Quantifying forces

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,force_buoyancy 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.

force_{grip 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.

force_drag = 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: force_gravity = G  ×  mass₁  ×  mass₂separation² , where G is a rather small constant.

For electrical forces: force_gravity = 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.