Diagrams that show the forces on one object

Push on something and change its motion. Your action has had an effect in the world: something (the object) changes its motion. There's only you and the other in this simple world. (In more complex environments there would be other forces to find from other interactions.)

In the SPT: Forces topic in this interaction between you and the other, you were encouraged to isolate the other from its environment, including you, so that you could begin to predict its motion. The agents, animate and inanimate (including you), interacting with that object, are replaced by forces – one for each kind of interaction with the environment. That process was refined in the earlier episodes of this topic and so up until now the guiding principle has been:

Teacher Tip: Deal with one object at a time. In any one diagram draw only forces that act on the (one) selected object.

Introducing interaction diagrams

In this episode we're gong to look at interacting objects, as a rather different strategic approach. As you can imagine, this means more complexity, with diagrams that represent forces on different objects all present in the same diagram. (It's not a fresh start, and you don't want to throw out all the hard-won skills that enable you to predict the motion of objects gained in episodes 01 and 02.) We suggest you keep track of the interactions between each object and its environment, matching them up with the forces. To keep things manageable, each facet of the environment is now another object. So now you'll have multiple interacting objects on one diagram, with all the interactions represented by forces.

There's a lot to keep track of, so we suggest very disciplined diagrams, with rather strict conventions to guide you in an analysis and to keep track of different elements (objects, forces and – vitally – which forces act on which objects). You might refer to such diagrams as interaction diagrams.

Teacher Tip: Take the steps necessary to represent more than one object on a diagram showing forces. Each force acts on only one object.

Keeping tabs on the links between force pairs

This is a sample diagram. We hope you'll see plenty of familiar elements, and a new one. More on that new element and its significance follows.

The new element you'll notice is the dashed lines linking pairs of forces. It happens that forces always turn up in pairs. A few moments thinking about how we've insisted on interactions being replaced by forces should make this seem reasonable. Each interaction of the object with the environment is replaced by a force.

Now swap your point of view, so that you're focused on the second object: the first object now becomes the environment. One ball can't squeeze another without itself being compressed, so there's also a force on the second object. The force-pairs are linked by dashed lines in alternating colours, linked to the coloured terminating squares. (These colours are simply to help you keep track of the pairings.)

In this simple case, there are only two balls, so only two objects, and only slightly more complex diagrams than you've been used to. Nevertheless, it's really important to emphasise that the pair of forces connected by the dashed line acts on two different objects. Each object has a resultant force, so the motion of both will change. The balls will spring apart.

Three interacting objects

A more complex example has three objects, or even four. In each case you should expect pairs of forces, representing the interaction. You're already good at identifying forces if you've worked through episode 01, and there's more still in the SPT: Forces topic.

Here's an old favourite, of a teapot sitting on a table. Even in the complexity represented here, we've assumed that the gravitational force between the table and the teapot is so small that it can be ignored (it's negligible). The diagram also shows extended objects for clarity – that means you can choose not to draw the forces acting through the centre for clarity. However, when you draw single objects, with all the forces acting on them, we do suggest that you draw the forces acting through the centre, as in the SPT: Forces topic.

Four interacting objects as a bicycle accelerates

Here's another example from the SPT: Forces topic, of a bicycle accelerating. The air is a significant object here – just ask any cyclist about the drag force at higher speeds – and so is not negligible. But we've still missed out the gravitational forces between the road and the cyclist, just like the previous example (the teapot and the table). The air is an interesting object. Not only have we decided not to draw in the gravity forces, but the buoyancy forces are missing as well: neither are likely to affect the outcome of the model, if our purpose is to model the motion. In fact, the air everywhere around the cyclist is approximately the same density, so there is probably as much mass of air in every direction, at least locally. The resultant gravitational force from the contributions of this evenly distributed mass would be zero in any case. As for buoyancy, it's again down to density – but on this occasion the low density of the air. Not much air is displaced by the cyclist, so the buoyancy force is small.

