# Openphysics/Energy

## Contents

# Energy

Energy is one of the most central concepts in all of science; the entire universe is made up of matter and energy. Matter is easily understood; it is what we can touch, taste, see, smell, and hear. Energy on the other hand can be rather difficult, as it is an abstract concept; our sense don’t allow us to examine energy in the same way as matter, and it was even unknown to Isaac Newton, with its existence being debated into the second half of the 19th century. Although it is now more common knowledge, and such a huge part of science, it is hard to define energy, as it is both a thing, and a process – a noun and a verb. Objects (such as people or vehicles and plants etc) have energy, we almost only observe energy when it is transforming (between different forms of energy). Energy can be found in various different forms; it can be transformed from electromagnetic waves being sent off by the sun into heat waves we feel; as well as being harnessed by plants to bind molecules together in photosynthesis; as well as being in the food we eat, which is then harvested when we digest it. It can be found in the tiniest of stationary or moving particles, or possessed by an entire planet.

The best way to start learning about energy is to start by learning about work.

## Work

We’ve learnt about impulse with momentum, where the impulse was the how long a force acted on an object – but ‘how long’ doesn’t need to be measured in time – it can also be measured in distance – and this is where we find work.

W = Fd

Where W = work done (measured in Joules, or Kilojoules) F = force (measured in newtons) and d = the distance (in metres) the object has moved.

When we push a block 10 m across the floor, we are doing work, as a force is being applied to move an object over a distance. However, if we push a much larger block with the same force, but it doesn’t move – no work is done – there is no change in distance (F*0 = 0) so for work to be done, some visible changes must usually take place, or something has to be achieved.

A good example of when work is, and isn’t being done, can be found with weightlifters. Bob, a weightlifter, chooses a 2,000N weight, and hoists it off the ground, to a position above his head – a movement of 2 metres. When he is lifting the weight, he is doing work, and when is fully lifted, he has done 4,000 Joules (2*2000) however, once fully lifted, and he is holding the weight above his head, he is doing no work is done on the weight, as it isn’t moving – there is no change in distance – he may get very tired from holding such a heavy object, but he isn’t ‘achieving’ anything by simply holding it.

Note: work is a scalar quantity, and the direction of work done doesn’t matter (although vectors can but used for calculations, the resultant vector’s direction isn’t important.)

#### Example1:

librarian picks up a stack of books with a mass of 3kg, and stacks them on a shelf 170cm higher than their original position. is work being done, and if so, how much?

Work is being done, because the librarian is moving the object’s mass over a distance

the work done is calculated:

W = Fd

W = 30 N * 1.7m (converting Kg into newtons, and cm into m)

W = 51 joules

by lifting and stacking the books, the librarian does 51 joules of work.

#### Example2:

## Power

Power is the rate at which work is done – it essentially measures the rate at which energy is transferred.

Like acceleration measures the change in velocity over time; power measures the change in work done over time:

P = W / t

Where time is measured in seconds, work done in joules, and power in watts (1 watt is 1 joule per second.)

#### Example1:

A weightlifter lifts a weight of 75 kg off of the ground, and holds it at a position of 200 cm above the ground. The lift takes 3 seconds, calculate the power of the weightlifter.

P = W/t

so first we need to find W (work done)

W = Fd

W = 750 * 2 (converting Kg into newtons, and cm into m)

W = 1,500 joules

P = W/t

P = 1,500 / 3

P = 500 watts

The weightlifter’s power is 500 watts (or joules per second).

## Potential energy

Objects can store energy because of their position, or their current states; this stored energy is called potential energy (Ep, or PE) because it has potential to do work. Stretched and compressed springs have potential kinetic or elastic energy (this is gone into with more detail in Hooke’s law) as when they are let go of (eg, uncompressed or stretched) they will move and gain kinetic energy. The same applies to bows, slingshots and elastic bands.

Objects such as people, food, or fuels also hold chemical potential energy, as on a microscopic point of view, the atoms are in position to do work; the energy is available to transferred when electrical charges between molecules is altered – when a chemical reaction takes place.

Gravitational potential energy is possessed by objects that are elevated above and against earth’s gravity, as work is required for them to reach this position. Skydivers obviously have a great amount of gravitational potential energy, as when they jump out of aircraft, this potential energy is quickly transferred into kinetic energy. The amount of gravitational potential energy possessed by an object is equal to the work done against gravity for it to reach that position. The work done equals the force required to move it upwards multiplied by the vertical distance it is moved ( W = Fd) the upward force equals the weight (mass multiplied by gravity, or, m*g) so the work done in lifting it through a height is given by the product mass multiplied by gravity multiplied by height; m*g*h.

## Kinetic energy

When objects move they posses kinetic energy – due to their motion – and when they slow down or speed up, the kinetic energy is transferred into different forms of energy; when a car slows down by breaking it produces heat and sound, and when it has stopped it has potential energy, and this energy can be converted back to kinetic when it starts moving again.

An object’s kinetic energy is determined by its mass, and the velocity of which it is travelling, shown by the formula:

Ek = ½*m*v2

Where Ek is measured in joules, mass in kilograms, and velocity in metres per second.

## Conservation of mechanical energy

In any and every situation where work is done, energy is transferred from one form into another; sometimes several different forms, such as heat, sound and light. What this means is that energy can never be created, and never be destroyed; it is always conserved and transferred. This helps solve many energy related problems, as energy before work is done must equal the total of all forms of energy afterwards (although it can be hard to calculate all the different forms of energy that are produced; for example the energy turned into heat and sound when something loses kinetic energy!)

## Hooke’s law

Work is done in compressing or stretching a spring. When this is done, energy is stored in the spring and can be released later. Clocks and catapults use this form of elastic potential energy.

The force needed to extend or stretch an ordinary spring, eg, the spring in a ball point pen, increases as the spring extends. This is also true is the spring is compressed rather than stretched.

A graph of the extending force against extension shows a linear relationship, if F is the extending force in N, and x is the extension in m, then:

F = k*x

Where k, the slope of the graph – called the spring constant – is measured in N m^{-1}.

The spring’s extension is the extra distance extended. It does not include the spring’s natural length. The elastic potential energy, Ep, stored when the spring is extended (or compressed) a distance x, is equal to the area under the graph from zero to x.

The area under the graph is measured in joules. This comes from multiplying the unit of force (newtons) by the unit of distance extended (metres). The elastic potential energy is shown as the shaded area in the graph below:

Area = ½ * base * height

Ep = ½ F*x

F = k*x

Ep = ½ (k*x)*x

Ep = ½ kx^{2}

The x^{2} in the formula Ep = ½ k*x^{2} shows that proportionally much more (a squared function) elastic potential energy is stored in a spring as it is stretched or compressed further.

#### Example1:

A mass of 2kg hangs from the end of a spring, extending the spring by 80 cm. Calculate the spring constant, and find how much energy is stored in the spring.

F = m*g

m = 2kg, g = 10

F = 20 N

F = k*x

20 N = k* 0.80

20 / 0.80 = k

k = 25

Ep = ½ k*x^{2}

Ep = ½ 25*0.82

Ep = 8 joules

The spring constant is 25N m^{-1}, and the elastic potential energy is 8 joules.