Work, Energy & Power

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수업 노트

Principle of Conservation of Energy

Energy cannot be created or destroyed; it can only be transferred from one form to another.
The total amount of energy in a closed system remains constant, although the amount in each form may change.
A system is an object or a group of objects; defining it narrows the focus to what is relevant.
When a system is in equilibrium, nothing changes and nothing happens; a change means energy is transferred.
Kinetic energy, gravitational potential energy and elastic potential energy are collectively known as mechanical energy types.
No energy transfer is 100% efficient — some energy is always dissipated to the surroundings.
Dissipated energy usually ends up as thermal energy transferred to the surroundings and is regarded as wasted energy.

Energy transfer and conservation

Energy Transfers and Dissipation

In a kettle, electrical energy is transformed into thermal energy in the heating element, which is transferred to the water.
Thermal energy transferred to the plastic casing or surrounding air is wasted energy; energy heating the water is useful.
A falling object in a vacuum transfers gravitational potential energy into kinetic energy with no dissipation.
A horizontal mass on a spring transfers elastic potential energy into kinetic energy.
A battery or cell transfers chemical energy into electrical energy; a car transfers chemical energy from fuel into kinetic energy.
A person bouncing on a trampoline transfers energy from elastic potential to kinetic to gravitational potential.
If an object travels up a rough inclined surface: loss in kinetic energy = gain in gravitational potential energy + work done against friction.

Energy transfer by heating from a hot coffee mug to cold hands

Sankey Diagrams

Sankey diagrams represent energy transfers using arrows whose width is proportional to the amount of energy going to each store.
The arrow pointing to the right represents the useful energy output; arrows pointing down represent wasted energy.
Conservation of energy gives: Total energy in = Useful energy out + Wasted energy.
A more efficient light bulb has less wasted energy, shown by a smaller downward arrow representing energy transferred by heating.
The same conservation principle applies to power: Total power in = Useful power out + losses + wasted power.
When drawing a Sankey diagram, plan the widths of the input, useful output and wasted arrows before drawing, and mark the wasted arrow carefully.

Work Done

Work done by a force is equivalent to a transfer of energy; its units are newton metres, where 1 N m = 1 J.
The work done by a resultant force on a system equals the change in energy of that system.
Mechanical work is the transfer of energy when an external force causes an object to move over a certain distance.
For a constant force parallel to the displacement: W = Fs, where W is work done (J), F is force (N) and s is displacement (m).
If the force is at an angle θ to the displacement: W = Fs cos θ, where θ is the angle between the force and the motion.
When θ = 0, cos θ = 1 and W = Fs; only the component of the force parallel to the displacement does work.
For horizontal motion use cos θ; for vertical motion use sin θ — always consider the horizontal and vertical components of the force.
On a graph of average force against displacement, the area under the graph equals the work done.

Work is done when a force is used to move an object over a distance

Kinetic Energy

Kinetic energy (Ek) is the energy an object has due to its motion; the faster it moves, the greater its kinetic energy.
Kinetic energy is calculated using Ek = ½mv², where m is mass (kg) and v is velocity (m s⁻¹).
Only the speed is squared, not the mass or the ½.
When an object falls, it gains kinetic energy transferred from the gravitational potential energy it loses.
An object maintains its kinetic energy unless its speed or mass changes.
Kinetic energy can also be written in terms of momentum: Ek = p² / 2m, where p is momentum (kg m s⁻¹).
The p²/2m form is very useful in particle physics when comparing momentum and kinetic energy.
Energy is a scalar quantity, so a 'loss of kinetic energy' is stated without a negative sign.

Kinetic energy of a moving car

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy stored in a mass due to its position in a gravitational field.
If a mass is lifted up it gains GPE; if it falls it loses GPE.
Close to the Earth's surface: ΔEp = mgΔh, where g = 9.8 N kg⁻¹ and Δh is the change in height (m).
The potential energy at ground level is usually taken as zero, but any position can be taken as zero when calculating a change in GPE.
This equation is only relevant in a uniform gravitational field, such as near the Earth's surface.
A different potential energy expression is used in the gravitational fields topic because the field is no longer uniform outside the Earth's surface.
Gravitational potential energy and height have a linear relationship, shown by straight-line graphs of GPE against height or time.

