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Conservation of energy: 7.2 Energy transfer - efficiency calculations, Sankey diagrams and useful work and wasted energy - lots of practice exam questions with worked out answers

Doc Brown's Physics exam study revision notes

Energy transfer and efficiency - useful work output and wasted energy

Introduction- reminders via a 'car' example:

Law of conservation of energy - energy cannot be created or destroyed.

BUT, not all the energy in a system is useful, there is always some wasted or dissipated energy and their relative values can be quite different.

In practical terms: Energy input = Useful output (useful work)  +  wasted (dissipated) energy

e.g. an electric car can have an efficiency of over 80%, that is over 80% of the stored electrical energy actually gets used to move the car. Most energy loss is from friction from moving parts and air resistance.

However, the electricity must be generated - if its from fossil fuels, the efficiency of electricity production drops to ~35%, but from renewable energy resources like wind turbines the overall efficiency is much higher.

However, for a petrol/diesel fuelled car the efficiency is less than 20%, meaning over 80% of the chemical energy store of the fuel is lost as heat from the engine and friction from the moving parts (more so than in an electric car) and air resistance.

In many cases the wasted or dissipated energy increases the thermal energy store of the surroundings.

So, ....

• Know and understand that when energy is transferred only part of it may be usefully transferred, the rest is ‘wasted’.

• BUT there is no change to the total energy of the system.

• Know and understand that wasted energy is eventually transferred to the surroundings, which will become warmer - increasing its thermal energy store.

• Appreciate that the wasted (dissipated) energy becomes increasingly spread out and so becomes less useful.

• Be able to calculate the efficiency of a device using the equations:

• Efficiency can be expressed as a decimal fraction from 0.0 to 1.0, but it is usually quoted as a percentage.

• The equation is quite simple and can be calculated from energy or power data.

• efficiency = useful energy out / total energy in

• efficiency = useful power out / total power in

• So, expressed as a percentage from 0 to 100

• % efficiency = useful energy out x 100 / total energy in

• % efficiency = useful power out x 100 / total power in

• With the x100 the efficiency has a value between 0% and 100%.

• There are other ways to express the efficiency formula e.g.

• efficiency = useful energy transferred to device / total energy supplied to the device

• You must be able to calculate efficiency as a decimal fraction (i.e. omit the x 100) or as a percentage (0-100%).

• The efficiency formula triangle if you need to rearrange the equation to calculate energy in or energy out.

• It should be pointed out that virtually no device is 100% efficient, there is no such device as a perfect machine.

• Friction is one of the principal ways in which energy is wasted when machines are operating.

• This is because when any work is done mechanically, friction forces must be overcome because moving parts are rubbing against each other.

• Work is done against the resistive force of friction.

• The friction produces thermal energy (things heat up, and often some sound too) which is lost to increase the thermal energy store of the machine and its surroundings.

• This wasted energy, by definition, cannot contribute to the useful work output.

• The heat may be conducted away or radiated away to increase the thermal energy store of the surroundings - the machine itself or surrounding air.

• Wherever possible, lubricants like oil and grease, which flows easily, are used to minimise the friction between moving parts that touch each other - bicycles, cars and locomotives are good examples of the application of lubrication.

• Without the use of lubricants on moving parts, any machine can overheat causing damage - very expensive!

• So, more useful work is done, less wasted energy, efficiency increased and money save on fuel and repairs!

A good example is oiling the gears and axles on a mountain bike.

The same arguments apply to any moving machinery where one surface has contact with another.

• In terms of a working machine or device you can state the efficiency formula as

• % efficiency = 100 x useful energy transferred by device / total energy supplied to the device

• REMINDER - the efficiency is rarely 100%, there are usually energy losses - there is no such thing as a perfect machine!

• e.g. any machine with moving parts will experience friction somewhere and in most cases the wasted energy ends up in a useless thermal energy store e.g. the surroundings.

• However, there are circumstances when you can get an initial 100% efficiency e.g.

• An electric heater will convert all the energy from the electrical energy store into heat, all of which is transferred to the thermal energy store of the room - which is where you want the heat. You may get subsequent loss of heat from the room by conduction or convection etc. BUT they are other thermal energy store transfers!

