Force
Forces and Motion

Resultant force sets acceleration - Physics narrative

Physics Narrative for 14-16

A Physics Narrative presents a storyline, showing a coherent path through a topic. The storyline developed here provides a series of coherent and rigorous explanations, while also providing insights into the teaching and learning challenges. It is aimed at teachers but at a level that could be used with students.

It is constructed from various kinds of nuggets: an introduction to the topic; sequenced expositions (comprehensive descriptions and explanations of an idea within this topic); and, sometimes optional extensions (those providing more information, and those taking you more deeply into the subject). 

Core ideas of the Force and Motion topic:

  • Natural motion
  • Unnatural motion
  • Relative motions
  • Interactions modelled with forces
  • Quantities of motion
  • Force changes motion
  • Invariants and conservation laws

 

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A significant discovery

Force
Forces and Motion

A significant discovery

Physics Narrative for 14-16

Natural and unnatural motions

We're programmed so that we notice change. Birds that move are much easier to pick out from the background than those that don't. However, explaining motion has proved much harder to nail. Even figuring out what needs explaining has taken millennia of discussion and theorising.

Think carefully about the following processes:

  • A falling stone, getting faster and faster.
  • The Moon circling the Earth.
  • A stone slithering across an ice-covered pond.
  • A racing cyclist powering to a sprint finish.
  • A cyclist free-wheeling down a hill.

Which are natural – just what things do, if left to their own devices – and which are forced on the objects as a result of interactions with their environment? In other words, which processes need explaining and which are just what things do? And just what is a natural environment for things?

Everyday experience turns out not to be a reliable guide to more universal truths. In some ways where we live, learn and gain our intuitions is not a very typical part of the universe. So if you want to see the bigger picture, the wider view, then you'll have to re-home the objects to find out what they get up to in a natural environment. Then compare this environment with the one you live in, gradually adding complexity as you understand the additions that you make.

It's this necessity for re-homing the objects that is at the root of why so many find the description of the world that is developed in this episode so alien. The lived-in world seems to consist of objects whose natural motion is to come to rest, or to fall, or to rise, or some combination of these motions. Pretty much nothing seems to keep on moving unless there is some agent or agency that maintains that motion. For many centuries the theories initiated by Aristotle, based on careful observation of what things just did, described motions in terms of natural motions that differed for different objects. There was no simple overall pattern. That had to wait until people learnt to ask different questions about the world, seeking more unified patterns of behaviour. This required more sophisticated mathematical descriptions than were available to Aristotle, or throughout the Middle Ages, and a significant leap of the imagination. This combination of painstaking, careful analytic thinking and imaginative leaps to see the world as it really is make this topic rather hard. It's not one to be rushed, or skimmed over. Newton was the one who built on the work of others to provide a truly comprehensive synthesis, so a shorthand for this new way of describing the world is to describe it as a Newtonian view. It's proved to be extraordinarily effective, underpinning much of the successes of the engineered environment humans have created over the last few centuries.

The motion of objects in their natural state

To arrive at a natural state of motion you need to do two kinds of stripping away:

  • Stripping away appearances, so the objects are not skinned in any way.
  • Stripping away contingent interactions of the object with its environment.

The idea of skinning is a common metaphor in the technical world, from buying new cases for your mobile phone to altering the appearance of your web browser or chat client. In neither case do you alter the functionality of the appliance or software, and that's what's so useful about the metaphor. In stripping away the inessential appearances of the objects we get to their functional essence. And since physics is concerned with building a model that mimics the functions of the lived-in world, rather than producing a photo-realistic depiction, that helps.

Isolating objects from their background influences is something at which you've already developed an expertise if you've worked through the SPT: Forces topic. Every time there is an interaction, you replace it with a force as you gradually extract the object from its environment. To find out how it'll move of its own accord you need only find an environment where there are no forces – none at all. So no slip, drag, grip, tension, compression, buoyancy, gravity, electric or magnetic forces. That'll be a very empty universe. But that's good.

If you can find out how it behaves in that very simple situation then you can add in complexity later. Physics is simple: don't start by developing a complete description. It's almost always better to add in the complication later. Work out what can be left out, not what must be put in. What are the bare necessities – the essence? As it turns out – not much.