Dragging a truck with a moving engine

This is as complicated as we're going to tackle. But there are still significant simplifications – no buoyancy forces; no drag forces; only the most significant of the gravity forces. Always simplify with a purpose in mind.

You can see that these diagrams will get rather complicated, and that you ought to have a clear purpose in mind when constructing them. This one is most likely to be about the forces on the coupling, connecting the engine and truck. Or else why represent it? You could choose to make a simpler diagram, and not have the coupling inserted, but this one could easily be modified to describe other situations where, say, a husky pulls a sled, or a stationary engine drags a truck up an incline.

Dragging a truck with a moving engine, simplified

Here the coupling is missed out, producing a much simpler diagram as you've been able to reduce the number of objects by one. As each object interacts with all the others, the first step in starting a force description is to decide on the smallest number of objects that you can get away with, and still produce a functioning model. To adapt a phrase, physics is about doing the mostest with the leastest. That is, we aim for economy – the simplest characters, and the fewest players on the stage. Here you have only (point) masses and forces.

These diagrams might seem complicated, but they are one way to force yourself to consider the possibilities when trying to model a system of interacting objects. Perhaps now you see why, in the SPT: Forces topic, we started off by focusing on only a single object, and then treating every other object as a part of its environment.

Connecting up all the stuff in the universe

Newton's grand synthesis of the work of Kepler, Galileo and others who thought about the movements in the solar system was an empirical force law. That is, the universal law of gravitation predicted how the force would change as you varied some physical quantities (masses and separations).

force_gravity = G  ×  mass₁  ×  mass₂r²

But it was more than that. It asserted that the same interactions happen everywhere. The same kinds of forces interact everywhere in the universe.

This is why it's worth meeting again here. It links two objects, showing how the force on one varies as a result of the masses of both and the separation of the pair. The constant fixes the size of every such force in this universe.

The relationship was also historically significant as it provided a concrete instance of a force law that worked for both astronomical objects and terrestrial objects. Therefore it further undermined the ancient separation of the unchanging heavens from the changing terrestrial sphere.

The significance of Newton's synthesis

Newton made two giant strides as he created his system of understanding that became known as Newtonian mechanics.

The first was to create a systematic way of conceptualising the world in terms of forces and material points. Once that apparatus was in place, and the re-description completed, then he could put his newly invented calculus of fluxions to work. As we've suggested before, there's a sequence:

acceleration = forcemass

The acceleration accumulates velocity:

Δ velocity = acceleration ×  Δ (time interval)

The velocity accumulates position:

Δ position = velocity ×  Δ (time interval)

The accumulations are a modern way of writing statements linking rates of change: Newton's fluxions were his way of managing relationships between quantities that varied with time.

The second was to invent a successful, universally applicable and empirically tested force law.

From now on, quantified predictions would be universal. Newton's synthesis really broke down the centuries-old divide of the here (what happens all around us on the Earth) from the there (what happened in the heavens). The universalism implied made a stunning difference to how we were able to imagine the very space which we inhabit. In the older, Aristotelian, system, space was very inhomogeneous. In the new, Newtonian, system, it was not only homogeneous, but also characterless – space is simply a blank in which things happen. Contrast this with the older idea, where everything had a natural motion that restored the object to its proper place: so space contained a set of hidden gradients, along which different components (earth, air, fire and water) were inclined to move, according to their character. The Newtonian world is a much simpler place.

There are two very significant implications of this new way of describing the world, still as significant today as when newly minted:

• Earth is not so special a place: the same rules apply everywhere.
• You're connected physically, and demonstrably, to everything. And you can now figure out by how much.
• Given a current situation, you can predict with some precision how it will evolve.

Interactions modelled as pairs of forces

Pairs of forces replace each interaction. The interaction diagram, introduced earlier in this episode, is a thorough way to keep tabs on all the force pairs (remember: one per interaction).