Sankey diagrams and efficiency - wasted energy and energy analysis of using electrical appliances

These are quantitative diagrams to show how the energy is distributed in two or more ways - useful and waste energy.

A generalised Sankey diagram

E = the energy input,  U = useful work or energy output, W = wasted (dissipated) energy

From the law of conservation of energy: E = W + U

(Remember W is usually one form of energy, but U could involve several forms of energy.)

Be able to interpret and draw a Sankey diagram.

• You should be able to use a Sankey diagram to calculate the efficiency of an appliance.

• From a Sankey diagram you can see quite clearly in a visual way what happens to the energy input into a device ie what proportion of energy was useful and how much energy was wasted.

• The breadth of the base of each arrow is proportional to the percentage of that energy output.

• The greater the width of the 'arrow' the greater proportion of energy it represents.

• Sankey diagram for an electric motor e.g. in a domestic appliance.

• An electric motor illustrates the idea of a Sankey diagram.

• You will find an electrical motor in devices such as an electric drill, washing machine, food mixer, electric car etc. etc. A lot of our lives runs on electrical power!

• The Sankey diagram above analyses what happens to every 100 J of electrical energy that is used by the electric motor.

• The total energy input and outputs can be expressed as ...

• total energy in = total energy out = T (J) = K (J) + S (J) + H (J)

• total energy in = total energy out = T% = 100% = K% + S% + H% = 56% + 17% + 27%

• % Efficiency = 56 x 100 / 100 = 56%

• This electric motor only has an efficiency of 56% useful energy out as kinetic energy with losses of 17% sound from friction-vibration and 27% heat energy loss from moving parts friction or warm electrical wiring in the motor.

• In this example the numbers are easy, but whatever the numbers are, from the Sankey diagram, you need to be able to get to the proportion of useful energy and covert it to a percentage of the total energy input.

• Sankey diagram for a cooling fan

• This is a more accurate Sankey diagram drawn to scale on 'graph paper'.

• To interpret it correctly, you need to be able to !

• The Law of conservation of energy states that energy cannot be created but only changes from one form to another.

• Know that appliances transfer energy but they rarely transfer all of the energy to the place we want, but whatever happens to the energy, its neither lost nor gained, but not all of it is usefully transferred.

• The Sankey style diagram above illustrates how you can represent what happens to the energy when some kind of energy consuming device is in operation.

• The input energy quantity indicated by the purple arrow (left) must equal the useful energy output of the blue arrow (right) plus the wasted energy output of the red arrow (down).

• If you can decrease the energy wasted you can increase the efficiency of the device.

• Appreciate that we need to know the efficiency of appliances so that we can choose between them, including how cost effective they are, and how to improve them.

• Devices eg industrial machines, household appliances, light bulbs etc. can only be useful if they can efficiently transform one form of 'source' energy ('total input') into an appreciable percentage of a 'useful' energy ('useful output').

• Such devices should be designed to 'waste' as little energy as possible, the less waste energy, the greater the efficiency of the device.

• See the energy flow diagram and efficiency formula 'triangle' above and the calculation of efficiency and Sankey diagram further down the page.

• It should be pointed out that virtually no device is 100% efficient, there is no such device as a perfect machine.

• One of the most efficient 'energy converters' is an electric heater were nearly all the input electrical energy is converted into useful output heat energy

• It is sometimes stated that an electrical heat is 100% efficient, BUT if the electric bar is glowing and visible, then there must be some energy loss as light.

• Whether of not the energy outputs are useful or waste, most of the energy input in a device ultimately ends up as heat energy.

• The most useful energy sources are in a sense 'highly concentrated' like a battery or a fuel like petrol, but as the energy is used you cannot recover the waste energy - it has been dissipated to the surroundings and become useless heat energy in this 'diluted' form.

A Sankey diagram for a cooling fan (above and below) presented in terms of absolute energy values in kJ or % useful output and % wasted energies.

The Sankey diagram for an electric motor.