Approximating natural motion in the everyday world

Where can we find things that approximate to this simplified natural state, to make it seem more approachable?

On ice is one place. Once you're moving in that situation, it's very hard to stop. And if you're not moving it's certainly hard to get going.

In a more urban environment, in-line skates, bikes and skateboards are engineered to move as freely as possible.

Indoors, air hockey tables are one place where the not-moving remain immobile, and the moving just keep on moving. Linear air tracks in the school laboratory provide a similar experience in one dimension. Specialist dynamics trolleys, also found in the school laboratory, provide another resource.

Not very close to home, experiences in free-fall, where gravity has its way, provide rather specialised laboratories where you can see objects falling alongside making their natural moves.

Deep space is even further from the everyday, yet may seem familiar through science fiction movies or writing.

All natural motions are somewhat like a lop-sided pendulum

In the imagined, simple world of the physicist, things just keep on going – all by themselves. Here's a rather famous thought-experiment that supports this line of thinking. It starts from something that you can do in the laboratory: making changes that show a trend. The final step is an extrapolation to something that's just not possible in the laboratory, but which does follow the line of reasoning established there. It's a kind of reasoning by extremes, and since extremes are often agreeably simple, that can be a very profitable way to get a feel for the behaviour of a system. So the pattern of reasoning, as well as the particular result, is of value here. As the original is attributed to Galileo, who was famous for his dialogues, the reasoning is presented in dialogue form.

Pensatrice: Here's a ball. I'll release it from this height, and let it roll. Now another from the same height, down the opposite slope. Which will go faster by the time it reaches the bottom?

Cercatore: Well, both sides are the same. They're symmetrical, so I suppose it'll go at the same speed. But maybe not quite – there are bound to be some differences.

Pensatrice: OK, so without any differences in the slope, then it'll go at the same speed. Now I'm going to make a change: make the two sides different. Now I'll make one slope steeper – but still release the balls from the same height.

Cercatore: I can see some things have changed, but nothing really important so I suppose they'll reach the same speed again. We could even think about the conservation of energy to support that conclusion.

Pensatrice: So, even if I make the two sides very different, you'll still agree that it goes to the same speed – even if it has to go farther. We could even run it down one side and back up the other: the speed it loses on the way up will be exactly equal to the speed it gains on the way down, whatever the slopes. Now what if I made one side very different – a very gentle slope – so that it had to travel a very long way as it climbed and lost the speed?

Cercatore: And still very smooth surfaces? I suppose still the same height.

Pensatrice: What about if I made it travel to the other side of the universe before it started the climb?

Cercatore: Well, I suppose it would keep on going at top speed – the speed it was going at the bottom of the ramp – right up until then.

Pensatrice: And if the universe were without end?

Cercatore: Then it'd keep going at the same speed for ever, I suppose.

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Newton's second law

Force
Forces and Motion

Newton's second law

Physics Narrative for 14-16

Force changes motion in a simple world

When the motion changes, that's when we need to provide an explanation.

Natural motion – occurring in a stripped-down environment – is simple: there is no acceleration. And physics seeks to understand the simple first, by refining the model or description of the world, then – and only then – adding complexity to the model to predict how the object will behave in a less simple environment. We'll return to this extreme simplicity in a while, but for now we suggest you consider the next simplest environment for the object: one where there is a single force acting on the object. (You'll recall that we isolated objects from their environments in the SPT: Forces topic, and re-described the interactions with the environment as a single force exerted by the environment on the object.)

This – perhaps the simplest interesting environment – leads to a very deep truth about the world: force, mass and acceleration are linked.

Resultant force sets acceleration: determining the change in motion. The greater the mass of the object, the more this acceleration is impeded. That's how the object's internal property and it's environment jointly determine the change in motion.

There is only one internal property that turns out to be important, and all the environmental factors can be reduced to the resultant force.

Here is how we suggest you express this great empirical discovery:

acceleration = forcemass

The change in motion is the result of two factors, and two factors only:

  • The mass of the object – an intrinsic property of the object.
  • The force acting on the object – a single quantity that represents the interaction of the environment with the object.

