Well, that’s a title that will be nonsensical to anyone unused to the technical, physical sciences sense of the word ‘work’ (but we will explain on this page).
This section contains some quite specific content, in contrast with the previous sections which are quite generally applicable. However, we are going to use that specific content to exemplify some general points and:
- Illustrate that sometimes two techniques can be equally good at solving a problem
- Introduce you to the idea of vectors in the physical sciences (not the life sciences, we don’t know anything about that kind of vector…)
- Show you that you thought you understood the idea of ‘energy’ but you don’t (sorry, couldn’t resist that, but it’s OK – we don’t really understand it either…)
And again, just behind those last two flippant comments (although we really don’t know much biology) there are serious points.
- So much (mis-)understanding comes through technical terminology. Two sets of people use the same word, like ‘vector’, in different ways… Or worse, nobody really explains a definition to you, so that your understanding of the concept is sketchy from the start… You should always ask what a word means – never feel inadequate just because you haven’t met some terminology
- The energy thing might come as a shock in a moment. But we learn things at school by rote, and if we are lucky then we ‘achieve mastery’ of them. Then when they become assimilated into our automatic toolkit, we tend to stop questioning them. This is normal because it avoids having to work everything out from first principles. But it can be really illuminating to go back and question your understanding of the basics.
So the conservation of energy is obvious, right?
“Energy can’t be created or destroyed, just transformed from one form to another…” We probably all learned that at school. It’s called the Law of Conservation of Energy.
Conservation just describes the value of a quantity staying constant over time. See Section 09 for some more insights into conservation of other quantities.
Many of you will have used the Law of Conservation of Energy to work things out. One example might be movement in a gravitational field, like the ‘how fast will the ball hit the ground?’ in the ‘falling objects’ part below, by knowing that:
- Kinetic energy, [latex]E_K[/latex], is given by [latex]E_K=\frac{_1}{^2}mv^2[/latex], where [latex]m[/latex] is the mass of the object and [latex]v[/latex] is its speed
- A change in gravitational potential energy, [latex]\Delta E_P[/latex], is given by [latex]\Delta E_P=mg \Delta h[/latex], where [latex]g[/latex] is the gravitational field strength, and [latex]\Delta h[/latex] is the change in height.
And some of you may even have realised that in the list of ‘forms’ of energy that we are taught at school: kinetic, potential, elastic, chemical, nuclear, light, sound, etc., many are really just manifestations of kinetic and/or potential energy.
And yes, [latex]E=mc^2[/latex] is relevant here, but there’s enough content in this Section as it is…! When we get round to developing Section 27 (see the map), we’ll address that then…!
So what’s energy? No, really. Go on – what is it? In one sentence. Take two if you like. We know that it’s conserved, but what is it?
Now that’s a really interesting question, because we all, I imagine, have an image in our mind. Probably a series of images, because in everyday life, ‘energy’ is a term we hear in a huge variety of contexts. But the more you think about it, the harder that image becomes to pin down (well, that’s what we find, anyway). We know how it behaves, and can make predictions based on it, and yet it is hard to say what it is. Apologies, but you won’t find the answer on this page.
In fact, energy as a concept is so hard to pin down that it is often defined in textbooks as “the capability to perform work.” Maybe that will help us understand what it is. So what’s ‘work’? No really. Go on – what is it? In one sentence. Sorry, we are getting carried away… But it does say something about how abstract a concept energy is, that it gets defined in terms of another, single abstract quantity…
There’s a brilliant analogy of how weird energy is as a concept in ‘The Character of Physical Law’ by Richard Feynman in the chapter called ‘The Great Conservation Principles’. A suitable search in YouTube will also find the lecture version.
What is work and why is it useful?
So what is work?
Work is often defined in textbooks (depending on the level) as ‘force multiplied by the distance travelled in the direction of the force’. That definition is often accompanied by a statement that ‘an amount of work done is accompanied by an equal energy change’, and so work has the units of energy (Joules in the SI).
But let’s get away from dry definitions that you can find anywhere and get to the nub of why work is useful. Work provides a link between forces and energy.
Some problems can be solved easily using forces, some using energy. The idea of work allows us to ‘translate’ from the language of forces to the language of energy, and vice versa.
Well see an example next, by returning to the falling bodies of Section 01.
There’s a really interesting thing going on here, which is that the technical terms ‘work’ and ‘energy’ are both words with many meanings in everyday life. Does the use of everyday terms make the technical concepts easier to understand by providing analogies from life experience, or does it automatically make us think in a non-technical way and introduce layers of misconception and imprecision? It’s a good question; we may have helped write your PhD proposal…! Good and bad use of analogy (with concepts, rather than just appropriating words from everyday life) is something we have discussed in our accompanying blog.