You can identify the interactions by paying close attention to the physical situation, noticing certain changes, and using your developed skills as a force-spotter to identify certain forces acting on objects according to their environment. The difference now is that you're also paying attention to the other end of the interaction, so treating the environment as a second object.

Now you have a universe of a pair of interacting objects, rather than an object isolated from its environment (as you did in the SPT: Forces topic and in episode 01 of this topic).

Reduce multiple interactions to sets of pairs

Where there are multiple interacting objects, you simply consider them in pairs. One interaction per pair. Two forces per interaction.

Once you've finished noticing all the interactions, and recording all the pairs, then simply inspect each object. Identify all the forces acting on it and then add these to calculate the resultant force.

The resultant force, acting on that single object, changes the motion of the object.

Ramming the point home

Research shows you can't say it enough, so we'll risk that error here.

Teacher Tip: The two forces in the force-pair of Newton's third law each act on two different objects.

Or, if you really must:

Teacher Tip: In identifying force-pairs, action is a force acting on one object and reaction is a force acting on a different object.

Talking sense about force-pairs

Drawing an interaction diagram provides a way to generate a representation of all the interacting objects in a process, connected as interacting pairs. Each interaction results in a force acting on each of the pairs of objects.

Newton's third law provides a way to switch views between each member of a pair of linked objects. Suppose you've carefully worked out a force on one such object in an interacting pair. Newton's third law then functions as a kind of short-cut, to allow you to immediately write down the corresponding force on the other member of the pair.

This law supports the facility to switch your point of view, and to focus on the objects in turn, allowing one to form the environment for the object you're focusing one, or vice versa.

Fully fledged interaction diagrams are one way to analyse a situation. Isolating an object from its environment and identifying the forces acting on it was the approach adopted in the SPT: Forces topic. Now you can move around amongst the objects in the situation without having to start from scratch each time: Newton's third law enables you to carry some information over from one point of view to another.

Alternatively, you could approach Newton's Third Law as the principle that underlies the construction of interaction diagrams – a way of gluing together the interaction diagram, as all forces appear in pairs and the necessary connections between these force-pairs (equal in magnitude, but opposite in direction: →F_AB = -→F_BA) are given by the law.

A simple argument for momentum conservation

A billiard ball rolls in, and strikes another identical stationary ball. What happens next? Many billiard players will know. But why does this rather surprising event happen? We suggest you start with a simpler situation and then use your skills at changing your point of view to find a rather simple and satisfying explanation.

Start with two balls, both stationary. They'll both continue to be stationary: there's nothing to disturb the symmetry and no reason for them to do anything else. For the next step, consider two balls moving towards each other with identical velocity. Again you have a symmetric situation, with nothing to break the symmetry, so there are two extremal possibilities: the balls stick together (the balls don't bounce at all), or they bounce back with the same velocity (perfectly bouncy balls). There's a spectrum of possibility between these two, depending on the internal construction of the balls. Let's focus on the perfectly bouncy case and shift our point of view to moving alongside one ball.

That ball is co-moving with us and is therefore at rest. You'll record the other ball as moving in with twice the velocity that it had (as a consequence of your chosen point of view). What will you notice after the collision? The ball that was travelling in towards you now appears at rest, whilst your ball shoots off with the same initial velocity that the other ball had. This is exactly the billiard ball collision.

Taking it further

You can repeat the analysis for the non-bouncy balls, and find that, from your new point of view, they'll move off with half the velocity that the inbound ball had. These results are just the conservation of momentum: the momentum before an interaction is the same as the momentum afterwards, assuming that there are no external forces acting – so nothing to break the symmetry.

How momentum is exchanged in some collisions

Pairs of objects interact, by collision or explosion.

As there is only one interaction, the contact time is the same for both objects. That's the duration for which the force exerted by one acts on the other.

Newton's third law links the force-pair, one member of which acts on one object, and one member of which acts on the other object. The actions of the forces are opposite in direction but equal in magnitude.