Acceleration really is the kingpin: it links all the work on forces with possible motions – being able to calculate acceleration is an end point for all that work on forces.

Acceleration, force and mass

This is a fantastic intellectual achievement: it's a great empirical discovery.

It's best written like this:

acceleration = forcemass

By choosing to write it in this way you emphasise that the dependent variable, the acceleration, is the result of other choices that you make. We like using mathematics to emphasise the physics.

However, the relationship presupposes that you understand the world to which it applies: that is, the Newtonian world. It happens that this world mimics the lived-in world particularly well, so long as you fully understand how to describe things in the lived-in world in terms that are Newtonian. You have to get good at translating between the two languages, or seeing the world in a different way. The SPT: Forces topic and the SPT: Motion topic were preparations for that. Here we're suggesting that you think a little more about exactly what's being done so that students can see that it's useful but not obvious or necessary.

Being able to predict how the motion of objects changes (that is, calculating the accelerations of the objects) is very useful.

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Isolating objects

Force
Forces and Motion

Isolating objects

Physics Narrative for 14-16

Looking for interactions; performing extractions

Putting on forces spectacles was used as a metaphor for seeing the world in a new way in the SPT: Forces topic. Now, in the SPT: Force and motion topic, this idea of seeing differently is developed and made more formal.

Identify the object – here a rock resting on the bottom of a pond. Then look for clues in the environment that there are significant interactions between the selected object and the environment. The ultimate aim is to extract the object from this messy environment of the lived-in world and to get it into a different, more sterile, but more predictable environment.

Here you are taking a step towards constructing a functional model, re-describing the object in a way that enables you to predict its changes in motion.

The extraction procedure

The very first step in extracting an object from its environment and building an idealised model of the situation is to isolate it. This isolation strips away the environment, leaving a bare object, characterised only by its mass, and with only the forces acting on it as a reminder of the environment it has left behind.

indicative interactionforce acting on object
stretchingtension
compressingcompression
displacing fluidbuoyancy
gripping a surface grip
slipping over a surfaceslip
moving in a fluiddrag
mass attractinggravity
charge attracting or repellingelectric
magnets attracting or repellingmagnetic

In this example, a child pushes a sledge over a rough surface.

The child's hands are compressed and provide a driving force  →  compression force.

The surface is rough and provides a retarding force  →  slip force.

The object being pulled is on the Earth and has mass  →  gravity force.

As you've not added the forces together yet, you can't say what the resultant force is. In particular, you can't say whether it's zero.

More interactions leads to more forces post-extraction

Here's a more complex situation – and more complex in a significant way. That is, the object is in a messier environment than our first example. It is hanging underwater. Yet the procedure to re-imagine it in a more natural environment is exactly the same.

And the process generates a physical description: it respects the phenomenal world. Every interaction of the object-to-be-isolated with its environment in the lived-in world results in a force acting on the pure mass that represents the object in it's natural world. This natural world, far removed from our everyday world, and yet capable of somehow explaining much of it, is the world of a mass and the forces acting on it. There are no other things in this very simple world. It's the world that it required the creative and powerful imagination of Newton to create.

Later, you'll see just how fruitful it is to imagine objects in this kind of world. Because it is their natural environment, objects can express their natural behaviour, and we can predict exactly how they'll move. Comparing such predictions with their behaviour in the lived-in world results in a match (or otherwise) between the model and our measurements of the phenomena. The quality of the match is a measure of the worth of the model. It might be very useful to know what will happen, and our ability to predict this with reasonable certainty may convince you that there is a case to be made that this very simple environment is actually a good description of how the world actually is.

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Identifying forces

Force
Forces and Motion

Identifying forces

Physics Narrative for 14-16

Three kinds of forces: identifying forces to replace interactions

As you isolate an object from its environment – in order to place the object in a completely natural environment – so you'll find the need to categorise the kinds of interactions with the environment, in order to identify the forces. These interactions can be of two kinds: interactions with the local environment and interactions with the non-local environment. In the object's new stripped-down environment, these are replaced by, respectively, contact and non-contact forces.

