Work, Energy & Power

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课程笔记

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 distribution among different forms may change.
A system is defined as an object or a group of objects; defining the system helps focus on what is relevant.
When a system is in equilibrium, nothing changes; when there is a change, energy is transferred.
Kinetic energy, gravitational potential energy, and elastic potential energy are collectively known as mechanical energy.
No energy transfer is 100% efficient; some energy is always dissipated to the surroundings, usually as thermal energy.

Energy transfer and conservation

Sankey Diagrams

Sankey diagrams represent energy transfers using arrows whose widths are proportional to the amount of energy.
The arrow pointing to the right represents useful energy output; downward arrows represent wasted energy.
The total energy in equals the useful energy out plus the wasted energy: Total energy in = Useful energy out + Wasted energy.
More efficient devices have smaller wasted energy arrows; for example, a modern light bulb compared to an old filament bulb.
Sankey diagrams can also represent power transfers, with the same conservation principle applied to power.

Work Done

Work done by a force is equivalent to a transfer of energy; its units are newton metres (N m) or joules (J).
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 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; only the component of force parallel to motion does work.
The area under a force–displacement graph equals the work done.
When work is done against friction, energy is transferred to heat and sound.

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; it depends on mass and speed.
The kinetic energy equation is: Ek = ½ m v², where m is mass (kg) and v is velocity (m s⁻¹).
Kinetic energy can also be written in terms of momentum: Ek = p² / (2m), where p is momentum (kg m s⁻¹).
When an object falls, gravitational potential energy is transferred to kinetic energy.
Kinetic energy is a scalar quantity; a loss of kinetic energy is expressed as a positive value.

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.
The equation for GPE near the Earth's surface is: ΔEp = mgΔh, where m is mass (kg), g is gravitational field strength (9.8 N kg⁻¹), and Δh is change in height (m).
If a mass is lifted, it gains GPE; if it falls, it loses GPE.
The zero of GPE can be chosen at any convenient position; only changes in GPE are physically meaningful.
GPE and height have a linear relationship; a graph of GPE against height is a straight line.

Gravitational potential energy of a lifted mass

Elastic Potential Energy

Elastic potential energy is the energy stored within a material when it is stretched or compressed.
For a material obeying Hooke's Law: EH = ½ k Δx², where k is the spring constant (N m⁻¹) and Δx is the extension (m).
It can also be written as EH = ½ F Δx, where F is the restoring force (N).
When a stretched wire breaks, its elastic potential energy is converted into kinetic energy: ½ k Δx² = ½ m v².
The greater the extension Δx, the greater the speed v of the fragments when the wire breaks (v ∝ Δx).

Conservation of Mechanical Energy

Mechanical energy is the sum of kinetic energy, gravitational potential energy, and elastic potential energy: Mechanical energy = Ek + ΔEp + EH.
In the absence of frictional or resistive forces, the total mechanical energy of a system is conserved.
For a falling object (no air resistance): Loss in gravitational potential energy = Gain in kinetic energy.
For a horizontal mass on a spring: Loss in elastic potential energy = Gain in kinetic energy.
A non-conservative force (such as friction) dissipates energy away from the system, reducing mechanical energy.
For an object moving up a rough incline: Loss in kinetic energy = Gain in GPE + Work done against friction.
In a vertical spring system, energy converts between elastic potential, kinetic, and gravitational potential energy, but the total remains constant.

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

Energy & Power

Power is the rate at which energy is transferred or work is done: P = ΔW / Δt.
Power can also be calculated as P = Fv, where F is a constant force and v is constant velocity.
The watt (W) is the unit of power: 1 W = 1 J s⁻¹.
Two motors lifting the same weight through the same height: the one that does it faster has more power.
Power is required to produce acceleration; the force must be applied in the direction of velocity.
Appliances have power ratings indicating the energy transferred per second.

Power and rate of energy transfer

Efficiency Formula

Efficiency is a measure of how successfully energy is transferred in a system.
Efficiency = (useful energy output / total energy input) × 100% or (useful power output / total power input) × 100%.
The symbol for efficiency is η (eta); it has no units and can be expressed as a ratio (0 to 1) or a percentage (0% to 100%).
A high-efficiency system transfers most of its energy usefully; a low-efficiency system wastes most of it.
Determining which energy transfers are useful or wasted depends on the purpose of the system.
To convert efficiency from a ratio to a percentage, multiply by 100%.

Energy Density

Energy density is a measure of the amount of energy per unit volume of a fuel.
Energy density is measured in J m⁻³ (or MJ L⁻¹).
Different fuels have different energy densities, making them suitable for different uses.
Examples of energy densities: coal 38 MJ L⁻¹, diesel 39 MJ L⁻¹, wood 3 MJ L⁻¹.
Fuels are chosen based on energy density, safety, and pollutants released.