Falling objects: forces, work and energy
Assuming that:
- The gravitational field strength, [latex]g[/latex], is constant (which will be pretty much fair enough if the object falls normal, everyday distances), and
- Air resistance can be neglected,
we can find the speed an object reaches after falling a given distance.
This is useful in all sorts of circumstances. Maybe we want to test the urban myth that a jam sandwich dropped from the top of the Empire State Building will kill you… So we will learn how to do the calculation in the next video.
From the calculations in the video, we see that the ‘air-resistanceless jam sandwich’ will hit the ground, after falling 381 m (this sandwich is falling off the roof…), at a speed of
[latex]\displaystyle v=\sqrt{2gh}=\sqrt{2 \times 9.81 \; \mathrm{m \; s^{-2}} \times 381 \; \mathrm{m}}=86.5 \; \mathrm{m \; s^{-1}}[/latex]
But it doesn’t matter whether we choose to work that out using forces or energies, we end up with the same calculation and the same answer.
Apologies to those of you who want to know the answer to the question about the likely outcome for the person underneath. It would need further information about the damage caused by jam sandwiches as a function of speed – information to which we are not privy… Note also that, in reality, the sandwich would reach a terminal velocity as discussed in Sections 01 – 04.
If we relax the condition that [latex]g[/latex] is constant we have the work done being found by integrating force with respect to distance, as we saw in Section 03. What we saw there is summarised in the diagram below.
There is a convenient feature of this situation that helps to prevent the maths being any harder – that is, the object is travelling in the same direction as the force. So why does this matter? When would this not be true? Ah, well, all the time, it turns out. Here are some examples…
- When we throw a ball anything other than directly upwards, the force on it is vertical, even though its trajectory takes it through a range of directions
- When planets orbit the Sun, the force causing them to remain in orbit is directed towards the Sun, even though they are travelling around it
- When using a lawn roller, the force is transmitted along the handle of the roller, which is usually orientated ‘diagonally’ during use, even though the roller is moving horizontally
And what impact does this have? Well, to explain that, we will need to introduce vector and scalar quantities.
Vectors and scalars
Vectors are quantities that have a magnitude (an amount, size or value) and a direction. Scalars are quantities that have magnitude only, with no direction.
For example, in the context of this section:
- Force is a vector quantity. It has a magnitude (commonly expressed in Newtons), and a direction. In the diagram above, the magnitude can be calculated from [latex] \textstyle F=\frac{Gm_1 m_2}{r^2} [/latex] and its direction could be stated as ‘towards the centre of the Earth’. Another example of a vector that we have met before is gravitational field strength.
- Energy is a scalar quantity. It has a magnitude (commonly expressed in Joules), but has no direction. Things do not have 20 J of energy upwards, they just have 20 J of energy. Notice that this is true for kinetic energy too – even though motion itself has a direction, the energy of motion does not. Two other examples of scalars are mass, and temperature.
There are various mathematical notations to express the fact that a quantity is a vector. Three of the most common (we know, we know, it would be nice if everyone used the same one…) are the use of:
- Bold text, such as [latex]\mathbf{F}[/latex] for a force vector
- An arrow above the symbol, as in [latex]\overrightarrow{F}[/latex]
- A line below the symbol (commonly used in handwritten script, where bold text is hard to achieve), as in [latex]\underline{F}[/latex]
Then the symbol [latex]F[/latex] without any of these notations means the magnitude of the vector. In the case of a force, that would be how many Newtons of force, but with no directional information. The magnitude of a vector can also be written with the modulus sign, as in [latex]|F|[/latex].
In the diagram above, it didn’t matter that we expressed the force as its magnitude, [latex]F[/latex], without any directional information, because it was acting along the direction of motion. Let’s look at one of our situations in which that isn’t true.
Reflections on lawn rollers: when two vectors are not aligned
Apologies to those of you who live in polar or equatorial regions, and may have no need of a lawn roller, but it is a classic example for what we are trying to show. The diagram shows a roller being pushed in a straight line three different times/ways.
The reason we haven’t chosen to show, say, a supermarket trolley is that it is more or less possible to apply a horizontal force to one of those, and since the trolley travels horizontally, we are then in the position of the force being in the same direction as the motion, which is what we need our example to avoid!
In contrast, with a roller, the force is transmitted down the handle, so the force is at an angle [latex]\theta[/latex] to the direction of motion. Hopefully, you can see that the acceleration achieved will be the greatest in case P where most of the force is acting in the direction of motion, and least in case R where the majority of the force is helping to squash the grass, rather than propelling the thing forwards. This is why work done (which in this case will transfer energy into the KE of the roller) is defined as
- ‘force multiplied by the distance travelled in the direction of the force’, rather than just
- ‘force multiplied by the distance travelled’.
The ‘… in the direction of the force’ part takes into account that ‘the ‘better’ the alignment between the force and the direction of motion, the more work will be done’. Mathematically that is achieved using the cosine of the angle , so that rather than [latex]W=Fx[/latex], we actually have [latex]W=Fx \cos{\theta}[/latex].
