The other half

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.