Table of Contents
Work, Energy and Power
1. Introduction
Work, energy and power are fundamental concepts in physics. They explain how forces transfer energy, how objects perform activities and how quickly energy is transferred or transformed.
These concepts are important in everyday life, engineering, transportation, electricity generation, industrial production and the study of machines.
2. Work
In physics, work is done when a force causes displacement of an object.
Work depends on three factors:
- The magnitude of the applied force.
- The displacement of the object.
- The angle between the force and displacement.
For a constant force:
[
W=Fs\cos\theta
]
Where:
- (W) = work done
- (F) = applied force
- (s) = displacement
- (\theta) = angle between force and displacement
The SI unit of work is the joule (J).
[
1\ J=1\ N\cdot m
]
One joule of work is done when a force of one newton produces a displacement of one metre in its direction.
2.1 Positive work
Positive work occurs when the force has a component in the direction of displacement.
For example, when a person pushes a trolley forward and the trolley moves forward, the applied force does positive work.
2.2 Negative work
Negative work occurs when the force has a component opposite to displacement.
For example, friction generally does negative work on a sliding object because it acts opposite to the direction of motion.
2.3 Zero work
Work is zero in either of the following situations:
- There is no displacement.
- The force is perpendicular to the displacement.
For example, a person carrying a bag horizontally at constant height applies an upward force to support the bag, while its displacement is horizontal. The supporting force therefore does zero mechanical work on the bag in the idealised situation.
2.4 Work done by a variable force
When force changes with displacement, work is calculated by integrating the force over the displacement.
[
W=\int_{x_1}^{x_2}F(x),dx
]
Graphically, work is the signed area under a force-displacement graph.
For a spring obeying Hooke’s law, the force varies with displacement, so the work required to stretch or compress it is:
[
W=\frac{1}{2}kx^2
]
Here, (k) is the spring constant and (x) is the displacement from the natural length.
3. Energy
Energy is the capacity of a system to produce change or perform work.
The SI unit of energy is the joule (J).
Energy exists in several forms, including:
- Kinetic energy
- Potential energy
- Thermal energy
- Chemical energy
- Electrical energy
- Nuclear energy
- Radiant or electromagnetic energy
- Sound energy
Energy can be transferred from one object or system to another and transformed from one form into another.
4. Kinetic Energy
Kinetic energy is the energy possessed by an object due to its motion.
For an object moving at a non-relativistic speed:
[
KE=\frac{1}{2}mv^2
]
Where:
- (KE) = kinetic energy
- (m) = mass
- (v) = speed
4.1 Factors affecting kinetic energy
Kinetic energy depends on:
- Mass: for a given speed, kinetic energy is directly proportional to mass.
- Speed: for a given mass, kinetic energy is proportional to the square of speed.
If the speed of an object doubles, its kinetic energy becomes four times greater.
If its speed triples, its kinetic energy becomes nine times greater.
4.2 Example: Calculation of kinetic energy
A vehicle of mass 1000 kg travels at 20 m/s. Calculate its kinetic energy.
[
KE=\frac{1}{2}mv^2
]
[
KE=\frac{1}{2}\times1000\times20^2
]
[
KE=500\times400
]
[
KE=200,000\ J
]
The kinetic energy is 200,000 joules or 200 kJ.
5. Potential Energy
Potential energy is stored energy associated with the position, configuration or state of an object or system.
5.1 Gravitational potential energy
Gravitational potential energy near Earth’s surface is the energy an object possesses because of its height above a chosen reference level.
[
PE=mgh
]
Where:
- (PE) = gravitational potential energy
- (m) = mass
- (g) = gravitational acceleration
- (h) = vertical height above the reference level
Example:
A 5 kg object is raised to a height of 10 metres. Calculate its gravitational potential energy, taking (g=9.8\ m/s^2).
[
PE=5\times9.8\times10
]
[
PE=490\ J
]
The object gains 490 joules of gravitational potential energy.
The reference level is important. Gravitational potential energy is measured relative to a chosen position.
For large distances, such as those involved in planetary motion, the simple expression (mgh) is not sufficiently accurate. The gravitational potential energy of two masses is described using the universal law of gravitation.
5.2 Elastic potential energy
Elastic potential energy is stored in an object when it is stretched or compressed within its elastic range.
For an ideal spring:
[
PE=\frac{1}{2}kx^2
]
Examples include:
- A compressed spring.
- A stretched elastic band.
- A drawn bow.
6. Work-Energy Theorem
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy.
[
W_{\text{net}}=\Delta KE
]
[
W_{\text{net}}=\frac{1}{2}mv^2-\frac{1}{2}mu^2
]
Where (u) and (v) are initial and final velocities.
This theorem explains how work changes the motion of an object.
Example:
A 2 kg object increases its speed from 3 m/s to 7 m/s. Calculate the net work done.
[
W_{\text{net}}=\frac{1}{2}m(v^2-u^2)
]
[
W_{\text{net}}=\frac{1}{2}\times2(49-9)
]
[
W_{\text{net}}=40\ J
]
The net work done is 40 joules.
7. Conservation of Mechanical Energy
Mechanical energy is the sum of kinetic energy and potential energy.
[
ME=KE+PE
]
The law of conservation of mechanical energy states that the total mechanical energy of a system remains constant when only conservative forces do work.
[
KE_i+PE_i=KE_f+PE_f
]
In practical situations, friction and air resistance can transform mechanical energy into thermal energy and sound. Total energy remains conserved, but mechanical energy alone may decrease.