SET A Questions - further examples of energy transfer analysis

(BUT not using a Sankey diagram, the problems are set out as a data table to solve using the law of conservation of energy)

Q1 is based on using a hair dryer

 electrical energy => useful/wasted energy outputs energy input transfers as a 3 J/s of wasted sound energy of 600 J/s to hair dryer 590 J/s of useful heat energy the appliance is being used ? J/s of ? energy

Q1 (a) The table above summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the hair dryer is in use?

 electrical energy => useful/wasted energy outputs energy input transfers as a 5 J/s of useful kinetic energy of 400 J/s to hair dryer 393 J/s of useful heat energy the appliance is being used ? J/s of ? energy

Q1 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the hair dryer is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 4 J/s of useful kinetic energy of 300 J/s to hair dryer 1 J/s of wasted sound energy the appliance is being used 95 J/s of wasted heat energy ? J/s of ? energy

Q1 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the hair dryer is in use?

Q2 is based on using a food mixer

 electrical energy => useful/wasted energy outputs input energy transfers as a 50 J/s of wasted sound energy of 1000 J/s to food mixer 400 J/s of useful kinetic energy the appliance is being used ? J/s of ? energy

Q2 (a) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the food mixer is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 80 J/s of wasted sound energy of 1200 J/s to food mixer 620 J/s of wasted heat energy the appliance is being used ? J/s of ? energy

Q2 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the food mixer is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 370 J/s of useful kinetic energy of 800 J/s to food mixer 400 J/s of wasted heat energy the appliance is being used ? J/s of ? energy

Q2 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the food mixer is in use?

Q3 is based on using a CD-HiFi system

 electrical energy => useful/wasted energy outputs input energy transfers as a 2 J/s of useful kinetic energy of 200 J/s CD-HiFi system 192 J/s of wasted heat energy to the is being 5 J/s of useful sound energy appliance used ? J/s of ? energy

Q3 (a) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 4 J/s of useful kinetic energy of 500 J/s CD-HiFi system 2 J/s of useful light energy to the is being 464 J/s of wasted heat energy appliance used ? J/s of ? energy

Q3 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 3 J/s of useful kinetic energy of 400 J/s CD-HiFi system 1 J/s of useful light energy to the is being 20 J/s of useful sound energy appliance used ? J/s of ? energy

Q3 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 287.5 J/s of wasted heat energy of 300 J/s CD-HiFi system 0.5 J/s of useful light energy to the is being 10.0 J/s of useful sound energy appliance used ? J/s of useful ? energy

Q3 (d) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

Q4 is based on using a washing machine

 electrical energy => useful/wasted energy outputs input energy transfers as a 1200 J/s of useful kinetic energy of 3000 J/s to washing machine 10 J/s of wasted sound energy the appliance is being used ? J/s of ? energy

Q4 (a) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the washing machine is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 5 J/s of wasted sound energy of 2000 J/s to washing machine 1485 J/s of useful/wasted heat energy the appliance is being used ? J/s of ? energy

Q4 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the washing machine is in use?

 electrical energy => useful/wasted energy outputs input energy transfers as a 900 J/s of kinetic energy of 2500 J/s to washing machine 1593 J/s of useful/wasted heat energy the appliance is being used ? J/s of ? energy

Q4 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the washing machine is in use?

Set B Questions - examples of how to solve work out efficiency calculation questions

(see also Types of energy store, mechanical work done and power calculations

You need to use other formulae apart from the equation for efficiency.

e.g. energy transferred = worked done E (J) = force (N) x distance (m)

power (W) = energy transferred (J) / time taken (s)

Exemplar 'efficiency' questions worked out for you

Set B Questions on efficiency and energy transfer analysis

Q1 An electric kettle transfers 50 000 J of electrical energy into an electric kettle in 40 seconds.

Measurements of the temperature rise, mass of water and its specific heat capacity enable you to calculate that 40 000 J of energy were added to the water's thermal energy store.

(a) What was the percentage efficiency of the electric kettle?

(b) What was the measured power rating of the electric kettle.

(c) What was the useful power output of the electric kettle?

Q2 An electrical appliance includes an electric motor, is found to be 75% efficient.

If the appliance has a maximum power input of 800 W

(a) What is the useful power output?

(b) If the appliance runs for two minutes how many joules of energy are wasted?

(c) If the motor of another electrical appliance transfers 120 J of useful energy for every 150 J of electrical energy supplied.

What is the efficiency of the motor?

Q3 After winding up, a clockwork toy car stores 400 J of elastic potential energy.

