# 16.21 Energy Before you read this post, I suggest you read posts 16.13, 16.16, 16.17, 16.19 and 16.20.

Most people know that “energy” has a precise meaning in science, for example when applied to nutrition or in the phrase “renewable energy”; unfortunately, several people who I’ve met completely misunderstand what this meaning is.

Energy is simply the ability to do work, as defined in post 16.20. Because we measure work in joules (abbreviated to J), we measure energy in joules too. (There is also a non-SI unit called the calorie that is sometimes used, see post 16.12 if you’re not sure about SI units.)

If I lift an object of mass m to a height h, I am applying a force mg, where g is the strength of the gravitational field, and so do work mgh (post 16.20). If I do not support the object it falls back down to the ground. It falls a distance h downwards because gravity exerts a force mg, on it, so the force does work because it moves the object.

Before it fell, the object had the ability to do work, so we say that it had energy. Its energy had a value of mgh.

A moving object also has energy. When a rolling ball collides with a stationary ball, the stationary balls begins to move. So the rolling ball must have exerted a force on it. Because the second ball (originally stationary) moves, work is done by the first (rolling) ball. If the first ball can do work, it must have energy.

How much energy does a moving object have? We can work out the force it exerts (rate of change of momentum, see post 16.13) and the distance it travels (average speed multiplied by time, see post 16.12). The mathematics involved in doing the calculation properly is more complicated than simple algebra, so I’ve added it as an appendix to this post for anyone who is interested. The result is that the kinetic energy of an object with mass m moving at a speed v is mv2/2.

An object that can do work because it is moving is said to have kinetic energy. The object that was lifted had the ability to do work before it started to move; this type of energy is called potential energy.

What is the speed an object of mass m, fallen from a height h, when it hits the ground? When it hits the ground it has no potential energy left – it has all been converted into kinetic energy. Since the kinetic energy must then be equal to the potential energy, mv2/2 = mgh which means that v2 = 2gh or v= √(2gh). Notice that the result does not depend on the mass of the object. Since g has a value of about 9.8 m.s-2 (see post 16.16) an object that falls from a height of 10 m hits the ground at a speed of √(2 × 9.8 × 10) = √(196) = 14 m.s-1. (If this isn’t clear, the √ sign means “square root”. √9 is the number which when multiplied by itself gives 9. So √9 = 3. We could also write this as 91/2 = 3.)

The gravitational energy of a lifted object is not the only form of potential energy. Let’s think about compressing a spring by a distance x. The force required to compress the spring is given by F = kx, where k is called the stiffness of the spring. (Since k = F/x, it is measured in N/m or N.m-1; see post 16.13 for more explanation of how we work out units.) Before the spring was compressed, the force that was applied was zero. As the spring is compressed, the force that is needed increases uniformly until it reaches kx when the spring is compressed a distance x. So the average force required to stretch the spring is kx/2. Since this force compresses the spring by a distance x, the work done is given by (kx/2) × x = kx2/2. When the force is removed the spring returns to its original dimensions – this return process is called recoil. The spring exerts a force to move back to its original position and does work because it moves. When it was compressed, it had the ability to do this work (just as the lifted object had the ability to do work) and so it too has potential energy.

So we could define potential energy as the energy that something has because of its position or shape.

Let’s suppose we make a toy that uses a compressed spring to shoot a ball of mass m vertically upwards. If the spring has a stiffness k, how high will the ball rise? The toy converts the potential energy of the compressed spring into potential energy of the ball. So mgh = kx2/2, where h is the height that the ball will rise. So h = kx2/(2mg)

We have done two simple calculations:

1. How fast is a falling object moving when it hits the ground?
2. How high can a compressed spring shoot a ball?

Both of these use the principle of conservation of mechanical energy.

Mechanical energy is not the only form of energy. The principle of conservation of mechanical energy is only valid when mechanical energy is not converted into other forms. When fuel burns a chemical reaction occurs that produces heat. Heat can be used, for example in a car engine, to do mechanical work. So heat is a form of energy. When fuel burns, the chemical energy that holds it together is released as heat. Similarly, food contains chemical energy that our bodies can use to do mechanical work. However, despite what many people believe, our bodies don’t burn foods like fats – if we did, we’d burn the tissues that make up our bodies at the same time!

If we consider all forms of energy, it appears that energy is neither created not destroyed in any process. This is called the principle of conservation of energy or the first law of thermodynamics. However, under certain circumstances (for example in a nuclear reactor), energy can be created by annihilating mass. The energy, E, created by annihilating a mass m is given by Einstein’s famous equation E = mc2, where c is the speed of light. A nuclear reactor uses this energy to do work on a generator to make electricity.

So there’s nothing mystical or magical about energy – it’s just the word we use for the ability to do work!

Related posts

Follow-up posts

16.23 Power
16.35 Heat
16.37 Solids, liquids and gases
17.45 Electrical energy
18.29 Reverse osmosis (for information about Free energy)
19.6 Cells and batteries
19.8 Wave energy The mathematics used in this appendix is explained in post 17.23.