Forces (in the SPT: Forces topic) were first introduced as acting on objects to support those objects – so performing the same function as a human agent. This kind of agency on the part of the environment is as a result of interactions between the object and the environment that can be replaced by a force exerted by the environment. That force can support an object, and it's at right angles to the surface of the object – a normal force. These normal forces can also accelerate or retard objects, so perhaps, now that you know a little more about normals (perhaps from the SPT: Light topic), we can comfortably call this set normal forces.

The contact forces are of two further kinds: those that are normal to the surface of the object in contact with the environment and those that retard. In the SPT: Forces topic these were called forces that can support and frictional forces. Now we'd suggest using somewhat more apposite and precise collective terms: normal forces and retarding forces (these forces always act so as to reduce motion).

Then there are the three forces that act without contact: gravity, electrical and magnetic.

So there are three by three – that is, nine different kinds of forces that might appear. Here is a table of the forces and the interactions.

interaction of object with environmentreplaced by force acting on object
solid environment stretchedtension
solid environment compressedcompression
fluid environment displacedbuoyancy
not moving past a solid part of the environment grip
moving past a solid part of the environmentslip
moving through a fluid part of the environmentdrag
massive object in an environment with massgravity
charged object in a charged environmentelectric
magnetic object in a magnetic environmentmagnetic

And here are the nine forces, in their three groups:

normalretardingnon-contact
tensiongripgravity
compressionslipelectric
buoyancydragmagnetic

Transforming a description: from interaction to force

As you move from from the messy complex world to the simple natural world for un-skinned objects, focus on particular facets of the environment to check if these are such that the interactions require a force arrow to be added in the new world in order to reproduce the effect of the interaction.

In moving from the lived-in world to the object's natural environment, you really are re-imagining the world – you're creating a parallel universe where the rules are different.

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Quantifying forces

Force
Forces and Motion

Quantifying forces

Physics Narrative for 14-16

Just how long are the force arrows to be?

The SPT: Forces topic was focused on identifying whether there was a force or not – so on identifying the forces. Now we expect more, as the model of motion changed by forces becomes more sophisticated. We'd like to know just how big the force is: the magnitude of the force.

The model is more functional, and better mimics reality: the predictions are precise enough that we can see whether the model is well fitted to predict and describe particular phenomena. It's empirically checkable – but only if we can quantify. You need to be able to add forces and find the resultant (some detail on that in the SPT: Forces topic, more to follow here), and you can only do that if you know both the magnitude and the direction of the forces.

So you'll need to look more carefully at the interactions with the environment that the forces replace so that we can find the magnitude of the forces.

Normal forces

Warp forces arise because particles in solids in the very local environment of the object are distorted by the object. The more the distortion, the greater the force – up to a maximum amount, at which point the environment yields. You can feel this: stretch some thin wire, or squash a tennis ball in your hand. Warp the environment, and a force acts on the object that does the distorting.

The solid environments of objects can consist of structures or of materials. Both, when distorted (stretched or compressed), can exert forces on the object. How the forces vary with the distortion depends on the nature of the structure or of the material. However, for a good range of examples, and for comparatively small distortions, the force exerted increases linearly with the distortion. A coil spring is one example that you're likely to have met in the school laboratory. Up to a certain limit the stiffness is constant: the force you need to extend or compress it by a certain length is constant. Some solid materials are rather like the spring; others less so.

warp = k × distortion

Buoyancy forces arise whenever fluid is displaced. The more fluid you displace, the greater the force. We hope you spotted the similarity with springs. Just as compressing a spring from its rest position results in a force, so displacing a fluid from its rest position results in a force acting on the thing doing the displacing. For a boat to float, it has to make a hole in the water: the larger the hole, the greater the buoyancy force – until you run out of hull and the water pours over the gunwales. Try it, and think about what the maximum force provided by your chosen object can be – that is, just how hard do you have to push to make it sink?

buoyancy = k × volume displaced

Just as with springs, where the spring matters, so with displacing fluids, the fluid matters. So does the gravitational field strength (it's a complicated story). This leads to the more conventional, but less helpful,forcebuoyancy is equal to the weight of fluid displaced}

Retarding forces

Trying to drag an object with a rough surface over a similarly rough surface in the environment is hard work – in an everyday sense and perhaps also in a more careful sense (in that energy will be dissipated if the surfaces do move past each other).