When allowed to run, 320 J of the toy's energy store is released as useful kinetic energy.

(a) What is the % efficiency of the clockwork motor of the toy car?

(b) If the toy runs for 20 seconds, what is the actual useful working power of the toy?

Q4 An electrical machine has a useful power output of 3.0 kW from a total power input of 4.0 kW

(a) What is the efficiency of the machine?

(b) How much electrical energy is transferred to the machine in 5.0 minutes?

(c) If, in a given time, 6.0 x 106 J of electrical energy are transferred to the machine, how many joules of useful work are obtained in that same time?

Q5 A sprinter's body applies a force of 80 N for a sprint distance of 100 m.

In the process the runner used 50 000 J of chemical energy from the body's food store.

(a) What is the efficiency of the sprinter?

(b) What has happened to the rest of the energy?

Q6 Suppose (i) an electrical car is 80% efficient in its use of electrical energy and (ii) the electricity is generated from a fossil fuelled power station with an efficiency of 30%.

(a) What is the overall efficiency of the energy transfer of original chemical energy into the kinetic energy store of the car?

(b) How much chemical energy from a fossil fuel energy store is needed to provide a car with 450 kJ of kinetic energy?

Q7 An electric motor is supplied with 2000 J of electrical energy per minute.

If 500 J of the electrical energy is lost as thermal energy from the circuits to the surroundings, what is the efficiency of the electric motor?

Q8  An LED lamp has an efficiency of 0.90.

If the LED lamp is supplied with 500 J of electrical energy, how much energy is converted to light energy?

Q9 based on the Sankey diagram for a cooling fan

Above is the Sankey diagram for the energy input and outputs for a working cooling fan.

Assume 6000 J of electrical energy is supplied to the fan every minute.

(a) How much useful energy is transferred per minute?

(b) What is the % efficiency of the cooling fan?

(c) (i) What % of the input energy is wasted through a heating effect?

(ii) What is this loss in J/min?

(iii) What causes the friction and what happens to the energy?

(d) What is the rate of energy loss due to sound vibrations in J/s?

(e) How can the energy losses be minimised.

(f) What is the power rating of the fan?

Q10 An electric motor in a toy train does 0.2 J of work to accelerate it to a speed of 25 cm/s.

(a) If the toy train has a mass of 3500 g, what is the efficiency of the motor?

First calculate the kinetic energy store of the train.

(b) Why is the efficiency much less than 100%?

For more power calculations see

Types of energy & stores and calculations of mechanical work done and power

More examples of Sankey diagrams

A Sankey diagram for an electric motor

Sankey diagram for an electric cooling fan

Keywords, phrases and learning objectives on energy conservation

Be able to do energy transfer efficiency calculations by interpreting Sankey diagrams in terms of joules (kJ) of energy or percentage (%) energy output from a 100% input.

Be able to describe and discuss the useful work compared with wasted energy from a Sankey diagram or give data.

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Q1 is based on using a hair dryer

 electrical energy => useful/wasted energy outputs energy input transfers as a 3 J/s of wasted sound energy of 600 J/s to hair dryer 590 J/s of useful heat energy the appliance is being used ? J/s of ? energy

Q1 (a) The table above summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the hair dryer is in use?

Answer: 7 J/s of useful kinetic energy

Comment: A fan has to operate to blow the warm air out.

 electrical energy => useful/wasted energy outputs energy input transfers as a 5 J/s of useful kinetic energy of 400 J/s to hair dryer 393 J/s of useful heat energy the appliance is being used ? J/s of ? energy

Q1 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the hair dryer is in use?

Answer: 2 J/s of wasted sound energy

Comment: Sound is of no obvious use.

 electrical energy => useful/wasted energy outputs input energy transfers as a 4 J/s of useful kinetic energy of 300 J/s to hair dryer 1 J/s of wasted sound energy the appliance is being used 95 J/s of wasted heat energy ? J/s of ? energy

Q1 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the hair dryer is in use?

Answer: 200 of J/s of useful heat energy

Comment: There will be losses due to friction of the moving parts e.g. the electric motor and fan blower

Q2 is based on using a food mixer

 electrical energy => useful/wasted energy outputs input energy transfers as a 50 J/s of wasted sound energy of 1000 J/s to food mixer 400 J/s of useful kinetic energy the appliance is being used ? J/s of ? energy

Q2 (a) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the food mixer is in use?

