# Work, Energy and Types of Energy.

In this Physics Tutorial, you will learn:

1. What is Work?
2. What is Energy?
3. What are the units of Work and Energy?
4. How is Energy classified?
5. The names of all types of energy you will encounter during your study of Physics

## Introduction

All of us have heard about the terms "Work" and "Energy" in many situations during our daily activities. We often say "I'm going to take a rest as I have done a lot of work today" or "I have no energy left to play football as I'm very tired" and so on. However, only a few people are able to define these terms correctly, relying on scientific basis. For this reason, we will provide some generic explanations on the Physics underlying behind the terms "Work and "Energy".

## What is Work in Physics?

The term "Work" in Physics is slightly different from the concept of "Work" in daily life. Thus, if you are employed as a supervisor in a company, your boss may pay you only to observe what happens or how the other employees are behaving or working. You may do this job sitting on your chair, without moving at all and still being paid.

In Physics however, the concept of "Work" is different. If you are not able to move something from its place, you have not done any work despite your efforts. No work is done in a Physics context if you only sit still and observe what happens around. Also, if something is moving but this movement is not because of your action, you are not doing any work as this process doesn't involve you at all. Thus, in Physics, a system, person or object is doing work if both the abovementioned conditions simultaneously meet, i.e.:

1. If a force is exerted on a certain object, and
2. If the object changes its place due to the action of the given force.

Mathematically, we have:

Work = Force × Displacement

In symbols,

### Equation 1

W = F × ∆x

Attention!

1. Despite the Force and Displacement are both vector quantities, Work is scalar as it is calculated by the dot product of two vectors, which gives a scalar.
2. Do not confuse the terms "Displacement" and "Distance". The Displacement (AB) represents the shortest path from point A to the point B while the Distance AB represents the length of the true path when an object has moved either from A to B or from B to A. Look at the figure below. In this figure, there are three possible paths to move the circular object from A to B. The shortest one, i.e. the straight line (or the vector) that connect these two points represents the Displacement Δx while the two other paths are longer as magnitude. Hence, they simply represent two Distances AB. To calculate the Work required for moving the object from A to B, we must multiply the force F needed to move the object and the Displacement Δx, not the Distance d. Thus, W = F ∙ Δx, (not F ∙ d).

## What is Energy?

When one gets tired or is sick, we often say phrases words like "I have no energy left today to go at work" or "I cannot move from the bed as I lack energy". When we move, we must carry our body, we exert a force F at our feet. In this case, we can displace ourselves by Δx metres. Therefore, it is obvious that we must do some work in order to go somewhere even though we only carry our own body weight.

However, everybody knows from practice that we need to eat well if we want to walk long distances (displacements). We can conclude that we need a considerable amount of Energy supply from food in order to convert it into work while walking. From here, we can deduce the meaning of the Energy concept. Thus,

"Energy (E) is the ability of an object or system to do work."

This means more energy an object possesses more work it can do. If no energy is left, the object cannot do work anymore. Hence, the maximum work an object or system can do is equal to the energy it possesses.

Mathematically, we can write:

### Equation 2

E = Wmax

In general, objects and systems (especially living organisms) keep some energy for other processes besides for boing work. They do not consume all their energy only for doing work. Therefore, we can write:

### Equation 3

W = ∆E = Einitial - Efinal = E1 - E2

From the above equation, it is easy to understand that when all the initial energy goes for doing work, there is no energy left at the end (E2 = 0). Therefore, we have

### Equation 4

W = ∆E = Einitial - Efinal
= E1 - 0
= E1
= Wmax

The last part of the equation 4 is simply the equation 2. Hence, we can say that eq. (2) represents a special case of eq. (4).

Remark!

Work and Energy are both scalar quantities, i.e. they are physical quantities that don't have direction but only magnitude.