Whether an object moves past its environment or not, there is a simple connection between the retarding force and the normal force between the surfaces. The greater the normal force, the greater the retarding force. Of course, the connection may not be linear – that is, the frictional force may not increase in a straightforward way as the normal force increases. That applies to both slip and grip forces.

forcegrip or slip = k × normal force

Drag forces (not illustrated here) also vary – both with the frontal (or cross-sectional) area of the object pushed through the fluid and with the shape of the object. Again, the variation in drag force with shape is not simple. Very careful measurements on the particular shapes to be pushed through the fluid give empirical constants which can then be used to predict the retarding forces at different speeds, or for different areas: making predictions for different shapes is even harder.

forcedrag = k × speed2 (Drag forces are very complicated because the interaction is very complicated – and the force may vary between speed2 and speed3, depending on the speed, and on the medium.)

The simpler environment of non-contact forces

Non-contact forces are introduced in a very rarefied and simple environment: for gravity forces, there are often only two masses, and nothing else; for electrical forces, only two charges.

The rules are precise, and simple.

For gravitational forces: forcegravity = G  ×  mass1  ×  mass2separation2 , where G is a rather small constant.

For electrical forces: forcegravity = k  ×  charge1  ×  charge2separation2 , where k is a rather large constant.

k is much larger than G, and it's this difference that leads to the statement that electrical forces are much stronger than gravitational forces.

For magnetic forces, the rules are a bit more complex (no-one has yet found an isolated magnetic pole, so you have to deal with dipoles), but the same kind of analytic relationships can be written, relating the force between the magnets to the strength of the magnets and the separation between them.

How forces vary

To begin to decide on magnitudes of force acting on the object within each of the three groups, think about how different interactions with the environment lead to different forces in the object's new natural environment. Here we'll summarise how the kinds of forces vary as you alter factors affecting the interaction with the lived-in environment.

Normal forces vary in size, depending on the interaction between the object and the environment. You can see how by varying the factors, using simple models of the materials with which the objects might interact. For warp forces, how the force varies with the distortion will vary with the ways in which the particles of the material are connected – measured, at a macroscopic level, by the stiffness of the material. Buoyancy forces depend on just how many particles are doing the bombarding, so on the ambient pressure of the fluid that is displaced. This in turn depends on the density of the fluid.

The second set of forces are retarding forces, and here there are even more simplifications. Simply put, there is a whole science of tribology, that studies the interactions of objects moving through their environment – it's not only drag that's complicated. The models here are good enough for many purposes but they are not the final word.

The third set of forces are non-contact forces, and these are in some ways simpler, because we characterise them in simple situations, then represent more complex situations as superpositions of these simple states. That's the reverse of the approach taken for the first two kinds of forces. So the relationships given here are precise, but only apply to rather special circumstances.

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Newton's first law

Force
Forces and Motion

Newton's first law

Physics Narrative for 14-16

Motion in a natural environment

We've talked a lot about putting the object into its natural environment – one where there is only the mass, and none of the messy interactions affecting real objects in the lived-in world. So having finally managed the extraction, how does the object move in the simplest case – that is, without any forces acting on it?

We suspect that you already know the answer, but here's an illuminating way of approaching the situation – all in very simple universes.

Start off by sitting next to a ball in a universe with just you and it (it's an imaginary universe, so you can switch gravity off as well, thus making it even simpler). It's stationary – that is, the separation between you and the object is unchanging. You can watch it for a long time, but symmetry arguments suggest that it'll sit there for just as long as you watch it: it has no reason to move off in any one direction. Stationary things remain stationary. That's a very simple rule.

Now Bob whizzes past, at constant velocity, so that the separation between you increases at 3 metre inverse second. He's also watching a stationary ball, whizzing along with him at 3 metre inverse second. What will Bob see his ball doing? It'll be stationary. There's no reason for it to do anything else.