Answer: 550 J/s of wasted heat energy

Comment: Friction of the moving parts of the mixture - sticky food mix will increase it.

 electrical energy => useful/wasted energy outputs input energy transfers as a 80 J/s of wasted sound energy of 1200 J/s to food mixer 620 J/s of wasted heat energy the appliance is being used ? J/s of ? energy

Q2 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the food mixer is in use?

Answer: 500 J/s of useful kinetic energy

Comment: Useful energy to do the mixing.

 electrical energy => useful/wasted energy outputs input energy transfers as a 370 J/s of useful kinetic energy of 800 J/s to food mixer 400 J/s of wasted heat energy the appliance is being used ? J/s of ? energy

Q2 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the food mixer is in use?

Answer: 30 J/s of wasted sound energy

Comment: Sound energy is of no use.

Q3 is based on using a CD-HiFi system

 electrical energy => useful/wasted energy outputs input energy transfers as a 2 J/s of useful kinetic energy of 200 J/s CD-HiFi system 192 J/s of wasted heat energy to the is being 5 J/s of useful sound energy appliance used ? J/s of ? energy

Q3 (a) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

Answer: 1 J/s of useful light energy

Comment: You need to be able to see what is going on!

 electrical energy => useful/wasted energy outputs input energy transfers as a 4 J/s of useful kinetic energy of 500 J/s CD-HiFi system 2 J/s of useful light energy to the is being 464 J/s of wasted heat energy appliance used ? J/s of ? energy

Q3 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

Answer: 30 J/s of useful sound energy

Comment: Needed to hear the output

 electrical energy => useful/wasted energy outputs input energy transfers as a 3 J/s of useful kinetic energy of 400 J/s CD-HiFi system 1 J/s of useful light energy to the is being 20 J/s of useful sound energy appliance used ? J/s of ? energy

Q3 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

Answer: 376 J/s of wasted heat energy

Comment: There is usually heat losses in electrical circuits.

 electrical energy => useful/wasted energy outputs input energy transfers as a 287.5 J/s of wasted heat energy of 300 J/s CD-HiFi system 0.5 J/s of useful light energy to the is being 10.0 J/s of useful sound energy appliance used ? J/s of useful ? energy

Q3 (d) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the CD-HiFi system is in use?

Answer: 2.0 J/s of useful kinetic energy

Comment: The CD needs to rotate.

Q4 is based on using a washing machine

 electrical energy => useful/wasted energy outputs input energy transfers as a 1200 J/s of useful kinetic energy of 3000 J/s to washing machine 10 J/s of wasted sound energy the appliance is being used ? J/s of ? energy

Q4 (a) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the washing machine is in use?

Answer: 1300 J/s of useful heat energy and 490 J/s of wasted heat energy

Comment: You need kinetic energy to do the wash by rotating the drum and clothes, but heat energy loss from friction in moving parts, also conduction out of machine - though should be well insulated

 electrical energy => useful/wasted energy outputs input energy transfers as a 5 J/s of wasted sound energy of 2000 J/s to washing machine 1485 J/s of useful/wasted heat energy the appliance is being used ? J/s of ? energy

Q4 (b) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the washing machine is in use?

Answer: 510 J/s of useful kinetic energy

Comment: You need kinetic energy to rotate the drum and clothes to do the wash.

 electrical energy => useful/wasted energy outputs input energy transfers as a 900 J/s of kinetic energy of 2500 J/s to washing machine 1593 J/s of useful/wasted heat energy the appliance is being used ? J/s of ? energy

Q4 (c) The table summarises what happens to the energy input and output of an electrical appliance.

What is the most likely missing quantity and form of output energy when the washing machine is in use?

Answer: 7 J/s of wasted sound energy

Comment: sound is of no use and indicates the machine is vibrating, difficult to avoid this loss.

Set B Questions on efficiency and energy transfer analysis

Q1 An electric kettle transfers 50 000 J of electrical energy into an electric kettle in 40 seconds.

