Are there two competing definitions of "inertia"?

Question

The term inertia is often introduced by stating Newton's first law: An object stays at rest or moves with $\vec{v}=const.$, if the resultant force is zero. This feature of masses is called "inertia".

Another way of describing inertia is by using Newton's second law: Here, inertia is understood as a measure of resistance of a mass against changes of its velocity. And mass is understood as the quantitative representation of inertia.

Both approaches are usually taught side by side. I just wonder if those two definitions have a common root. To me, they seem to be independent of each other. And they do not mean the same thing, or do they?

For example: A book laying still on a table, would be a perfect example of inertia in terms of the 1st law. But you couldn't use this situation to explain "inertia" in terms of the 2nd law. For that you would need two books with different mass and try to accelerate them with the same force. The larger book would obviously accelerate more slowly due to its increased mass/inertia.

Answer 1

First, lets look at Newton's definition of inertia in his own words (as translated by Andrew Motte [1]):

The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest or of moving uniformly forward in a right line.

So, the original definition of inertia is more closely inline with the second definition you gave - inertia is the resistance of a mass to changes in its linear velocity. However, this is not inconstant with the first definition you gave. To see this, lets look at Newton's definition in light of his 2$^{\textrm{nd}}$ law of motion: \begin{equation} \mathbf{F} = \dot{\mathbf{P}}=m\dot{\mathbf{v}}+\dot{m}\mathbf{v}, \end{equation} where the over dot means first order, total time derivative (e.g. $\dot{A}=\frac{dA}{dt}$) and the bold symbols implies they are vectors. If the mass itself is not changing, then \begin{equation} \begin{split} \mathbf{F}&=m\dot{\mathbf{v}}. \end{split} \end{equation} This relationship shows us two things: (1) to change $\mathbf{v}$ (i.e. $\dot{\mathbf{v}}\ne 0$), a force must be applied, and (2) how big that force needs to be to achieve the a given change in the objects linear velocity directly depends on $m$ and $m$ alone. So, it is also safe to say that $m$ is a measure of an objects resistance to change its velocity.

Both statements about inertia are consistent, however, the definition of inertia is a measure of resistance to changes in an object linear velocity. The statement about inertia being the "feature of masses" that keeps them at rest or moving with constant velocity is a consequence of the definition of inertia and Newton's 2$^{\textrm{nd}}$ law.

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[1] I. Newton, N. W. Chittenden, D. Adee, A. Motte, and T. P. Hill. Newton's principia: The mathematical principles of natural philosophy. Geo. P. Putnam, 1850. https://archive.org/embed/newtonspmathema00newtrich

Answer 2

Both approaches are usually taught side by side. I just wonder if those two definitions have a common root. To me, they seem to be independent of each other. And they do not mean the same thing, or do they?

One can consider the law of conservation of momentum as a common root between the first and second law, and the third law as well. Conservation of momentum is a fundamental law of physics which states that the momentum of a system is constant if there are no net external forces acting on the system.

Newton's Second Law:

Newton's second law, in its most recognizable mathematical form is $$F_{net}=ma$$

But in its most general form is

$$F_{net}=\frac{dp}{dt}$$

Where $\frac{dp}{dt}$ is the rate of change in momentum of the system.

If there is no net force acting on the particle, then $\frac{dp}{dt}=0$, meaning $p$ is constant and momentum is conserved.

Newton's First Law:

Newton's First Law states that bodies at rest will remain at rest and a body in motion in a straight line at constant speed will remain so, as long as no net external force act upon the body. It can be considered a special case of the second law when no net forces act upon the system ($p$=constant). So the first law is a statement of conservation of momentum.

Newton's Third Law:

Newton's third law states that for every force there is an equal and opposite force. This can be derived from conservation of momentum. For example, in the collision of two objects consisting of an isolated system a change in momentum in one of two objects colliding is equal and opposite to the change in momentum in the other colliding object, for a total change in momentum of zero and momentum is conserved.

For example: A book laying still on a table, would be a perfect example of inertia in terms of the 1st law. But you couldn't use this situation to explain "inertia" in terms of the 2nd law.

The force of gravity downward on the book equals the upward reaction force of the table on the book, for a net force of zero and no acceleration (change in momentum) of the book per the second law.

Hope this helps.