But what will you see? Both Bob and the ball moving at constant velocity at 3 metre inverse second. So once the ball is moving, it'll keep moving at the same speed. What will Bob record, if he looks at you and your ball? Your ball's separation from him will be increasing at the same rate as his separation from you (3 metre inverse second). Otherwise you'd see it move. So if it is stationary for you, it is moving at constant speed for Bob.

Alice zips past at another constant speed. You can repeat the argument.

So Newton's first law is a consequence of two things only:

  • A simple rule that things that are stationary, remain stationary. (If they've no reason to start moving, they won't.)
  • The requirement that what you record will also be recorded by anyone who performs the same experiment at a constant speed is the principle of relativity.

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Motion with forces

Force
Forces and Motion

Motion with forces

Physics Narrative for 14-16

The next simplest situation

The simplest addition to the natural environment is to to have a single force acting. In practice we do this by adding a single force to the situation. This makes a big difference, because the symmetry is now broken. You cannot arbitrarily choose a point of view without paying attention to this force.

The situation where an object has a single force acting on it is common in physics because we make it so, by adding all the forces representing different interactions to give a single resultant force.

It may be that this resultant force is zero. If so, then you'll have recovered the natural environment – or at least a situation that is not discernibly different from the natural environment.

Adding complexity: more forces acting on the object

Strictly speaking, this is unnecessary, as we can sidestep this analysis as shown. First calculate the resultant force, and then analyse as before. This is strongly recommended, otherwise you find yourself drawn into conversations like this:

Befuddled: The acceleration that would be caused by this driving force, and the acceleration that would be caused by this retarding force.

Better to add the forces to a single resultant force and then find the acceleration for the object. An object has only one acceleration (although this might have components in different directions).

Teacher Tip: Too many hypotheticals are bad for the digestion.

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A resultant force

Force
Forces and Motion

A resultant force

Physics Narrative for 14-16

Combining multiple forces to give a single force

To get to a single resultant force you need to be able to add forces. The rules are the same as for adding any other vectors, and are particularly easy to apply if you are strict about keeping the action of all the forces along a single line. Just place one vector, then align the tail of the second on the tip of the first by sliding the second vector across, without rotating it. Repeat for as many vectors as there are. The resultant is calculated by drawing in a vector from the tail of the first to the tip of the last.

The process is identical if carried out with forces that are not along one line. The principle is the same – you can translate, but not rotate, the vectors (because it's both magnitude and direction that are important). And it doesn't matter in which order you add them.

A number of quantities are best described using vectors, so knowing how to perform this addition will be useful throughout this topic (for example, velocity and momentum are both most straightforwardly dealt with as vectors).

Here you're clearly dealing with a very abstract set of representations, carefully developed through several steps from the original messy physical interactions. It's a good idea to start a new diagram on which to carry out these kinds of manipulations, and not to try to do all of the reasoning with a near photo-realistic depiction of the original situation.

How forces are combined, and the combination used

Teacher Tip: We'd suggest always modelling a situation using resultant force as a stepping stone to finding the acceleration.

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Forces at right angles

Force
Forces and Motion

Forces at right angles

Physics Narrative for 14-16

Adding forces at right angles

Forces just add as vectors, tip to tail. So if they don't happen to all be along one line, then it really doesn't matter. Just so long as you place the vectors to be added tip to tail, the resultant can always be found by going from the tail of the first to the tip of the last.

There's a case which is both simple and special: the vectors to be added are at right angles to each other. We expect you'll realise why it's special if you remember working with vectors on gridded paper – how far you move along the paper doesn't affect how far you move up the paper.

You can rotate the graph paper without changing the vector, just as you can use different co-ordinate systems to describe where one village is with respect to another (for example, Ordnance Survey grid, magnetic bearing and distance, or latitude and longitude) without affecting how you have to travel to get from one to the other. The values of the ordered set of numbers representing the vector can change without changing the vector: just so long as you only translate, and don't rotate.

Vectors are ordered sets of numbers

Put more formally, components at right angles don't interact. All vectors are ordered sets of numbers, and only numbers in identical positions within the vectors interact.