Gravitational potential energy of a lifted mass

Elastic Potential Energy

Elastic potential energy is the energy stored within a material (e.g. a spring) when it is stretched or compressed.
For a material obeying Hooke's Law: EH = ½kΔx², where k is the spring constant (N m⁻¹) and Δx is the extension (m).
It can also be written as EH = ½FΔx, where F is the restoring force given by F = kΔx.
It is very dangerous if a wire under large stress suddenly breaks because its elastic potential energy is converted into kinetic energy.
When all elastic potential energy becomes kinetic energy: ½kΔx² = ½mv², so v ∝ Δx.
The greater the extension Δx, the greater the speed v of the wire when it breaks.

Conservation of Mechanical Energy

Mechanical energy = Ek + ΔEp + EH — the sum of kinetic, gravitational potential and elastic potential energy.
The change in total mechanical energy of a system is interpreted in terms of the work done by any non-conservative force, such as friction.
In the absence of frictional or resistive forces, the total mechanical energy of a system is conserved throughout its motion.
For a vertical spring oscillating, energy converts between EPE, KE and GPE while the total energy stays constant.
Vertical spring positions: at maximum height GPE is maximum and KE is zero; at the equilibrium position KE is maximum; at maximum extension EPE is maximum and KE is zero.
For a horizontal mass on a spring, GPE is constant and the spring only converts between kinetic and elastic potential energy.
Using conservation of energy with negligible drag: loss in gravitational potential energy = gain in kinetic energy.
Examples of energy transfer include a swinging pendulum, objects in freefall, and sports involving falling such as skiing and skydiving.

The principle of conservation of energy applied to a bat hitting a ball

Energy & Power

The power of a mechanical process is the rate at which energy is transferred, i.e. the rate of work done.
Power is calculated using P = ΔW / Δt = Fv, where F is force (N) and v is velocity (m s⁻¹).
The Fv form is only relevant where a constant force moves a body at constant velocity, with the force in the same direction as the velocity.
Two cars may do the same work to accelerate, but the one with more power transfers that energy in a shorter time.
Of two motors lifting the same weight by the same height, the one that lifts it faster has more power.
Power is required to produce an acceleration.
Power is measured in watts (W), where 1 W = 1 J s⁻¹ — a transfer of 1 joule of energy in 1 second.
Appliances are given a power rating (e.g. 1000 W) indicating the energy transferred per second to the appliance.

Power and rate of energy transfer

Efficiency Formula

Efficiency is a measure of how successfully energy is transferred in a system.
It is defined as the ratio of the useful power or energy transfer output to the total power or energy transfer input.
Efficiency is calculated using η = Eout / Ein = Pout / Pin.
To express efficiency as a percentage, multiply the ratio by 100%.
In words: η = useful work out / total work in = useful power out / total power in.
High efficiency means most of the energy transferred is useful; low efficiency means most is wasted.
Which energy is useful or wasted depends on the system — for a kettle, heating the water is useful, heating the casing and air is wasted.
Efficiency has no units because it is a ratio of quantities with the same units; it can be given as a ratio (0 to 1) or a percentage (0% to 100%).

Energy Density

A fuel is anything that can be burned to produce heat, which can be used for an engine to work.
Energy density is a measure of the amount of energy per unit volume of a fuel, measured in J m⁻³.
Different fuels contain different amounts of energy, making them suitable for certain uses, e.g. petrol for running vehicles.
Example energy densities (MJ L⁻¹): coal 38, diesel 39, biodiesel 33, vegetable oil 30, liquid hydrogen 9, wood 3, methane 0.3.
1 litre = 0.001 m³.
We can get more energy per unit volume of coal than of wood.
Fuels are chosen for specific uses based on factors including energy density, safety of use and pollutants released in combustion.

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1.Which statement is the principle of conservation of energy?
Easy
- AEnergy cannot be created or destroyed; it can only be transferred from one form to another
- BThe total amount of energy in a system always decreases over time
- CEnergy can be created when a force does work on an object
- DEnergy is always transferred as thermal energy
2.In a Sankey diagram, what does the width of each arrow represent?
Easy
- AThe amount of energy going to each store
- BThe temperature of the energy store
- CThe speed at which energy is transferred
- DThe direction of the energy transfer
3.A constant force of 20 N is applied to a box at an angle of 45° to the horizontal, moving it 5 m horizontally. What is the work done on the box?
Medium
- A71 J
- B100 J
- C50 J
- D35 J
4.A spring has a spring constant of 92 N m⁻¹ and is extended by 0.3 m. What is the elastic potential energy stored in the spring?
Medium
- A4.1 J
- B8.3 J
- C27.6 J
- D13.8 J
5.A car engine exerts a thrust of 200 N while the car travels at a constant speed of 27 m s⁻¹. What is the power of the car?
Medium
- A5400 W
- B7.4 W
- C200 W
- D2700 W
6.Which of the following are forms of wasted energy transfer when a petrol car's engine converts chemical energy? (Select all that apply)
Medium
- ASound from the engine
- BThermal energy from the engine
- CKinetic energy of the car
- DElectrical energy from the battery
- ELight from the headlights
7.A ball is dropped from a height of 2 m on Jupiter, where the acceleration of free fall is 24.58 m s⁻². Which statements are correct? (Select all that apply)
Hard
- AThe gravitational potential energy at the drop point equals the kinetic energy just before impact
- BThe speed just before impact is about 9.9 m s⁻¹
- CThe speed just before impact is about 4.9 m s⁻¹
- DThe relationship between GPE and KE is mgh = ½mv²
- EThe kinetic energy just before impact is greater than the gravitational potential energy at the drop point
8.In a closed system, the total amount of energy remains constant, but the amount of each form of energy may change.
Easy
True or false?