### Units of Work and Energy

It is obvious that since Work represents a change (difference) in Energy, they must both have the same unit. Since Force is measured in Newtons [N] and Displacement in metres [m], the unit of Work is [N ∙ m]. It is otherwise known as Joule [J]. Hence,

1J = 1N × m

As stated before, Work and Energy have are measured using the same unit. Therefore, the unit of Energy is Joule [J] as well. This is the official (SI) unit of Energy.

In Thermodynamics (especially when dealing with food (chemical) energy and thermal energy, another unit is often used. It is known as Calorie [cal]. The conversion factor between Joule and Calorie is

1cal = 4.18J

Not always, the direction of force is in the direction of motion. In this case, we have to use only the component of force that lies in the direction of motion. Look at the figure below: In this case, only Fx = F ∙ cos α contributes to the motion. Therefore, the work done by the force F will be

W = Fx × ∆x = F × cos α × ∆x

For example, if the force F in the above figure is equal to 50 N and the angle is 300 (cos 300 = 0.86), the work W done by the force F to move the object by Δx = 40 m is

W = Fx × ∆x
=F × cos α × ∆x
=50N × 0.86 × 40m
=1720 J

Remark!

This is still a 1-D motion regardless the fact that the direction of force F is not in the direction of motion. You must consider only the force component that lies in the direction of motion. You can use the 1D Motion Work Calculator to calculate and check your own results when performing this Physics equation.

#### Example

A 50 kg girl took 1200 kcal energy from food. What linear distance can she travel using this amount of energy? Take g = 9.81 m/s2. Round up the result to the nearest whole number.

#### Solution

Here the linear distance represent the displacement. First, we have to convert cal to J. Also it is known that the prefix (k) stands for kilo, i.e. thousands. Thus,

E1 = 1200 kcal
= 1 200 000 cal
= 1 200 000cal × 4.18 J/cal
= 5 016 000 J

Also, it is known that Weight = Mass ∙ Gravity. Here Weight represents the force to be exerted by the girl in order to make herself move. Thus,

F = W = m × g
= 50kg × 9.81 m/s2

= 490.5 N

(From the Newton's Second Law, it is known that 1N = 1 kg × m / s2)

Hence, from the equation (1)

W = F × ∆x

after rearranging, we obtain:

∆x = W/F
= 5 016 000 J/490.5 N
= 10 226 m

In reality, most of the energy generated by the consumption of foods goes for vital processes such as for keeping the body temperature constant, maintaining a normal blood flow, for breathing etc. Only a small portion is available for doing work.

## Calculation of Work in 2 and 3 Dimensions

When the object is moving in two or three dimensions, we use the equation of the distance r between two points

r = √(∆x)2 + (∆y)2 + (∆z)2

Or

r = √(xf - xi)2 + (yf - yi)2 + (zf - zi)2 We can also write the components of the force F according the three main directions. The figure is the same as well. The only difference is that we can write F instead of r. Thus, if we denote the angles of the force with these directions as α, β and γ respectively, we can write for the force components

Fx = |F| × cos α
Fy = |F| × cos β
Fz = |F| × cos γ

where |F| is the magnitude of the force F. It is calculated by the equation

|F| = √F2x + F2y + F2z
= √( |F| × cos α )2 + ( |F| × cos β )2 + ( |F| × cos γ )2
= √ |F|2 × (cos2α +cos2β+cos2γ
= √|F|2
= |F|

The last expression is true because

cos2 α+cos2 β+cos2 γ = 1

Therefore, if we want to calculate the work done by the force in each direction, we can write

Wx = Fx × ∆rx
= Fx × (xf - xi )
= |F| × cos α × (xf - xi )
Wy = Fy × ∆ry
= Fy × (yf - yi )
= |F| × cos β × (yf - yi )
Wz = Fz × ∆rz
= Fz × (zf - zi )
= |F| × cos γ × (zf - zi )

All the above-mentioned equations contribute in generating the equation of work W done by a force F in 3 dimensions (in space) in the most complicated scenario:

