The Law of Acceleration

The acceleration of a body is equal to the force which acts on it divided by its mass.

Can this law be verified by experiment?

If so, we have to measure the three magnitudes mentioned in the enunciation: acceleration, force, and mass.

Acceleration may be measured, because I pass over the difficulty arising from the measurement of time.

But how are we to measure force and mass?

We do not even know what they are.

What is mass?

Newton replies:

Thomson and Tait answer:

What is force?

• Lagrange says that it is that which moves or tends to move a body.
• Kirchoff says that it is the product of the mass and the acceleration.

Then why not say that mass is the quotient of the force by the acceleration?

These difficulties are insurmountable.

When we say force is the cause of motion, we are talking metaphysics.

This definition is absolutely fruitless as it would lead to absolutely nothing.

For a definition to be of any use it must tell us how to measure force. That is quite sufficient, for it is by no means necessary to tell what force is in itself, nor whether it is the cause or the effect of motion.

We must therefore first define what is meant by the equality of two forces.

When are 2 forces equal?

We are told that it is when they give the same acceleration to the same mass, or when acting in opposite directions they are in equilibrium.

This definition is a sham.

A force applied to a body cannot be uncoupled and applied to another body as an engine is uncoupled from one train and coupled to another.

It is therefore impossible to say what acceleration such a force, applied to such a body, would give to another body if it were applied to it.

It is impossible to tell how two forces which are not acting in exactly opposite directions would behave if they were acting in opposite directions.

It is this definition which we try to materialise when we measure a force with a dynamometer or with a balance.

Two forces, F and F0 are acting vertically upwards. They are respectively applied to two bodies, C and C0.

I attach a body weighing P first to C and then to C0.

If there is equilibrium in both cases, I conclude that the two forces F and F0 are equal, for they are both equal to the weight of the body P.

But am I certain that the body P has kept its weight when I transferred it from the first body to the second?

Far from it. I am certain of the contrary.

Weight varies from one point to another. It is greater at the pole than at the equator.

The difference is very small, and we neglect it in practice. But a definition must have mathematical rigour.

This rigour does not exist.

What I say of weight would apply equally to the force of the spring of a dynamometer, which would vary according to temperature and many other circumstances.

We cannot even say that the weight of the body P is applied to the body C and keeps in equilibrium the force F.

What is applied to the body C is the ACTION of the body P on the body C.

On the other hand, the body P is acted on by its weight. By the reaction R of the body C on P the forces F and A are equal. This is because they are in equilibrium.

The forces A and R are equal by virtue of the principle of action and reaction.

Finally, the force R and the weight P are equal because they are in equilibrium.

From these 3 equalities, we deduce the equality of the weight P and the force F

Thus, we are compelled to bring into our definition of the equality of two forces the principle of the equality of action and reaction.

Hence, this principle can no longer be regarded as an experimental law but only as a definition.

To recognise the equality of two forces we then have 2 rules:

1. The equality of two forces in equilibrium
2. The equality of action and reaction

But these are not sufficient. We are compelled into a third rule: We admit that certain forces—the weight of a body, for instance—are constant in magnitude and direction.

But this third rule is an experimental law.

It is only approximately true. It is a bad definition.

We are therefore reduced to Kirchoff’s definition: force is the product of the mass and the acceleration.

This law of Newton in its turn ceases to be regarded as an experimental law, it is now only a definition.

But as a definition it is insufficient, for we do not know what mass is.

It enables us to calculate the ratio of two forces applied at different times to the same body.

But it tells us nothing about the ratio of 2 forces applied to 2 different bodies.

To fill up the gap we must have recourse to Newton’s third law, the equality of action and reaction. That is still regarded not as an experimental law but as a definition.

Two bodies, A and B, act on each other. The acceleration of A, multiplied by the mass of A, is equal to the action of B on A; in the same way the acceleration of B, multiplied by the mass of B is equal to the reaction of A on B.

As, by definition, the action and the reaction are equal, the masses of A and B arc respectively in the inverse ratio of their masses.

Thus, the ratio of the 2 masses is defined. It is for experiment to verify that the ratio is constant.

This would do very well if the 2 bodies were alone and could be abstracted from the action of the rest of the world.

But this is by no means the case.

The acceleration of A is not solely due to the action of B, but to that of a multitude of other bodies, C, D, . . . .

To apply the preceding rule, we must decompose the acceleration of A into many components, and find out which of these components is due to the action of B.