더 많은 문제

9.Dissipated energy is usually regarded as wasted energy because it cannot easily be used for another purpose.
Easy
True or false?
10.In a vertical spring-mass system, the total mechanical energy is the sum of kinetic energy, gravitational potential energy, and elastic potential energy.
Medium
True or false?
11.Match each energy type to its correct definition.
Medium
- Kinetic energy
- Gravitational potential energy
- Elastic potential energy
- Energy stored in a mass due to its position in a gravitational field
- Energy an object has due to its motion
- Energy stored within a material when it is stretched or compressed
12.Place the energy transfers in the correct order for a person bouncing on a trampoline, starting from the moment they are at the lowest point (maximum elastic potential energy).
Medium
- Elastic potential energy is converted to kinetic energy
- Kinetic energy is converted to gravitational potential energy
- Gravitational potential energy is converted back to kinetic energy
- Kinetic energy is converted back to elastic potential energy
13.Which of the following are forms of energy that are collectively known as mechanical energy? (Select all that apply)
Easy
- Akinetic energy
- Bgravitational potential energy
- Celastic potential energy
- Dthermal energy
- Echemical energy
14.Energy cannot be created or destroyed; it can only be transferred from one form to another.
Easy
True or false?
15.A box of mass 5.8 kg with initial speed 7.2 m s⁻¹ moves up a smooth incline and is momentarily brought to rest after colliding with a spring of spring constant 210 N m⁻¹. It stops a vertical distance of 0.65 m above its initial position. What is the initial kinetic energy of the box?
Medium
- A150 J
- B75 J
- C300 J
- D37 J
16.A spring of spring constant 92 N m⁻¹ is extended by 0.3 m. Calculate the elastic potential energy stored in the spring.
Medium
- A4.14 J
- B8.28 J
- C27.6 J
- D2.76 J
17.Match each fuel to its approximate energy density.
Medium
- coal
- liquid hydrogen
- methane (natural gas)
- diesel
- 38 MJ L⁻¹
- 9 MJ L⁻¹
- 0.3 MJ L⁻¹
- 39 MJ L⁻¹
18.A skier descends 750 m along a slope at 25° to the horizontal, starting from rest. 15% of the initial gravitational potential energy is not transferred to kinetic energy. Which of the following statements are correct? (Select all that apply)
Hard
- AThe vertical height fallen is 750 sin 25°
- BOnly 85% of the gravitational potential energy becomes kinetic energy
- CThe final speed depends on the skier's mass
- DThe final speed is about 73 m s⁻¹
- EThe final speed is about 120 m s⁻¹
19.A car engine exerts a constant thrust of 200 N while the car travels at a constant speed of 27 m s⁻¹. What is the power developed by the engine?
Medium
- A7.4 W
- B5400 W
- C113.5 W
- D10800 W
20.A box of mass 5.8 kg moves up a smooth incline with an initial speed of 7.2 m s⁻¹ and is momentarily brought to rest after colliding with a spring of spring constant 210 N m⁻¹. It stops a vertical distance of 0.65 m above its initial position. What is the initial kinetic energy of the box?
Medium
- A150 J
- B20.9 J
- C75 J
- D37 J
21.A box of mass 5.8 kg moves up a smooth incline with an initial speed of 7.2 m s⁻¹ and is momentarily brought to rest after colliding with a spring of spring constant 210 N m⁻¹. It stops a vertical distance of 0.65 m above its initial position. By how much is the spring compressed?
Hard
- A1.20 m
- B0.37 m
- C0.65 m
- D0.75 m
22.A kite pulls a ship with a tension of 350 kN at an angle θ to the horizontal. The ship travels at a steady speed of 3.9 m s⁻¹ and the work done by the kite over 5 minutes is 355 MJ. What is the angle θ?
Medium
- A0.086°
- B5.0°
- C85°
- D30°
23.A kite pulls a ship with a tension of 350 kN at 30° to the horizontal. The ship travels at a steady speed of 3.9 m s⁻¹. What power does the kite provide?
Medium
- A1.2 MW
- B0.68 MW
- C5.7 MW
- D2.4 MW
24.A golf club head is pivoted and its centre of mass is raised by 75 cm before being released. What is the maximum speed of the club head at the vertical position?
Hard
- A3.8 m s⁻¹
- B14.7 m s⁻¹
- C7.4 m s⁻¹
- D1.5 m s⁻¹
25.A skier descends 750 m along a slope at 25° to the horizontal, starting from rest. 15% of the initial gravitational potential energy is not transferred to kinetic energy. Which statements are correct? (Select all that apply)
Medium