Measurements of the temperature rise, mass of water and its specific heat capacity enable you to calculate that 40 000 J of energy were added to the water's thermal energy store.

(a) What was the percentage efficiency of the electric kettle?

% Efficiency = 100 x useful energy output / total energy input

% Efficiency = 100 x 40 000 / 50 000 = 80%

(b) What was the measured power rating of the electric kettle.

power = work done / time taken = energy transferred (J) / time taken (s)

Power in watts = 50 000 / 40 = 1250 W (1.25 kW)

(c) What was the useful power output of the electric kettle?

Efficiency = 100 x useful power output / total power input

useful power output = efficiency x total power input / 100

useful power output = 80 x 1250 / 100 = 0.8 x 1250 = 1000 W

Original set B questions

Q2 An electrical appliance includes an electric motor, is found to be 75% efficient.

If the appliance has a maximum power input of 800 W

(a) What is the useful power output?

Efficiency = 100 x useful power output / total power input

useful power output = efficiency x total power input / 100

Efficiency = 100 x useful power output / total power input

useful power output = 75 x 800 / 100 = 0.75 x 800 = 600 W

(b) If the appliance runs for two minutes how many joules of energy are wasted?

power = 800 W = 800 J/s

energy input for 2 minutes = 800 x 2 x 60 = 96,000 J

since the appliance is 75% efficient, then 25% of the energy is wasted

therefore energy wasted = 25 x 96,000 / 100 = 0.25 x 96,000 = 24,000 J 'useless' energy

(c) If the motor of another electrical appliance transfers 120 J of useful energy for every 150 J of electrical energy supplied.

(i) What is the efficiency of the motor?

% Efficiency = 100 x useful energy output / total energy input

% efficiency = 100 x 120 / 150 = 80%

Q3 After winding up, a clockwork toy car stores 400 J of elastic potential energy.

When allowed to run, 320 J of the toy's energy store is released as useful kinetic energy.

(a) What is the % efficiency of the clockwork motor of the toy car?

% efficiency = 100 x useful energy output / total energy input

% efficiency = 100 x 320 / 400 = 80%

(b) If the toy runs for 20 seconds, what is the actual useful working power of the toy?

power = useful work done / time taken

power = 320 / 20 = 16 W

(If you ignore energy loss due to friction, the original maximum power output would be 400 / 20 = 20 W)

Q4 An electrical machine has a useful power output of 3.0 kW from a total power input of 4.0 kW

(a) What is the efficiency of the machine?

% efficiency = 100 x useful energy output / total energy input

% efficiency = 100 x 3.0 / 4.0 = 75% (0.75 as a decimal fraction)

(b) How much electrical energy is transferred to the machine in 5.0 minutes?

P = E / t, E = P x t, P = 4.0 kW = 4000 J/s, t = 5 x 60 = 300 s

therefore E transferred = P x t = 4000 x 300 = 120,000 J = 120 kJ

(c) If, in a given time, 6.0 x 106 J of electrical energy are transferred to the machine, how many joules of useful work are obtained in that same time?

efficiency = useful energy output / total energy input

useful energy output = efficiency x total energy input

useful energy output = 0.75 x 6.0 x 106 = 4.5 x 106 J

Original set B questions

Q5 A sprinter's body applies a force of 80 N for a sprint distance of 100 m.

In the process the runner used 50 000 J of chemical energy from the body's food store.

(a) What is the efficiency of the sprinter?

useful work done = applied force x distance

work done = 80 x 100 = 8000 J

% efficiency = 100 x useful energy output / total energy input

% efficiency = 100 x 8000 / 50 000 = 16%

(b) What has happened to the rest of the energy?

When you take exercise, your body becomes temporarily hotter. A lot of the chemical energy being transferred ends up increasing the thermal energy store of your body.

Q6 Suppose (i) an electrical car is 80% efficient in its use of electrical energy and (ii) the electricity is generated from a fossil fuelled power station with an efficiency of 30%.

(a) What is the overall efficiency of the energy transfer of original chemical energy into the kinetic energy store of the car?

The answer is basically 80% of 30%

Overall efficiency is 30 x 100 / 80 = 24%

(b) How much chemical energy from a fossil fuel energy store is needed to provide a car with 450 kJ of kinetic energy?