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练习题

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1.Which of the following best describes the principle of conservation of energy?
Easy
- AEnergy cannot be created or destroyed, only transferred from one form to another
- BEnergy can be created and destroyed in a closed system
- CThe total energy in a closed system always decreases
- DEnergy can only be transferred from kinetic to potential stores
2.Mechanical energy is the sum of kinetic energy, gravitational potential energy and elastic potential energy.
Easy
True or false?
3.Which of the following are considered mechanical energy types? (Select all that apply.)
Medium
- AKinetic energy
- BGravitational potential energy
- CElastic potential energy
- DThermal energy
- EChemical energy
4.Match each energy type with its correct description.
Easy
- Kinetic energy
- Gravitational potential energy
- Elastic potential energy
- Energy stored due to position in a gravitational field
- Energy an object has due to its motion
- Energy stored when a material is stretched or compressed
5.Which equation correctly defines work done when a constant force acts at an angle θ to the displacement?
Easy
- AW = F s cos θ
- BW = F s sin θ
- CW = F s tan θ
- DW = F s
6.A runner of mass 52 kg maintains a constant velocity of 7 m s⁻¹ by exerting a forward force of 15.2 N against air resistance. What is her power output?
Medium
- A106.4 W
- B53.2 W
- C212.8 W
- D7.6 W
7.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 is usually regarded as wasted energy.
- DDissipated energy can always be easily used for another purpose.
- EEnergy is destroyed when it is dissipated.
8.A car transfers chemical energy from its fuel into kinetic energy. Which of the following are wasted energy transfers in this system? (Select all that apply.)
Medium
- AThermal energy transferred to the surroundings
- BSound from the engine
- CKinetic energy of the car
- DChemical energy stored in the fuel
- EThermal energy transferred to the engine casing