- AThe vertical height descended is 750 sin 25°
- BThe final kinetic energy is 0.85 × mgh
- CThe final speed depends on the skier's mass
- DThe final speed is about 73 m s⁻¹
- EThe energy not transferred to kinetic energy is dissipated as thermal energy
26.A vertical spring hangs with a mass attached and oscillates. Which statements about the energy at the lowest point of the oscillation are correct? (Select all that apply)
Medium
- AElastic potential energy is at a maximum
- BKinetic energy is at a maximum
- CGravitational potential energy is at a minimum
- DKinetic energy is zero
- EGravitational potential energy is at a maximum
27.In a Sankey diagram, the width of each arrow is proportional to the amount of energy transferred to that store.
Easy
True or false?
28.If a force acts at an angle to the direction of motion, the work done is calculated using only the component of the force parallel to the displacement.
Medium
True or false?
29.The force-extension graph for a spring is a straight line from the origin to the point (0.20 m, 12 N). What is the work done in extending the spring by 0.20 m?
Medium
- A2.4 J
- B1.2 J
- C0.60 J
- D4.8 J
30.A ball is dropped from rest in a vacuum and falls through a vertical height h. Which single equation gives its speed just before impact?
Hard
- Av = √(2gh)
- Bv = 2gh
- Cv = √(gh)
- Dv = mgh
31.A student uses a force meter to pull a block along a bench at constant speed and records the force and the distance moved. Which of the following would improve the reliability or accuracy of the work done measurement? (Select all that apply)
Medium
- ARepeat the measurement several times and average the force
- BKeep the force meter parallel to the direction of motion
- CUse a larger unit for the distance so the numbers are smaller
- DMeasure the distance travelled with a ruler fixed along the bench
- EPull the block as fast as possible to reduce timing errors
32.In a spring test, the extension should be measured from the natural length of the spring to its stretched length, not from the top of the spring to the bottom.
Easy
True or false?
33.Match each measurement or graph feature to the quantity it determines for a spring obeying Hooke's law.
Medium
- Gradient of a force-extension graph
- Area under a force-extension graph
- Force divided by extension
- Maximum extension before the line curves
- Spring constant k
- Work done / elastic potential energy
- Spring constant k (same as gradient)
- Elastic limit
34.A student carries out a spring experiment to determine the elastic potential energy stored at a given extension. Place the steps in the correct order.
Medium
- Measure the natural length of the spring
- Hang known masses from the spring and record the new length each time
- Calculate the extension for each mass and plot a force-extension graph
- Find the gradient (spring constant) and the area under the graph up to the chosen extension
35.Which of the following is the correct definition of a system in physics?
Easy
- AAn object or group of objects
- BA closed container with no energy transfer
- CThe total energy of an object
- DA force acting on an object
36.When a system is in equilibrium, nothing changes, and so nothing happens.
Easy
True or false?
37.Which of the following energy types are collectively known as mechanical energy?
Easy
- AKinetic energy, gravitational potential energy, and elastic potential energy
- BChemical energy, thermal energy, and nuclear energy
- CElectrical energy, sound energy, and light energy
- DThermal energy, kinetic energy, and elastic potential energy
38.A kettle transfers electrical energy to thermal energy. Which of the following is a wasted energy transfer in this process?
Medium
- AThermal energy transferred to the surrounding air
- BThermal energy transferred to the water
- CElectrical energy transferred to the heating element
- DThermal energy transferred to the heating element
39.Which of the following statements about energy dissipation are correct? (Select all that apply)
Medium
- ANo energy transfer is 100% efficient.
- BDissipated energy usually ends up as thermal energy transferred to the surroundings.
- CDissipated energy can always be easily used for another purpose.
- DDissipated energy is usually regarded as wasted energy.
40.Which of the following is the correct definition of the watt?
Medium
- AA transfer of 1 joule of energy in 1 second