So both the force and the duration for which the force acts are identical for both objects. The product of force and duration is the impulse, and is equal to the change in momentum.

The impulses are equal and opposite so the change in momentum of one object is equal in magnitude but opposite in direction to the change in momentum of the other. What is taken from one is necessarily added to the other and so momentum is conserved. But you already knew that because you had reached the same conclusion using a separate argument, using symmetry.

Note that there is no requirement that the distances over which the forces act is equal, so the energy shifted to or from kinetic stores may (an elastic collision), or may not (an inelastic collision), be conserved. Energy is conserved, but we may need to look in other stores in addition to kinetic stores.

Is Newton's third law always true?

If space and time are entangled, as suggested in the special theory of relativity, we'd expect there to be some differences, as the argument for invariance depends on Charlie, Bob and Alice recording events without taking account of the time for the information about that event to reach each of them. (In fact, the argument assumes the Galilean theory of relativity, where everyone agrees on the passage of time.)

If you change how you record the measurements of space and time then you'll also change how the forces appear.

In contact forces it seems reasonable to postulate that the forces are the same at all instants of the interaction, but for the kinds of interactions that are modelled by non-contact forces this is less clearly the case.

In fact, any non-contact force presents an interesting challenge – make a change at the emitter, and you won't know about it until the photons are absorbed by your hand (1 metre or 3 nanosecond away). These are difficult ideas and not for classrooms, but it's important that you're clear that the world as imagined by Newton is a very good mimic of some aspects of the lived-in world, but that it's not the final imagining.

So Newton's third law is the one of the three laws of motion that has been shown to be inadequate at very high speeds.

Fortunately, the idea of momentum and its conservation is so useful that it does get reused in the special theory of relativity. The idea gets developed and extended to be more general, and even more powerful.

In the imagined world you've been investigating there are things on which Alice, Bob and Charlie all agree (the acceleration and the force, and the mass), so long as they're all travelling at constant velocities.

Knowing what does not change is a very valuable guide to understanding a new situation (the whole of the SPT: Energy topic is an example of the value of that kind of thinking).

Physicists, as they explored the new world that was opened up by Einstein's rethinking of time, looked for things that did not change. They found two:

• Duration and displacements might be recorded as being different by Alice and Bob, but they both agree on a new, generalised measure: the spacetime interval.
• Energy and momentum might be recorded as being different by Alice and Bob, but they both agree on a new, generalised measure: the momenergy.

Both of these unchanging entities live in four-dimensions, so are necessarily vectors.

These arise because of two more fundamental truths:

• There is one universal speed, on which Alice, Bob and Charlie will always agree, and light travels at that speed.
• The physical laws are agreed on by Alice, Bob and Charlie.

One such physical law is: d →p d t = →a.

You might remember this from episode 03.

An austere, but productive, view of the world

The world of Newton is extremely simple, and therefore extremely powerful. We can be certain of what to expect, because there are so few interacting entities. The world of Newton consists only of point masses and of forces acting on those point masses.

Given the force and mass, you can calculate the acceleration. From the acceleration you can calculate the accumulation in velocity – that is, how much the velocity changes. From the velocity you can calculate the accumulations in displacement – that is, how much the displacement changes (so giving changes in position).

When a pair of masses interact, you have choices to make in modelling the interaction.

• You can choose to calculate accelerations from forces (perhaps using Newton's third law as a short cut).
• You can choose to look at the momentum before and after the collision (perhaps using the conservation of momentum as a short-cut, or as a basis for deciding what the possible outcomes are – and so what is impossible).

This pair of alternatives has been the focus of this topic.

• You could also choose to look at how the energy is shifted from store to store, looking before and after the collision (perhaps using the conservation of energy as a short-cut, or as a basis for deciding what the possible outcomes are – and so what is impossible).

In later studies, students may combine these different approaches. For now, it is enough to know that there are different ways of modelling the same situation and that you should choose one that allows you to make predictions simply.