W = |F| × r
= |F| × √(xf - xi )2 + (yf - yi )2 + (zf - zi )2

Don't be afraid by this long formula as in general, most of these quantities are zero. For example when the object is moving only in one direction (for example only according the x-direction), all values of the coordinates in the other two directions are zero. Therefore, the expression for the displacement r becomes

r = rx = √(xf - xi )2
= xf - xi
= ∆x

and that of the work W is

W = Wx = Fx × ∆x

As you see, the known formula Δx = xf - xi represents only a special case of the general expression for the displacement. It is used when the motion is taking place in a single direction. You can use the 3D Motion Work Calculator to calculate and check your own results when performing this Physics equation.

## The graphical representation of Work

First, let's consider a constant force F newtons acting on an object. As a result, the object moves horizontally by Δx meters. If we take the initial position xi = 0, the final position of the object will be xf = Δx (as Δx = xf - xi).

Thus, if we put the position x at the horizontal axis and the force F at the vertical one, we will obtain the following graph: Thus, it is obvious that Work represents the magnitude of the area under the Force vs Position graph (here, the area of a rectangle). This method for calculating the Work is particularly helpful if the force is not constant. In such cases, we simply calculate the abovementioned area and the result represents the amount of Work in Joules. Look at the example below:

### Example

What is the work done by an object if the Force vs Position graph of the process is shown below? ### Solution

From the graph, we can see that the initial force used was 8N. This force constantly increased up to 24N during the 20m of displacement.

1. One method for calculating the Work would be finding first the average force < F > = (8N + 24N) / 2 = 16N and then using the equation W = < F > ∙ Δx = 16N ∙ 20m = 320J.
2. The other method (graph method) consists on calculating the area A of the trapezium formed by the graph and the two axes. Giving that the Area of trapezium is calculated by the equation:
Atrapezium = ( ( B + b ) × h ) ÷ 2

where B and b are the large and the small parallel sides of the trapezium respectively (here B = 24N and b = 8N) and h is its height (here h = 20m). Thus, substituting these values, we obtain

W = Atrapezium = ( 24N + 8N ) × 20m/2
= 32N × 20m/2
= 640J/2
= 320J

As you can see, the result obtained in both cases is the same.

## How is Energy classified?

Energy is classified in two main groups: I. Energy of state and II. Energy of transfer.

The first group (Energy of state) includes all types of energy that are possessed by the system regardless of what happens outside it. The second group (Energy of transfer), includes all types of energy exchanged between two or more systems when they enter in contact with each other.

In the following section, we will show a diagram where all types of energy are represented based on a hierarchic structure. This is only a version of Energy classification; it does not represent the absolute truth. For example, Heat energy can also be included as a sub-category of Thermal energy etc.

Now, let's take a closer look at all the above mentioned types of energy. Thus,

### Kinetic Energy

Kinetic Energy represents the energy a system possesses due to its motion. This means when a system is moving it possesses Kinetic Energy while when the system is at rest, it does not possess Kinetic energy. This energy can transfer from one place to another. Therefore, Kinetic Energy is an energy of transfer. You can view more detailed information on Kinetic Energy Here

### Potential Energy

Potential Energy represents the energy stored in a system when no change of system's structure is considered. This stored energy can be converted into other forms of energy in certain conditions. Therefore, Potential Energy is an energy of state.

There are many kinds of Potential Energy such as:

• Gravitational Potential Energy is the stored energy in an object when it is raised at a certain height from the ground. It is obvious that when the object is released, this Gravitational Potential Energy is converted into Kinetic Energy as the object approaches the ground and loses height because it moves faster and faster.
• Elastic Potential Energy is the stored energy in an elastic object (spring, rubber band, etc.) when the object is stretched or compressed. When the elastic object is released, the Elastic Potential Energy is converted into other forms of energy as well.

• Nuclear Potential Energy is the stored energy in the nuclei of atoms. When particles from outside hit the nuclei, this stored energy is released and as a result, it turns into other forms of energy.