- BA transfer of 1 joule of energy in 1 minute
- CA transfer of 1 newton of force in 1 second
- DA transfer of 1 watt of energy in 1 second
41.What is the name of the energy stored within a stretched or compressed spring?
Easy
42.What is the SI unit of power?
Easy
43.A runner of mass 52 kg maintains a constant velocity of 7.0 m s⁻¹. Calculate the kinetic energy of the runner. Give your answer in joules.
Medium
44.An action figure of mass 250 g is dropped from a height of 5 m. Calculate its gravitational potential energy relative to the ground. Use g = 9.8 N kg⁻¹. Give your answer in joules.
Medium
45.A violin string has a spring constant of 23 000 N m⁻¹ and is extended by 0.06 m before it breaks. Calculate the force applied to the string just before it breaks. Give your answer in newtons.
Medium
46.Complete the statement about energy conservation.
Easy
The principle of conservation of energy states that energy cannot be ____ or destroyed; it can only be ____ from one form to another.
47.Complete the equation for work done when a force acts at an angle θ to the displacement.
Medium
W = F s ____ θ
48.An electric motor has an efficiency of 35%. It lifts a 7.2 kg load through a height of 5 m in 3 s. Calculate the useful power output of the motor. Give your answer in watts to the nearest whole number.
Medium
49.Name the energy store that is at a maximum when a vertical spring is at its maximum extension.
Medium
50.A lorry of mass 3500 kg moves up a road inclined at 14.5° to the horizontal at a constant speed of 9.4 m s⁻¹. Air resistance is negligible. Calculate the useful power from the engine. Give your answer in kW to 2 significant figures.
Medium
51.A body travelling at 12 m s⁻¹ has kinetic energy 1650 J. Its speed is increased to 45 m s⁻¹. Calculate the new kinetic energy. Give your answer in joules.
Medium
52.Complete the equation for the work done by a constant force applied parallel to an object's displacement.
Easy
W = ____ × ____
53.State the principle of conservation of energy and explain what it means for the total energy in a closed system.
Hard
54.Explain, using the principle of conservation of energy, why the speed of a ball just before it hits the ground is independent of its mass when dropped from rest in a vacuum.
Hard
55.Explain, using the principle of conservation of energy, why the speed of a ball just before it hits the ground is independent of its mass when dropped from rest (assuming no air resistance).
Hard
56.A kettle transfers electrical energy to thermal energy. State two energy transfers that are wasted in this process.
Medium
57.A student measures the efficiency of a motor lifting a load by recording the electrical energy input and the gravitational potential energy gained by the load. Describe two practical steps the student should take to make the result more reliable, and explain how each improves the measurement.
Hard
58.A runner of mass 52 kg maintains a constant velocity of 7.0 m s⁻¹. Calculate the kinetic energy of the runner. Give your answer in joules.
Medium
- A1147 J
- B1401 J
- C1274 J
- D1528 J
59.An action figure of mass 250 g is dropped from a height of 5 m. Calculate its gravitational potential energy relative to the ground. Use g = 9.8 N kg⁻¹. Give your answer in joules.
Medium
- A14.25 J
- B13.25 J
- C12.25 J
- D11.25 J
60.A violin string has a spring constant of 23 000 N m⁻¹ and is extended by 0.06 m before it breaks. Calculate the force applied to the string just before it breaks. Give your answer in newtons.
Medium
- A1380 N
- B1242 N
- C1656 N
- D1518 N
61.An electric motor has an efficiency of 35%. It lifts a 7.2 kg load through a height of 5 m in 3 s. Calculate the useful power output of the motor. Give your answer in watts to the nearest whole number.
Medium
- A142 W
- B130 W
- C106 W
- D118 W
62.A lorry of mass 3500 kg moves up a road inclined at 14.5° to the horizontal at a constant speed of 9.4 m s⁻¹. Air resistance is negligible. Calculate the useful power from the engine. Give your answer in kW to 2 significant figures.
Medium
- A89 kW
- B97 kW
- C73 kW
- D81 kW
63.A body travelling at 12 m s⁻¹ has kinetic energy 1650 J. Its speed is increased to 45 m s⁻¹. Calculate the new kinetic energy. Give your answer in joules.
Medium
- A23203 J
- B20883 J
- C25523 J
- D27843 J

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