Since force (a vector) sets acceleration (also a vector), which in turn sets how velocity (again a vector) accumulates, this non-interaction leads to general patterns of considerable interest. These patterns are most prominent when the vector addition is iterated over several cycles, and we'll come back to these later in episode 02, when predicting motions.

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All about mass

Mass
Forces and Motion

All about mass

Physics Narrative for 14-16

Mass impedes acceleration

Once you have identified the forces, set the mass, then you can find the acceleration: acceleration = forcemass.

We've expended quite some effort on identifying forces; we'll spend some more showing how acceleration contributes to our abilities to predict motion. But what of the third quantity: mass? It is in many ways the hardest to get to grips with, because we can do least with it. It's just there. An object has a mass, and there is nothing we can do about it. By way of comparison, we can vary the interactions and change the forces, so getting some handle on what kinds of things they are. We can even imagine ourselves having the same kind of agency as the force – that is, acting in its place.

So how can we measure the mass of an object? Let's go to a place that's both very significant and is closer to the natural environment for any objects with mass – that is, with minimal interactions.

Astronauts are very concerned with body mass. In a cooped-up environment it's all too easy to gain or lose mass, because the inputs and outputs will be far from normal. How much of me is there? is a key question for an astronaut. With issues such as loss of bone mass to consider, a tape measure won't do! NASA solved this problem by designing a chair on springs (the image here is copyright NASA). It's as close to a pure mass measurer as you're likely to find, because it simply measures resistance to acceleration. The greater the mass, the lower the frequency of oscillation of the chair. If a more massive astronaut gets in, the greater mass reduces the acceleration that results from the force exerted by the springs: the average speed is lower; the trip time for a single out and back trip on the chair is longer; the frequency is reduced. Mass simply impedes acceleration. For a given force acting on you for a short while, if you're not moving, a larger mass results in a smaller top speed. If you are moving – a larger mass results in a smaller change in speed.

The mass of an object is a measure of its reluctance to change motion – the extent to which it impedes acceleration.

Mass on Earth

Here on planet Earth we don't need oscillating balances to measure masses. And it's a good thing – think of the cleaning up after a hurried measuring out of 400 gram of flour. Instead we rely on the gravitational field, which pulls with a steady 9.8 newton inverse kilogram (10 newton inverse kilogram is usually precise enough). Imagine dropping a 500 gram bag of flour. Earth pulls on it with a force of 5 newton. So the acceleration can be calculated:

acceleration = 5 newton0.5 kilogram

But a cunning action on our part prevents this fall. We place some kitchen scales to provide a support force – of 5 newton. Since scales are commonly said to weigh things, we might call this the weight. Just because Earth's gravitational pull is so (nearly) constant on the surface, manufacturers of the scales can write numbers on the scale, at the rate of 100 gram for every 1 newton of support force provided.

On other planets the suppliers of kitchen scales would have to write numbers on the scale at different rates. You can find out more in the SPT: Earth in space topic and in the SPT: Forces topic.

One advantage of calling the support force the weight is that in situations where we don't provide any support force the sensation of weightlessness (whether partial or total) is naturally dealt with. You might experience this as a lift starts to descend (the floor doesn't support you as much as it once did) or, more dramatically, in amusement park rides where sometimes, and for short durations, the floor doesn't support you at all.

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Acceleration determined by force and mass

Forces and Motion

Acceleration determined by force and mass

Physics Narrative for 14-16

A simple, but profound truth

We've moved from a lived-in world, perhaps of the everyday, perhaps of the remote, to an environment which is natural for un-skinned objects. Here the only property the object retains is its mass: everything else is lost in the process of the extraction.

Acting on the mass are forces. We've identified the forces by looking at the environment. Particular facets of the lived-in world interact with the object in ways that result in these facets being said to exert forces on the object. These forces are all that's left of the environment that the object left behind.

The forces acting on a single object can be added to give a resultant. Now you've an even simpler world where there is only an object, which has a mass, and a single force acting on it. It's now simple enough that you can predict changes in its motion, using:

acceleration = forcemass

How this acceleration affects the velocity and the motions that result are the subject of the next episode.

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