The decomposition would still be possible if we suppose that:

• the action of C on A is simply added to that of B on A
• the presence of the body C does not modify the action of B on A, or that the presence of B does not modify the action of C on A

If we admit that any 2 bodies attract each other, their mutual action is along their join, and is only dependent on their distance apart, if, in a word, we admit the hypothesis central

We know that to determine the masses of the heavenly bodies we adopt quite a different principle.

The law of gravitation teaches us that the attraction of two bodies is proportional to their masses; if r is their distance apart, m and m 0 their masses, k a constant, then their attraction will be kmm 0 /r 2.

What we are measuring is therefore not mass, the ratio of the force to the acceleration, but the attracting mass; not the inertia of the body, but its attracting power.

It is an indirect process, the use of which is not indispensable theoretically.

We might have said that the attraction is inversely proportional to the square of the distance, without being proportional to the product of the masses, that it is equal to f /r 2 but without having f = kmm 0.

If it were so, we should nevertheless, by observing the relative motion of the celestial bodies, be able to calculate the masses of these bodies.

But have we any right to admit the hypothesis of central forces?

And if we must abandon this hypothesis, the building which has been so laboriously erected must fall to the ground.

We have no longer any right to speak of the component of the acceleration of A which is due to the action of B.

We have no means of distinguishing it from that which is due to the action of C or of any other body.

The rule becomes inapplicable in the measurement of masses.

What then is left of the principle of the equality of action and reaction?

If we reject the hypothesis of central forces, this principle must go too.

The geometrical resultant of all the forces applied to the different bodies of a system abstracted from all external action will be zero.

In other words, the motion of the centre of gravity of this system will be uniform and in a straight line.

Here would seem to be a means of defining mass.

The position of the centre of gravity evidently depends on the values given to the masses. We must select these values so that the motion of the centre of gravity is uniform and rectilinear.

This will always be possible if Newton’s third law holds good, and it will be in general possible only in one way.

But no system exists which is abstracted from all external action. Every part of the universe is subject to the action of the other parts.

The law of the motion of the centre of gravity is only rigorously true when applied to the whole universe.

But then, to obtain the values of the masses we must find the motion of the centre of gravity of the universe.

The absurdity of this conclusion is obvious. The motion of the centre of gravity of the universe will forever be unknown to us.

Nothing, therefore, is left, and our efforts are fruitless.

There is no escape from the following definition, which is only a confession of failure: Masses are co-efficients which it is found convenient to introduce into calculations.

We could reconstruct our mechanics by giving to our masses different values.

The new mechanics would be in contradiction neither with experiment nor with the general principles of dynamics (the principle of inertia, proportionality of masses and accelerations, equality of action and reaction, uniform motion of the centre of gravity in a straight line, and areas).

But the equations of this mechanics would not be so simple.

Only the first terms would be less simple—i.e., those we already know through experiment; perhaps the small masses could be slightly altered without the complete equations gaining or losing in simplicity.

Hertz has inquired if the principles of mechanics are rigorously true.

“In the opinion of many physicists it seems inconceivable that experiment will ever alter the impregnable principles of mechanics; and yet, what is due to experiment may always be rectified by experiment.”

From what we have just seen these fears would appear to be groundless. The principles of dynamics appeared to us first as experimental truths, but we have been compelled to use them as definitions.

It is by definition that force is equal to the product of the mass and the acceleration; this is a principle which is henceforth beyond the reach of any future experiment.

Thus, action and reaction are equal and opposite.

But these unverifiable principles are absolutely devoid of any significance. They cannot be disproved by experiment, but we can learn from them nothing of any use to us; what then is the use of studying dynamics?

This somewhat rapid condemnation would be rather unfair. There is not in Nature any system perfectly isolated, perfectly abstracted from all external action; but there are systems which are nearly isolated.

If we observe such a system, we can study not only the relative motion of its different parts with respect to each other, but the motion of its centre of gravity with respect to the other parts of the universe.

We then find that the motion of its centre of gravity is nearly uniform and rectilinear in conformity with Newton’s Third Law.

This is an experimental fact, which cannot be invalidated by a more accurate experiment. What, in fact, would a more accurate experiment teach us?

It would teach us that the law is only approximately true, and we know that already.

Thus is explained how experiment may serve as a basis for the principles of mechanics, and yet will never invalidate them.