更多题目

9.Place the following energy stores in the order they reach their maximum during one complete oscillation of a vertical mass on a spring, starting from the highest point (A) and ending at the lowest point (C).
Medium
- Gravitational potential energy
- Kinetic energy
- Elastic potential energy
10.A pump has an efficiency of 20%. Water with 700 kg and speed 3.5 m s⁻¹ has its kinetic energy used to lift a smaller mass of water by 12 m. Which mass of water could be lifted?
Medium
- A6.2 kg
- B4.6 kg
- C7.3 kg
- D0.24 kg
11.In a Sankey diagram, the width of each arrow is proportional to the amount of energy going to each store.
Easy
True or false?
12.Which of the following statements about energy conservation are correct? (Select all that apply.)
Medium
- AEnergy cannot be created or destroyed
- BTotal energy in a closed system remains constant
- CEnergy can be transferred from one form to another
- DDissipated energy is usually thermal energy transferred to the surroundings
- EEnergy can be destroyed when it is wasted
13.A student uses a force meter to pull a block along a bench at a constant speed. Which measurement, together with the force reading, is needed to calculate the work done on the block?
Medium
- Athe distance the block is pulled along the bench
- Bthe mass of the block
- Cthe time taken to pull the block
- Dthe weight of the force meter
14.A student investigates a spring by hanging masses from it and recording the extension. Which of the following are good experimental techniques for this investigation? (Select all that apply.)
Medium
- Aread the extension from a ruler with the eye level with the marker to avoid parallax
- Badd masses one at a time and record the new extension after each addition
- Cmeasure the extension from the top of the spring to the bottom of the spring each time
- Dtake repeat readings of each extension and calculate a mean
- Eestimate the extension by eye without a ruler
15.In a graph of force against extension for a spring, the area under the straight-line section up to the elastic limit represents the elastic potential energy stored in the spring.
Easy
True or false?
16.A skier of mass 70 kg descends a slope, dropping a vertical height of 40 m. If 20% of the gravitational potential energy lost is dissipated as thermal energy, what is the skier's speed at the bottom? Use g = 9.8 m s⁻².
Hard
- A25 m s⁻¹
- B28 m s⁻¹
- C22 m s⁻¹
- D31 m s⁻¹
17.A ball is released from rest at the top of a smooth curved track and rolls down, then up a smaller hill, then along a flat section. Place the points in order of the ball's gravitational potential energy, from highest to lowest.
Medium
- at the top of the first hill
- halfway down the first hill
- at the bottom of the first hill
- at the top of the smaller hill
18.A box is pushed up a rough incline at constant speed. Which of the following statements are correct? (Select all that apply.)
Medium
- AThe work done by the pushing force equals the gain in gravitational potential energy plus the work done against friction.
- BThe kinetic energy of the box remains constant.
- CThe total mechanical energy of the box increases.
- DThe work done against friction is transferred to the thermal store of the surroundings.
- EThe gravitational potential energy of the box decreases.
19.A car engine provides a constant driving force of 600 N while the car travels at a steady 15 m s⁻¹. The engine has an efficiency of 25%. What is the rate at which energy is supplied by the fuel?
Hard
- A36 000 W
- B9 000 W
- C2 250 W
- D150 000 W
20.Which of the following best defines the principle of conservation of energy?
Easy
- AEnergy can be transferred from one form to another but the total energy in a closed system remains constant
- BEnergy can be created when a system is heated and destroyed when it is cooled
- CThe total energy of a system always decreases because energy is lost as heat
- DEnergy can be transferred between objects but only if the total kinetic energy stays the same
21.In a closed system, the total amount of energy remains constant, although the amount of each form of energy may change.
Easy
True or false?
22.Which of the following are examples of energy being transferred in a system? (Select all that apply.)
Medium
- AA battery transferring chemical energy to electrical energy
- BA car transferring chemical energy from fuel into kinetic energy of the car
- CA book resting on a table with no change in position
- DA person bouncing on a trampoline, transferring energy between elastic potential, kinetic and gravitational potential stores
- EA closed system in equilibrium
23.Match each energy store with its correct description.
Easy
- 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
24.Which of the following statements about dissipated energy is correct?
Easy
- ADissipated energy is usually transferred to the surroundings as thermal energy and is regarded as wasted
- BDissipated energy is always stored as elastic potential energy
- CDissipated energy can be easily reused for another purpose
- DDissipated energy is created during an energy transfer
25.In a Sankey diagram, the total energy in is equal to the useful energy out plus the wasted energy.
Easy
True or false?
26.Which equation correctly defines mechanical work done when a constant force F acts in the line of displacement s?
Easy
- AW = Fs
- BW = F/s
- CW = Fs²
- DW = F + s
27.A constant force of 25 N is applied to a box, moving it 4.0 m in the direction of the force. Calculate the work done on the box.
Medium
- A100 J
- B6.25 J
- C29 J
- D400 J
28.When a force is applied at an angle to the direction of motion, only the component of the force parallel to the displacement does work.
Medium
True or false?
29.Which of the following statements about kinetic energy are correct? (Select all that apply.)
Medium
- AKinetic energy is directly proportional to the square of the speed
- BKinetic energy is directly proportional to the mass of the object
- CKinetic energy is a scalar quantity
- DKinetic energy is directly proportional to the speed
- EKinetic energy can be negative
30.A body travelling at 12 m s⁻¹ has kinetic energy 1650 J. Its speed is increased to 45 m s⁻¹. What is its new kinetic energy?
Medium
- A23 200 J
- B6190 J
- C14 800 J
- D6600 J
31.A mass is lifted vertically through a height Δh. Which equation gives the change in gravitational potential energy?
Medium
- AΔEp = mgΔh
- BΔEp = mgh²
- CΔEp = ½mgΔh
- DΔEp = mg/Δh
32.A man of mass 74 kg climbs five flights of stairs, each of height 3.7 m. What is the approximate change in his gravitational potential energy?
