measured along a slope as Δh\Delta hΔh, because only the vertical rise changes this store.
  • Do not call ggg gravity, because the quantity is gravitational field strength measured in N/kg\text{N/kg}N/kg.
  • Do not leave a mass in grams, because the equation needs kilograms.
Self review
  • State what gravitational potential energy is and which two objects share the store.
  • Write the equation for the change in gravitational potential energy.
  • Give the unit of each quantity in that equation.
  • Explain why the vertical rise is used for a load pushed up a ramp.
  • Calculate the change in the store when a 2.0 kg2.0\ \text{kg}2.0 kg book is raised 1.5 m1.5\ \text{m}1.5 m with g=10 N/kgg = 10\ \text{N/kg}g=10 N/kg.

8.2.2 Kinetic energy

The kinetic energy store

Definition

Kinetic energy

Kinetic energy is energy stored by an object because it is moving.

  1. Kinetic energy is stored by an object because it is moving, so every moving object holds some.
  2. The size of the store depends on the object's mass and on its speed.
  3. Speed matters more than mass, because the store depends on the square of the speed.
  4. A stationary object has a speed of 0 m/s0\ \text{m/s}0 m/s, so its kinetic energy store holds 0 J0\ \text{J}0 J.
  5. The store fills as an object speeds up and empties as it slows down.
  6. Kinetic energy is measured in joules, J\text{J}J, like every other store.

The kinetic energy equation

  1. The store is calculated from KE=12×m×v2KE = \dfrac{1}{2} \times m \times v^{2}KE=21​×m×v2.
  2. Here KEKEKE is the kinetic energy in joules, mmm is the mass in kilograms and vvv is the speed in metres per second, m/s\text{m/s}m/s.
  3. Square the speed before multiplying, so a speed of 6.0 m/s6.0\ \text{m/s}6.0 m/s gives v2=36 m2/s2v^{2} = 36\ \text{m}^{2}/\text{s}^{2}v2=36 m2/s2.
  4. Writing the working as KE=0.5×m×v2KE = 0.5 \times m \times v^{2}KE=0.5×m×v2 makes the factor of one half harder to forget.
  5. Rearranging for speed gives v=2×KEmv = \sqrt{\dfrac{2 \times KE}{m}}v=m2×KE​​, and rearranging for mass gives m=2×KEv2m = \dfrac{2 \times KE}{v^{2}}m=v22×KE​.
  6. Doubling the mass at the same speed doubles the store, because the mass is not squared.
  7. Doubling the speed at the same mass makes the store four times larger, since 22=42^{2} = 422=4.
  8. Tripling the speed makes the store nine times larger, since 32=93^{2} = 932=9.
  9. Convert grams into kilograms, and convert any speed given in km/h\text{km/h}km/h into m/s\text{m/s}m/s, before substituting.

Kinetic energy in falling and braking

  1. A falling object empties its gravitational potential energy store and fills its kinetic energy store, so 12×m×v2=m×g×Δh\dfrac{1}{2} \times m \times v^{2} = m \times g \times \Delta h21​×m×v2=m×g×Δh when resistive forces are small.
  2. Cancelling the mass gives v=2×g×Δhv = \sqrt{2 \times g \times \Delta h}v=2×g×Δh​, which shows that the landing speed does not depend on how heavy the object is.
  3. A vehicle's kinetic energy store has to be emptied before it stops, and the work done by the braking force equals that store, so F×d=12×m×v2F \times d = \dfrac{1}{2} \times m \times v^{2}F×d=21​×m×v2.
  4. Because the speed is squared, doubling a car's speed makes its kinetic energy store four times larger and so needs about four times the distance to stop with the same braking force.
  5. Those joules are transferred into the thermal energy stores of the brakes, tyres and road, which is why brake discs become hot.
Example

Kinetic energy of a cyclist

  • A cyclist and bicycle have a combined mass of 80 kg80\ \text{kg}80 kg and travel at 6.0 m/s6.0\ \text{m/s}6.0 m/s.
  • The equation is KE=12×m×v2KE = \dfrac{1}{2} \times m \times v^{2}KE=21​×m×v2.
  • Squaring the speed gives v2=6.02=36v^{2} = 6.0^{2} = 36v2=6.02=36.
  • Substituting gives KE=0.5×80×36KE = 0.5 \times 80 \times 36KE=0.5×80×36.
  • The kinetic energy store holds KE=1440 JKE = 1440\ \text{J}KE=1440 J.
Example

