Newton's laws: background and limitations

• Newton's laws written in different ways
• Do Newton's laws apply? Inertial vs non-inertial frames
• Do Newton's laws work for different inertial frames?
• F = ma: Does it define F? define m? Is it a physical law? Can it possibly be all three?
• What is F?
• What is m?
• What if m varies?
• Newton's third law

Newton's laws written in different ways

The italic is a translation of Newton's own words, the roman text is my paraphrasing.

Newton's first law: Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

If the total force acting on a body is zero, then it travels in a straight line at constant speed. ie its velocity is constant and its acceleration is zero.

Newton's second law: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

A nett force F accelerates a body in the direction of the force. To any body may be ascribed a scalar constant, its mass m, such that the acceleration a produced in two bodies by a given force is inversely proportional to their masses, ie, for a given F, a₂/a₁ = m₁/m₂.

Newton's third law: To every action there is always opposed an equal and opposite reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Forces always come in pairs, and the sum of the pair is zero.

If we write the total force acting a body as F, its mass as m and its acceleration as a, then we may write

Newton's first and second law combined:    F = ma,

where setting F = 0 gives the first law.

As we've said, forces come in pairs that add to zero. For example, the force that I exert on my chair equals the force it exerts on me, or the gravitational force the earth exerts on the moon equals that exerted by the moon on the earth. Or, in general, for bodies 1 and 2, and for any interaction between them,

Newton's third law:    F_1,2 = - F_2,1 ,     or     F_1,2 + F_2,1  =  0.

For a body of finite size, one might ask what acceleration is meant. In this case, a is the acceleration of its centre of mass. We deal with this in one the centre of mass module on Physclips . We also deal with a more general version of Newton's laws in the modules on momentum and energy.

In a naive or very general sense, it appears that Newton's combined first and second law, F = ma , does not always work. On a fairground ride such as a merry-go-round, or on a turning bus, unrestrained objects have unexplained horizontal accelerations. In other words, relative to the ride, the bus etc, we observe accelerations for which there are no forces. What is happening here? (This animation from our site on Relativity, frames of reference are discussed in more detail.)

In the situation depicted here, if we neglect air resistance, both observers would agree that the ball, in flight, is subject to no horizontal forces. In the animation at left, Zoe, the observer on the merry-go-round, sees the ball accelerate in the horizontal direction, in violation of F = ma. In contrast, Jasper, the observer on the ground, sees no horizontal acceleration of the ball. He sees that the merry-go-round and the ball thrower are both accelerating due to their circular motion. So Jasper sees Newton's laws obeyed (to an excellent approximation), and explains Zoe's observation as due to the acceleration of Zoe's frame of reference.

So, which is/are the frame/s of reference for which Newton's laws hold? Such frames are called inertial frames. Frames (such as Zoe's) in which Newton's laws do not hold are called non-inertial frames. Apart from conducting Newtonian experiments, is there a way whereby we may tell whether a frame is inertial or not?

In fact there is: one looks at the distant stars. But this raises the subject of relativity, and the work of Galileo, Newton, Mach and Einstein. To pursue these ideas further, go to:

• Our site on Relativity, which presents and discusses the different theories of relativity in a range of detail.
• The Foucault pendulum, a site the demonstrates and explains how the Earth itself is not an inertial frame.
• Newton's laws and gravity, which also introduces inertial frames and relative accelerations.

To a level of approximation that is good enough for many experiments, the surface of the earth is an inertial frame. So Newton's laws apply, for example, on the platform of a railway station. Do they also apply to someone on a smoothly moving train? (Here, "smoothly moving" means not accelerating, which includes not turning and not bumping up and down.)

This animation is also from our site on Relativity, where relative motion is discussed in greater detail. Briefly, however, let us derive the important result here, using the motion of the ball as an example.

Suppose that the train is moving at velocity v, as measured by a person at rest in the station. Let's give this observer the x,y coordinates shown. With respect to that coordinate system (at rest with respect to the station), let the position of the ball be r. With respect to a point at rest with respect to the train (the origin of the x',y' coordinates), let the position of the ball be r'. Note that we use the dash to indicate measurements with respect to the train (eg x', y', r'). Let the origin fixed in the train have position p, as measured from the station in the x,y frame.

Now in the station frame, the velocity any point fixed in the train is v. Integrating with respect to time, its position r is given by

r  =  r₀ + vt ,
where r₀, the constant of integration, is its position at t = 0. (To revise, see the introductory pages on Calculus and Vectors.) In particular, we can write
p  =  p₀ + vt.
for the time dependent position of the origin of the reference frame on the train, as measured from the station. Now let's consider the position r of the ball, measured in the station frame, and r', its position measured in the train frame. The derivative of r with respect to time is the velocity in that frame, which we'll call u (remembering that we have already used v for the velocity of the train). And the time derivative of u is a, the acceleration in that frame.

Now let's relate the measurements r and r' and their derivatives, remembering that r' is measured with respect to the x',y' frame.

We assumed that, with respect to the station, the train is not accelerating, so the time derivative of v is zero. This gives us the important result

a  =  a'.