### Mechanical Energy

Mechanical Energy represent the sum of Kinetic and Potential Energy of a system.

### Thermal Energy

Thermal Energy is a part of the internal energy of a system (the other part is the chemical energy). Thermal energy is the energy possessed by the system due to the vibration of its particles (atoms and molecules). More vibrant the molecules of a system, greater the value of its thermal energy. Thermal energy is an energy of state.

Radiant Energy is the energy radiated in the environment by hot objects. Hotter the source object, greater the amount of the energy radiated around. When the radiant source is not very hot, we perceive the radiant energy as Heat and when the source is very hot, it becomes luminescent and therefore it emits radiation in the form of Light Energy. Radiant energy is an energy of transfer.

### Chemical Energy

Chemical Energy is the energy released (or absorbed) by a system during a chemical reaction, i.e. when the structure of substances involved in the process changes. This is the main difference between Potential and Chemical Energy (in potential energy there is no change in the structure of material or substance). Chemical energy is an energy of state when not activated yet. Then, during a chemical reaction it is transferred.

### Electric Energy

Electric Energy is the energy produced when the electrons of a conductor flow in a pre-defined direction. It is an Energy of transfer.

### Sound Energy

Sound energy is produced when a sound source disturbs the medium's molecules (usually air molecules) and make them vibrate. This disturbance is propagated in the medium in the form of sound waves until it reaches the receiver. Therefore, it is obvious that sound energy is a kind of transfer energy.

## Summary

In Physics, a system, person or object is doing work if

1. A force is exerted on a certain object, and
2. The object changes its place due to the action of the given force.

Mathematically, we have:

Work = Force × Displacement

In symbols,

W = F × ∆x

where:

• F is the force exerted
• Δx is the object's displacement.

Work is a scalar quantity. It is measured in Joules [J]. 1 J = 1 N × m.

In Thermodynamics (especially when dealing with food (chemical) energy and thermal energy, another unit known as Calorie [cal] is often used. The conversion factor between Joule and Calorie is

1cal = 4.18J

Not always, the direction of force is in the direction of motion. In this case, we have to use only the component of force that lies in the direction of motion.

In this case, only Fx = F ∙ cos α contributes to the motion. Therefore, the work done by the force F will be

W = Fx × ∆x
= F × cos α × ∆x

This is still a 1-D motion regardless the fact that the direction of force F is not in the direction of motion. You must consider only the force component that lies in the direction of motion.

When the object is moving in two or three dimensions, we use the equation

W = |F| × r
= |F| × √(xf - xi )2 + (yf - yi )2 + (zf - zi )2

To calculate the work done by the force F. Here, the expression within the square root represents the displacement r in 3 dimensions.

Geometrically, Work is represented by the area under the Force vs Position graph. This method is particularly useful when the force used is not constant.

Energy E is the ability of an object or system to do work. Therefore, Work is a kind of energy change in a system. We can write

W = ∆E
= Einitial - Efinal

Energy is also measured in Joules [J].

Energy is classified in two main groups: I. Energy of state and II. Energy of transfer. The scheme below shows the relationship between the most important types of energy with each other. ## Work, Energy and Types of Energy Revision Questions

1. A 120N horizontal force pushes an object for some metres. The work done by the horizontal force is 1560 J. What is the displacement of the object?

1. 13 m
2. 1440 m
3. 1680 m
4. 187 200 m

A 40 N force at 370 to the horizontal direction is used to move an object horizontally by 20 m. What is the work done by the force? (cos 370 = 0.8; sin 370 = 0.6) 1. 800 J
2. 640 J
3. 480 J
4. 0

What is the work done by the jet engine in the figure to move the jet from the landing position at the origin (0, 0, 0) to the given position if the constant force produced by the engine is 1200 N? 1. 1,080,000 J
2. 702,794 J
3. 108,000 J
4. 7,027,944 J