450 kJ only represents 24% of the energy from the fossil fuel.

You therefore need to scale up by a factor of 100/24.

Therefore energy required from the fossil fuel chemical energy store = 450 x 100 / 24 = 1875 kJ

(just as a check, as in maths lessons!, 24% of 1875 = 24 x 1875 / 100 = 450 kJ)

Q7 An electric motor is supplied with 2000 J of electrical energy per minute.

If 500 J of the electrical energy is lost as thermal energy from the circuits to the surroundings, what is the efficiency of the electric motor?

If 500 J is wasted energy, 2000 - 500 = 1500 J of useful work per minute.

Efficiency = useful work / total energy input = 1500 / 2000 = 0.75  (or 75% efficiency)

Q8  An LED lamp has an efficiency of 0.90.

If the LED lamp is supplied with 500 J of electrical energy, how much energy is converted to light energy?

Efficiency = useful energy output / total energy input

Rearranging gives: Useful light energy output = Efficiency x Total energy input

Light energy output = 0.90 x 500 = 450 J

Q9 based on the Sankey diagram for a cooling fan

Above is the Sankey diagram for the energy input and outputs for a working cooling fan.

Assume 6000 J of electrical energy is supplied to the fan every minute.

(a) How much useful energy is transferred per minute?

From the scale of the Sankey diagram you can see that:

10 squares equals the total input = 100 % ≡ 6000 J/min

On the right the useful kinetic energy driving the fan around is equal to 8 squares.

Therefore useful energy out put = 6000 x 8 /10 = 4800 J/min

(b) What is the % efficiency of the cooling fan?

% efficiency = 100 x useful energy transferred to device / total energy supplied to the device

Therefore the efficiency of this device = 100 x 4800 / 6000 = 80%

In other words 80% of the electrical energy input is transferred to the kinetic energy store of the rotating fan.

(c) (i) What % of the input energy is wasted through a heating effect? (ii) What is this loss in J/min?  (iii) What causes the friction and what happens to the energy?

(i) The energy loss as thermal energy equals 1.5 squares on the Sankey diagram.

The % wasted energy = 100 x 1.5/10 = 15%

(ii) This equates to 6000 x 15 / 100 = 900 J/min

(iii) What causes the friction and what happens to the wasted energy?

There are two sources of friction.

Friction between moving parts in the fan motor.

Friction between the fan and the air it is passing through.

In both cases work is done against the force of friction.

The energy lost from friction increases the thermal energy store of the surrounding air - wasted - dissipated. (Strictly speaking this includes the sound energy too)

(d) What is the rate of energy loss due to sound vibrations in J/s?

The energy loss due to sound is 0.5 of a square on the Sankey diagram.

Therefore the loss of energy = 6000 x 0.5 / 10 = 300 J/min  = 300 / 60 = 5 J/s

(The sound vibrations are caused by friction as the surfaces of moving parts rub against each other. This energy is eventually dissipated as thermal energy.

(e) How can the energy losses be minimised.

Quite simply, a drop of oil on the fan axle.

This allows surfaces to rub over each other more smoothly, reducing the friction effects.

The lubricant reduces friction and increases the output of useful energy, less wasted as heat or sound.

(f) What is the power rating of the fan?

6000 J is supplied per min. time 1 min = 60 s, power of 1 W = 1 J/s

Power = energy transferred (J) / time (s) = 6000 / 60 = 100 W

Q10 An electric motor in a toy train does 0.2 J of work to accelerate it to a speed of 25 cm/s.

(a) If the toy train has a mass of 3500 g, what is the efficiency of the motor?

First calculate the kinetic energy store of the train.

mass = 3500/1000 = 3.50 kg, speed = (25/100) = 0.25 m/s

KE = 1/2mv2  = 0.5 x 3.5 x 0.252 = 0.1094 J

Efficiency = useful energy out / total energy input = 0.1094 / 0.20 = 0.55 (2 s.f.)

Efficiency = 0.55 x 100 = 55% (2 s.f.)

(b) Why is the efficiency much less than 100%?

Apart from a little sound, most energy is lost (dissipated) from the friction of the moving parts of the electric motor and friction between the wheels and the rails - all energy lost ends up increasing the thermal energy store of the surroundings.

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