A mass given in grams

  • A cricket ball of mass 160 g160\ \text{g}160 g is thrown at 18 m/s18\ \text{m/s}18 m/s.
  • Converting the mass gives 160 g=0.160 kg160\ \text{g} = 0.160\ \text{kg}160 g=0.160 kg.
  • Squaring the speed gives v2=182=324v^{2} = 18^{2} = 324v2=182=324.
  • Substituting gives KE=0.5×0.160×324KE = 0.5 \times 0.160 \times 324KE=0.5×0.160×324.
  • The kinetic energy store holds KE=25.9 JKE = 25.9\ \text{J}KE=25.9 J to three significant figures.
Example

Speed of a falling stone

  • A stone is dropped from a bridge 20 m20\ \text{m}20 m above a river, with g=10 N/kgg = 10\ \text{N/kg}g=10 N/kg and air resistance small enough to ignore.
  • The gravitational store emptied equals the kinetic store filled, so 12×m×v2=m×g×Δh\dfrac{1}{2} \times m \times v^{2} = m \times g \times \Delta h21​×m×v2=m×g×Δh.
  • Cancelling the mass and rearranging gives v=2×g×Δhv = \sqrt{2 \times g \times \Delta h}v=2×g×Δh​.
  • Substituting gives v=2×10×20=400v = \sqrt{2 \times 10 \times 20} = \sqrt{400}v=2×10×20​=400​.
  • The stone reaches the water at v=20 m/sv = 20\ \text{m/s}v=20 m/s.
Example

Finding a speed from the store

  • A trolley of mass 2.5 kg2.5\ \text{kg}2.5 kg holds 45 J45\ \text{J}45 J in its kinetic energy store.
  • Rearranging gives v=2×KEmv = \sqrt{\dfrac{2 \times KE}{m}}v=m2×KE​​.
  • Substituting gives v=2×452.5=36v = \sqrt{\dfrac{2 \times 45}{2.5}} = \sqrt{36}v=2.52×45​​=36​.
  • The trolley is moving at v=6.0 m/sv = 6.0\ \text{m/s}v=6.0 m/s.
Exam technique

Working with the squared speed

  • Write KE=12×m×v2KE = \dfrac{1}{2} \times m \times v^{2}KE=21​×m×v2 and square the speed on its own line so the working can be followed.
  • Square only the speed, never the mass and never the one half.
  • Keep the full value in the calculator when a rearranged answer is wanted, then take the square root last.
  • Compare two kinetic energy stores by comparing the masses and the squares of the speeds rather than the speeds themselves.
  • Round to the same number of significant figures as the data in the question, which is usually two or three.
Common Mistake
  • Do not multiply by the speed once instead of squaring it, because that badly underestimates the store.
  • Do not leave the mass in grams, because using grams in place of kilograms makes the answer a thousand times too large.
  • Do not forget the factor of 12\dfrac{1}{2}21​, which is the most commonly dropped mark in this calculation.
  • Do not say that a heavier object always holds more kinetic energy, because a light object moving quickly can hold more.
Self review
  • State what kinetic energy is and give its unit.
  • Write the equation for kinetic energy and name each quantity in it.
  • Explain what happens to the kinetic energy store when the speed triples.
  • Calculate the kinetic energy of a 1200 kg1200\ \text{kg}1200 kg car travelling at 15 m/s15\ \text{m/s}15 m/s.
  • Explain why the landing speed of a dropped object does not depend on its mass.

8.2.3 Dissipation of energy

Dissipation of energy

Definition

Dissipation

Dissipation is the spreading of transferred energy into the thermal energy stores of the surroundings, which makes the energy less useful for further transfers.

  1. Every real change in a system transfers some energy into stores that were not wanted, and that energy is said to be dissipated.
  2. Dissipated energy almost always ends up in the thermal energy stores of the object and its surroundings.
  3. Conservation of energy still holds, so all of the joules are still there and none have been destroyed.
  4. Dissipated energy is described as less useful because it has been shared out between huge numbers of particles in the surroundings.
  5. Spread that thinly, the energy raises the temperature of the surroundings by only a tiny amount and cannot be gathered back to drive the change again.
  6. Whether a transfer counts as useful depends on the job the device is meant to do, so heating is the useful transfer in a kettle and an unwanted one in a drill.
  7. Sound waves also carry energy away from a system, and those joules finish in the thermal energy store of the surroundings as well.

Why mechanical processes waste energy

  1. A mechanical process is one in which forces do work, such as a motor turning a shaft or a wheel rolling along a road.
  2. Whenever two surfaces slide or rub across each other, friction acts between them and work is done against it.
  3. That work transfers energy into the thermal energy stores of both surfaces, so their temperature rises.
  4. A mechanical process becomes wasteful when it causes this rise in temperature, because the energy is then dissipated into the surroundings by heating.
  5. Moving through air or water means doing work against air resistance or drag, which warms both the fluid and the moving object.
  6. The rise in temperature is the signature of the waste, so a machine that runs hot is dissipating a large share of the energy supplied to it.
  7. The unwanted transfer also depends on how long the process runs, because a longer run dissipates more joules.