If this isn’t obvious to you, then don’t worry. We are building up to this in Section 11. It’s called ‘resolving vectors into components’ but it just the quantitative way to describe what was described in words in the previous paragraph.
Just to recap, then, the definition of work we have used so far is ‘the force multiplied by the distance moved in the direction of the force’. This corresponds to interpreting the equation
[latex] \displaystyle W=Fx \cos{\theta}[/latex]
as
[latex] \displaystyle W=F\times(x \cos{\theta})[/latex]
This interpretation is represented in the left hand part of the diagram below. However, you might find it conceptually simpler to think of work as “that portion of the force moving the object, multiplied by the distance moved. This corresponds to interpreting the equation
[latex]\displaystyle W=Fx \cos{\theta}[/latex]
as
[latex]W=(F\cos{\theta})\times x[/latex]
Both forms are equivalent, but conjure different mental images, and we suggest the second makes more conceptual sense, as represented in the right hand part of the diagram below. (The vertical portion of the force, which is not moving the roller, along with the weight, is balanced by the reaction force from the ground. It is also helping to flatten the grass, which is what makes a roller a roller, although to be fair, the weight of the roller is probably going to do that better than you can…)
Notice that a force at 90° to the motion does no work, because the cosine of 90° is zero. In fact, if a force remains at right angles to the motion, you have circular motion at constant speed, which we will examine in more detail in Section 13.
Velocity and speed
Although this page is primarily concerned with work done, as a link between forces and energy, now that we have met vectors and scalars it may help to broaden our scope a little and look at other examples.
Motion has a direction, and there are therefore vector quantities to describe it. We have met some of them before, so let’s summarise.
- Acceleration is the rate of change of velocity. And what’s velocity?
- Well, velocity is a vector quantity and it is the rate of change of displacement. The magnitude of velocity is called speed, which is therefore a scalar, with no direction. And what’s displacement?
- Well, it is a position vector from a defined fixed point, so that velocity is the rate of change of that position (and speed is the magnitude of that rate of change).
What all this means is exemplified by the following video, showing motion graphs for a ball being thrown vertically and then falling back to the starting position.
Notice that, because acceleration is the rate of change of velocity (which is a vector, thus having magnitude and direction), acceleration is also a vector. Acceleration can therefore be a change in speed or a change in direction, or both. This is important, because resultant forces cause accelerations (changes in velocity), and we know that forces can, indeed, change directions of motion as well as speeds. Another way of saying this is that objects subject to zero resultant force don’t just travel at constant speed (which could be a very meandering path); they travel at constant velocity.
How do we multiply vectors?
If [latex]F[/latex] is not constant over the distance [latex]x[/latex], we can find the amount of work by integrating, so that [latex]W=\int Fdx[/latex], rather than simply [latex]W=Fx[/latex]. And if the force is not in the direction of motion, then we can write [latex]W=\int Fdx[/latex] more generally as [latex]W=\int \mathbf{F}. d\mathbf{r}[/latex], where the bold symbols signify vectors, [latex]\mathbf{r}[/latex] is a position vector, and the ‘dot’ is a way of multiplying vectors to produce a scalar. (It is a good thing we have such a mathematical trick, because work is a scalar, so the integral needs to be the sum of an infinite number of tiny infinitesimal bits of scalar…) This way of multiplying vectors takes into account the angle between them, and is called the ‘dot product’ (because you write it with a dot) or ‘scalar product’ (because it results in a scalar).
You might well wonder how two quantities can be multiplied if they are not in the same direction. After all, when multiplying two ordinary numbers, each might be positive or negative. If one is positive and one is negative, you might visualise them as being in ‘opposite directions’, and you might say that they have an angle of 180° between them, but it’s hard to visualise multiplying numbers having any other angle between them…
You would be wise to wonder such a thing, and that will be dealt with in Section 11. In fact, there is more than one way to multiply vectors! The other way is called the ‘cross product’ or ‘vector product’ and, until you visit Section 11, we’ll let you draw your own conclusions as to what features that might have!
For now, we’ll finish by reminding you that of what we said near the top of this section: that Work provides a link between forces and energy. Using what we know now, we could equally have said that work provides a link between vector problems and scalar problems.
Link to Section 09 – Conservation laws (coming soon)
Where we will see that energy is one of a host of quantities that are (or appear to be…) conserved.
Link to Section 11 – Vectors
Where we will take a more general look at the properties of vectors, and see what they can do for us
Link to Section 13 – Circular motion and phasors
Where we will see that there is no work done in circular motion!
Link to Home Page
And you can always return here…
And finally, here is a link to a video from Nick Lucid, aka ScienceAsylum, which covers many of the same points, and some other cool stuff besides…