Medium
- A13 000 J
- B2700 J
- C1500 J
- D12 500 J
33.Which equation gives the elastic potential energy stored in a spring of spring constant k when it is extended by Δx?
Medium
- AEH = ½kΔx²
- BEH = kΔx
- CEH = ½kΔx
- DEH = kΔx²
34.A wire under large stress suddenly breaks. Why is this dangerous?
Medium
- AThe elastic potential energy stored in the wire is converted into kinetic energy, giving the wire a high speed
- BThe wire loses all its energy instantly and becomes weightless
- CThe wire's gravitational potential energy increases suddenly
- DThe wire's spring constant increases rapidly as it breaks
35.For a wire that breaks when extended by Δx, the speed v of the wire just after breaking is related to Δx by which proportionality?
Medium
- Av ∝ Δx
- Bv ∝ Δx²
- Cv ∝ 1/Δx
- Dv ∝ √Δx
36.In a vertical mass-spring system, at which position is the elastic potential energy at a maximum?
Medium
- AAt the lowest point, where the spring has maximum extension
- BAt the highest point, where the mass is momentarily stationary
- CAt the equilibrium position, where the speed is maximum
- DAt all positions, because total energy is constant
37.A rollercoaster travels along a track A → B → C → D. At A it is raised above the ground, then it descends to B, rises up a loop to C, and travels along flat terrain at D. Which sequence correctly describes the energy changes?
Medium
- AGPE → KE → GPE → KE
- BKE → GPE → GPE → KE
- CKE → GPE → KE → GPE
- DGPE → KE → KE → GPE
38.Which of the following statements about mechanical energy are correct? (Select all that apply.)
Medium
- AMechanical energy is the sum of kinetic energy, gravitational potential energy and elastic potential energy
- BIn the absence of frictional forces, the total mechanical energy of a system is conserved
- CMechanical energy is always conserved even when friction is present
- DFor a horizontal mass on a spring, gravitational potential energy does not need to be considered because it does not change
- EMechanical energy can be negative
39.When an object moves up a rough inclined surface, the loss in kinetic energy equals the gain in gravitational potential energy plus the work done against friction.
Medium
True or false?
40.Which equation correctly defines power?
Medium
- AP = ΔW/Δt = Fv
- BP = ΔW × Δt
- CP = Δt/ΔW
- DP = F/v
41.Complete the sentence about Sankey diagrams.
Easy
In a Sankey diagram, the width of each arrow is proportional to the ____ going to each store.
42.A spring with spring constant 92 N m⁻¹ is extended by 0.3 m. Calculate the elastic potential energy stored in the spring.
Medium
43.An action figure of mass 250 g is dropped from a height of 5 m. Calculate the gravitational potential energy of the action figure at this height. Use g = 9.8 N kg⁻¹.
Easy
44.Complete the equation for the efficiency of a system in terms of power.
Easy
η = ____ / ____
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 just before it breaks.
Medium
46.State what the area under a graph of force against displacement represents.
Medium
47.A spring with spring constant 50 kN m⁻¹ is compressed by 10 cm. Calculate the elastic potential energy stored.
Medium
48.A pushchair has a force of 20 N applied for a distance of 5 m at an angle of 45° to the horizontal. Calculate the work done on the pushchair, in joules.
Easy
49.An electric car engine provides a thrust of 200 N while the car maintains a constant speed of 27 m s⁻¹. Calculate the power of the car, in watts.
Medium
50.A runner of mass 52 kg is lifted vertically by a 500 W motor through a height of 2.0 m in 2.5 s. Calculate the efficiency of the motor, as a percentage.
Medium
51.A spring of spring constant 92 N m⁻¹ is extended by 0.3 m. Calculate the elastic potential energy stored in the spring, in joules.
Medium
52.An action figure of mass 250 g is dropped from a height of 5 m. Calculate the gravitational potential energy of the action figure at this height, in joules. Use g = 9.8 N kg⁻¹.
Medium
53.State the main form of energy used by each of the following toys: (i) a catapult, (ii) an action figure falling using a parachute, (iii) a wind-up bouncing rabbit, (iv) a train moving along a train track.
Medium
54.A ball is dropped from a height of 2 m towards the surface of Jupiter, where the acceleration of free fall is 24.58 m s⁻². Assume no energy is lost to the surroundings. (i) State the relationship, in symbols, between gravitational potential energy at the point the ball is dropped from and the kinetic energy just before it hits the surface. (ii) Rearrange the relationship to obtain an equation for v, the speed of the ball when it hits the surface. (iii) Hence, calculate v.
Hard
55.Explain why a wire under large stress is dangerous if it suddenly breaks.
Hard
56.State what is meant by the principle of conservation of energy.
Medium
57.Explain, in terms of energy transfers, why a car engine is not 100% efficient.
Hard
58.A spring with 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
- B5.14 J
- C3.14 J
- D6.14 J
59.An action figure of mass 250 g is dropped from a height of 5 m. Calculate the gravitational potential energy of the action figure at this height. Use g = 9.8 N kg⁻¹.
Easy
- A12.25 J
- B14.25 J
- C11.25 J
- D13.25 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 just before it breaks.
Medium
- A1380 N
- B1518 N
- C1656 N
- D1242 N
61.A spring with spring constant 50 kN m⁻¹ is compressed by 10 cm. Calculate the elastic potential energy stored.
Medium
- A225 J
- B300 J
- C275 J
- D250 J
62.A pushchair has a force of 20 N applied for a distance of 5 m at an angle of 45° to the horizontal. Calculate the work done on the pushchair, in joules.
Easy
- A71 J
- B64 J
- C78 J
- D85 J
63.An electric car engine provides a thrust of 200 N while the car maintains a constant speed of 27 m s⁻¹. Calculate the power of the car, in watts.
Medium
- A6480 W
- B4860 W
- C5940 W
- D5400 W
64.A runner of mass 52 kg is lifted vertically by a 500 W motor through a height of 2.0 m in 2.5 s. Calculate the efficiency of the motor, as a percentage.
Medium
- A98 %
- B82 %
- C74 %
- D90 %
65.A spring of spring constant 92 N m⁻¹ is extended by 0.3 m. Calculate the elastic potential energy stored in the spring, in joules.
Medium
- A5.14 J
- B3.14 J
- C6.14 J
- D4.14 J
66.An action figure of mass 250 g is dropped from a height of 5 m. Calculate the gravitational potential energy of the action figure at this height, in joules. Use g = 9.8 N kg⁻¹.
Medium
- A11.25 J
- B13.25 J
- C12.25 J
- D14.25 J

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