Tracking dissipation in real systems

  1. A bouncing ball rebounds to a lower height each time, because energy is dissipated at every bounce.
  2. During a bounce the ball squashes and then recovers, and the work done inside the rubber warms both the ball and the ground.
  3. The bounce also makes a sound, so a further share of the energy leaves the ball as sound waves.
  4. Less energy is therefore returned to the gravitational potential energy store each time, so each bounce is lower than the one before.
  5. A car left in neutral rolls to a stop because friction in the bearings and air resistance move its kinetic energy store into thermal energy stores.
  6. A filament lamp transfers most of the energy supplied to it by heating the surroundings rather than by light, so most of that transfer is unwanted.
  7. Brakes are the case that proves the rule, because they are designed to raise a thermal energy store, yet once those joules are spread through the brakes and air they can no longer move the car.
Example

Bounce heights of a dropped ball

  • A tennis ball is dropped from 1.0 m1.0\ \text{m}1.0 m onto a hard playground surface and rebounds to about 0.55 m0.55\ \text{m}0.55 m.
  • Falling empties the ball's gravitational potential energy store and fills its kinetic energy store.
  • At the moment of impact the ball squashes, so work is done inside the rubber and against the ground.
  • That work transfers energy into the thermal energy stores of the ball, the ground and the air, and some energy leaves as sound waves.
  • Only the energy still in the ball's kinetic store after the bounce can refill its gravitational store, so the ball rises to a lower height.
  • Every later bounce dissipates more energy, so the heights keep falling until the ball stays on the ground.
Example

Dissipation in an electric drill

  • A drill is used to make a hole in a wall, and its casing feels warm afterwards.
  • Energy is transferred electrically from the supply and increases the kinetic energy store of the rotating bit, which is the useful transfer.
  • Friction acts in the bearings, in the gears and between the bit and the wall, so work is done against friction.
  • That work transfers energy by heating into the thermal energy stores of the drill, the wall and the surrounding air.
  • The warm casing shows where the wasted joules have gone, and they cannot be recovered to turn the bit.
Exam technique

Explaining dissipation for full marks

  • Name the store that holds the energy at the start, such as the kinetic energy store of the car.
  • Name the cause of the unwanted transfer, which is usually friction, air resistance or the squashing of a material.
  • State that work is done against that force, so energy is transferred by heating.
  • Name the thermal energy stores that increase, and include the surroundings as well as the object itself.
  • Finish by saying that the energy has become spread out and is now less useful, which is the mark most answers miss.
Common Mistake
  • Do not write that energy is lost or destroyed, and write that it is dissipated or transferred to less useful stores instead.
  • Do not write heat energy, because energy is transferred by heating into a thermal energy store.
  • Do not claim the total energy falls during dissipation, because the total is unchanged and only its spread has altered.
  • Do not treat sound as a store, because sound waves are a way of carrying energy away from the system.
Self review
  • Define dissipation and name the store dissipated energy usually ends in.
  • Explain what less useful means for dissipated energy.
  • State what makes a mechanical process wasteful.
  • Explain why a bouncing ball does not return to its starting height.
  • Name two forces that cause unwanted energy transfers in a moving vehicle.

Recap questions

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A ball falls straight down and speeds up. Ignoring air resistance, which statement best describes the main energy transfer?

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Energy stores diagram of a falling ball showing a decreasing gravitational potential store, an increasing kinetic store, and a small transfer to thermal stores by air resistance within the system ball plus Earth plus surroundings

In GCSE Physics, energy is not "used up". We describe changes as transfers between energy stores inside a chosen system.

A system is the object or group of objects we are focusing on, such as the ball and the Earth. For a falling ball, the gravitational potential store decreases while the kinetic store increases, and air resistance can transfer some energy to thermal stores.

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An automated guided vehicle (AGV) used in a distribution centre has a total mass of 240 kg.

The AGV travels along a straight, horizontal corridor at a constant speed of 1.5 m/s.

Calculate the kinetic energy of the AGV.

Use the equation:

kinetic energy=0.5×mass×speed2 \text{kinetic energy} = 0.5 \times \text{mass} \times \text{speed}^2 kinetic energy=0.5×mass×speed2

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8.1 Energy stores, work done and energy calculations Revision Guide

  1. GCSE
  2. /Physics
  3. /8.1 Energy stores, work done and energy calculations

Revision notes for Edexcel GCSE Physics 8.1 Energy stores, work done and energy calculations. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Physics (1PH0) specification, so the content matches what's examinable rather than general Physics background.