7.1 Example: Falling object
An object is released from rest at a height above the ground.
Initially:
- Kinetic energy is zero.
- Gravitational potential energy is maximum relative to the ground.
As the object falls:
- Potential energy decreases.
- Kinetic energy increases.
Just before reaching the ground, its kinetic energy is greatest and its gravitational potential energy relative to the ground is approximately zero.
If air resistance is neglected, total mechanical energy remains constant.
8. Forms and Transformations of Energy
Energy transformation means the conversion of energy from one form into another.
| Device or process | Energy transformation |
|---|---|
| Electric bulb | Electrical to light and heat |
| Electric fan | Electrical to mechanical and heat |
| Electric heater | Electrical to thermal |
| Solar panel | Radiant to electrical |
| Battery powering a device | Chemical to electrical |
| Petrol engine | Chemical to thermal and mechanical |
| Hydroelectric plant | Gravitational potential to kinetic, mechanical and electrical |
| Microphone | Sound to electrical |
| Loudspeaker | Electrical to sound |
| Nuclear power plant | Nuclear to thermal, mechanical and electrical |
9. Law of Conservation of Energy
The law of conservation of energy states that energy cannot be created or destroyed; it can only be transferred or transformed from one form into another.
The total energy of an isolated system remains constant.
This law is one of the most fundamental principles of physics.
Examples include:
- A falling object converting potential energy into kinetic energy.
- A battery converting chemical energy into electrical energy.
- A generator converting mechanical energy into electrical energy.
- Food energy being transformed into movement and heat in the human body.
10. Power
Power is the rate at which work is done or energy is transferred.
[
P=\frac{W}{t}
]
Where:
- (P) = power
- (W) = work done
- (t) = time taken
The SI unit of power is the watt (W).
[
1\ W=1\ J/s
]
One watt is equal to one joule of work done or energy transferred per second.
10.1 Power in terms of force and velocity
For a constant force acting in the direction of velocity:
[
P=Fv
]
More generally:
[
P=\vec F\cdot\vec v
]
The power delivered by a force depends on the component of velocity in the direction of that force.
10.2 Example: Calculation of power
A machine performs 6000 J of work in 20 seconds.
[
P=\frac{6000}{20}
]
[
P=300\ W
]
The power of the machine is 300 watts.
11. Commercial Units of Energy
The joule is the SI unit of energy, but electrical energy is commonly measured in kilowatt-hours (kWh).
One kilowatt-hour is the energy consumed by a device operating at a power of one kilowatt for one hour.
[
1\ kWh=3.6\times10^6\ J
]
An electricity meter records energy consumption, usually in kWh.
Example:
A 2 kW heater operates for 3 hours.
[
E=Pt
]
[
E=2\times3=6\ kWh
]
The heater consumes 6 kWh of electrical energy.
12. Efficiency
Efficiency is the ratio of useful output energy or work to total input energy or work, expressed as a percentage.
[
\text{Efficiency}=
\frac{\text{Useful Output Energy}}{\text{Total Input Energy}}\times100
]
For machines, efficiency can also be expressed in terms of useful output work and input work.
No real machine is 100% efficient because some input energy is usually transferred into unwanted forms, such as heat and sound.
13. Work, Energy and Power: Comparison
| Concept | Meaning | SI unit |
|---|---|---|
| Work | Energy transferred by a force through displacement | Joule |
| Energy | Capacity to perform work or produce change | Joule |
| Power | Rate of work or energy transfer | Watt |
| Efficiency | Ratio of useful output to total input | Percentage |
36 Key Takeaways for Exams
- Work is done when a force produces displacement.
- Work is a scalar quantity.
- The SI unit of work is the joule.
- One joule equals one newton-metre.
- Work depends on force, displacement and the angle between them.
- Positive work occurs when force has a component in the direction of displacement.
- Negative work occurs when force has a component opposite to displacement.
- Zero work occurs when there is no displacement or force is perpendicular to displacement.
- Work done by a variable force is represented by the signed area under a force-displacement graph.
- Energy is the capacity to perform work or produce change.
- The SI unit of energy is the joule.
- Kinetic energy is the energy of motion.
- Kinetic energy is calculated using one-half mass multiplied by speed squared.
- Doubling speed increases kinetic energy fourfold for the same mass.
- Potential energy is stored energy associated with position or configuration.
- Gravitational potential energy near Earth’s surface is (mgh).
- Elastic potential energy in an ideal spring is one-half (kx^2).
- The work-energy theorem states that net work equals the change in kinetic energy.
- Mechanical energy is the sum of kinetic and potential energy.
- Mechanical energy is conserved when only conservative forces do work.
- Friction can transform mechanical energy into thermal energy.
- The law of conservation of energy states that energy cannot be created or destroyed.
- Energy can be transferred and transformed between different forms.
- A solar panel converts radiant energy into electrical energy.
- A generator converts mechanical energy into electrical energy.
- Power is the rate of doing work or transferring energy.
- The SI unit of power is the watt.
- One watt equals one joule per second.
- Power equals work divided by time.
- Power delivered by a force in the direction of velocity is the product of force and velocity.
- A kilowatt-hour is a unit of energy, not power.
- One kilowatt-hour equals 3.6 million joules.
- Efficiency is the ratio of useful output to total input.
- Efficiency is commonly expressed as a percentage.
- Real machines lose some useful energy through heat, sound and other effects.
- Work, energy and power are essential in engineering, transport and energy systems.