In other words, the two observers would obtain the same value for the acceleration of the ball. So, if the station is an inertial frame, then a = F/m, where m is the mass of the ball and F is the total force acting on it.

If the two observers agree about the forces and agree about the mass, then they will have the same values of F and m, and we've shown that the both get the same value of acceleration: a = a'. So a frame of reference moving at uniform velocity with respect to an inertial frame is also an inertial frame.

We have deliberately omitted mentioning one further assumption made above, and it has to do with taking time derivatives. We have tacitly assumed that the clocks in the two frames both measure the same time. If you did not notice this assumption, you are in good company -- it was largely unnoticed between the times of Newton and Einstein. However, this is one of the assumptions that must be considered when studying relativity. Which we do in the site on relativity.

Let's start by observing that acceleration is measured in terms of length and time, which are defined independently of force and mass. The module on Constant Acceleration is a good place to start.

F = ma really does define force and define mass. Either one of these seems at first glance to make it a tautology - something that is true by definition. And yet it is, as the name implies, a scientific law and therefore, in principle, falsifiable. In the module on Newton's Laws, whence comes the image at right, we explain how it can be all three. Consider a system for generating a force reproducibly, such as a wire carrying an electric current in a uniform magnetic field. We could use this to accelerate a collection of different objects. From the ratios of their measured accelerations, we should have the ratios of their masses, because their product, ma, is the same in each case. We could, if we wished, choose one of these objects as our standard mass, in which case we now know the masses of each object. Thus we can use F = ma to define the inertial mass. Now we could vary the force -- in our example, we could vary the current, the magnetic field, the length of the wire, or we could use totally different systems such as a spring or an electrical interaction to supply the force. We could use each different system of force production to accelerate the same mass. Measuring the acceleration now gives us the ratios of any pairs of forces, and gives a value for each force if we have chosen a standard mass. So yes, F = ma really does define F and ma, and does so independently. Finally, we could in principle falsify this law. If we have measured a number say nm masses and a number nF forces, as just described, then we have done nm+nF − 1 measurements. All possible combinations of masses and force cand give us nm.nF measurements of acceleration. For all of the new combinations, we have already measured F and m, so we can predict a, and see how it compares with the measured a. Further, as indicated in the sketch at right, we could take two similar masses (ie two masses that are accelerated at the same rate under the same conditions) and see whether they were together accelerated, by the same force, at half the rate. Finally, in every case we expect a to be parallel to F: any exception violates Newton's laws. So yes, F = ma is testable and, in principle, falsifiable.

We can measure the size and direction of forces using F = ma. We can also compare two forces by 'balancing them'. For instance, when we weigh an object on a balance, we are often adjusting a magnetic force until it is equal to and opposite the weight of the object. (In practice, there are often levers as well.) In a very convincing and pragmatic way, such measurements tell us all most of what we know about forces. What we know from observation is how an object interacts with its environment. It may accelerate or not.

Sometimes we can feel forces: we can feel, almost directly, contact forces against our feet or against our bums on the chairs. In most cases, however, we don't feel or observe forces directly. Forces appear in our models, ie in our explanations. They are usually somewhat removed from direct observation. Because forces are often a little removed from observation, their description could be said, philosophically, to be somewhat arbitrary.

In this picture (from a background page in electricity and magnetism, we explain the observation in terms of an electrical interaction, but that explantion itself depends upon models for gravity and for the tension in strings.

Sometimes forces are modelled as action at a distance and sometimes via fields (the two are logically almost equivalent). As we show in the module about energy, it is possible to make models that use distributions of potential energy and that avoid the use of forces altogether. Again, in most cases, these different models are logically almost equivalent.

However, as a general rule, forces are less helpful as a model on the very small scale, in part because position, velocity and acceleration become poorly defined in that scale. See the uncertainty principle for more detail.

Mass can have two different meanings, philosophically at least. Inertial mass m_i is that quantity that resists acceleration. It is defined by m_i = F/a. In a very convincing and pragmatic way, measurements of a in different conditions tell us what we know about inertial mass.

Gravitational mass, m_g, on the other hand, could be defined by m_g = W/g, where W is the object's weight and g is the local gravitational field.

Galileo's famous experiments, David Scott's version shown below, and some much more sophisticated and precise experiments since then demonstrate that m_g and m_i are proportional, and therefore may be set equal by a choice of units.

However, there is no philosophical reason why Aristotle had to be wrong and Galileo right: as everywhere in science, only experiment and observation tell us that. Philosophically, m_g and m_i could be different*. That the proportionality is not a coincidence is Mach's Principle. That the two are indistinguishable -- the principle of equivalence -- is a starting point for Einstein's theory of General Theory of Relativity, which is Einstein's theory of gravitation.

* The Pioneer 10 and 11 space probes are now further from the sun than Neptune. The Pioneer anomaly is a tiny difference between the apparent acceleration of these deep space probes during the later stages of their trajectories and the acceleration calculated from the gravitational effects of planets, tidal effects, pressure of light against them and some other small effects. Some researchers think that this difference is due to calculation errors, others think that new physical principles are involved. Some have proposed that non-Newtonian mechanics may be required to explain the difference.

Yes, but what is m really?

Okay, so they are experimentally indistinguishable and perhaps identical. So what are they? What is this property of objects that resists acceleration? At the time of writing, this is not thoroughly understood. However, there are well developed theories of the origin of mass that will soon be put to experimental test.

Many physicists suspect that an as yet undiscovered field, the Higgs field, plays a role in determining the mass of some fundamental particles. It is the particles' interaction with the Higgs field that would determine, in some cases, their masses. Associated with the Higgs field is a particle, the Higgs boson, which can only be created by energies that are just beyond the reach of nuclear experiments performed to date. The Large Hadron Collider is being built by the Centre Européen pour la Recherche Nucléaire (CERN) in part to create* Higgs bosons and to measure some of their properties. We may know more about mass when and if "the Higgs" is discovered. However, even with out the Higgs, we can say something about the origin of mass.

If this theory turns out to be correct, then it will explain the mass of the electron but only a small fraction of the mass of the neutron and proton (which are a couple of thousand times more massive than the electron). In fact, most of the mass of neutrons and protons (and thus most of the mass of ordinary objects) is probably due to an effect that, while still exotic, is much better understood than the Higgs field. Many physicists suspect that most an ordinary mass m is just E/c². Saying this is just a rearrangement of Einstein's famous equation E = mc². Mass and energy are either interconvertible (via the very large conversion factor c²) or, for some theorists, the same thing. Quantum chromodynamics (QCD) analyses how the 'ordinary' nuclear particles such as neutrons and protons are made from quarks, which interact via the chromoelectric field carried by gluons. (There is a limited analogy here with quantum electrodynamics, which analyses how protons and electrons interact via the electric field carried by photons.)

According to QCD, nearly all of the mass of neutrons, protons and most nuclear particles is due to the energy of the different distributions of quarks and antiquarks which, because of quantum uncertainty, cannot ever cancel out each other's colour charge completely. (See E = mc², Energy in Newtonian mechanics and in relativity and the uncertainty principle for further background).

Exactly how this works is not known. The energies depend on how quarks and gluons are confined, and QCD itself does not give a scale for confinement -- currently, this is introduced empirically. Perhaps this scale comes from the Higgs field interaction too. Some suspect that it may have something to do with dark matter. This may be resolved fairly soon. Nevertheless, it seems very likely that most of your mass is in fact the energy stored in neutrons and protons in the different configurations of their quarks and gluons.

Of course that doesn't tell us what energy is. Physclips has a module on energy, but there we relate it to forces, so, while I think that the module does tell a lot about energy, you might justifiably accuse me of circular reasoning if I claimed that this settled the matter. Further, QCD doesn't tell us the mass of the electron and it doesn't account for quite all of the mass of neutrons and protons either. Some readers may be helping here: if you pay taxes in Europe, then you are helping fund the Large Hadron Collider and thus the search for the Higgs boson.

Nevertheless, we can put particle physics aside and say that mass (and, at least to some extent, its other self, energy) are defined by Newton's second law, in one of its forms. Hence my perseverence in showing that F = ma really can define mass and force, as well as being a testable law. And if you find this unsatisfying, it's worth noting that there will always remain departure points in physics, just as there are postulates or axioms as the departure points in mathematics, just as there are postulates or axioms as the departure points in mathematics.

* So, how do you create a Higgs boson? The LHC will accelerate two beams of either protons or lead nuclei in opposite directions in storage rings 27 km long. At speeds of about 99.99999% of c, they will collide. Much of their kinetic energy will be converted into particle-antiparticle pairs. Including sometimes, it is hoped, the elusive (and massive) Higgs. It's a big project. For instance, the LHC needs 100 tonnes of liquid helium just to keep the magnets cool.

The simple versions of this law were given above:
To every action there is always opposed an equal and opposite reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Forces always come in pairs, and the sum of the pair is zero.

In the module on momentum, we introduce a more general law that includes all three of Newton's laws: it is the law of conservation of momentum:

In no external forces act on a system, its total momentum is conserved.

This more general law is, so far as we know, completely true. There is even an odd sort of argument that suggests why it should be true: see Newton's third law and the anthropic principle. However, just as F = ma does not apply if m is not constant, we should point out that Newton's third law does not apply if some effects are omitted.

For instance, if we consider the magnetic interaction between two moving charges, in the general case the forces do not add to zero. In such cases, relativistic effects and the momentum of radiation caused by accelerating charges must be included.

Conclusion

Before leaving Newton's laws, we should repeat the points made above. First, Newton's laws (expressed in terms of forces rather than energies) are not readily applied on the very small scale: As one goes to extremely low energies on the atomic scale, position and acceleration are not well defined, because of the the uncertainty principle. We should add, too, that relativistic effects complicate the dynamics at high speeds and high energies (see Energy in Newtonian mechanics and in relativity for a discussion).

These caveats made, however, we note that, on the macroscopic scale and over the vast range of practical energies and forces, Newton's laws work spectacularly well and are the foundation of a large fraction of physics and engineering. And, if you are going to work on the scale of very small or very high energies, then you'll simply have to learn some quantum mechanics and some relativity -- which will be fun, too!