In classical mechanics, there seems to be a need to distinguish between inertial and gravitational mass. Some texts show how the concept of mass may be defined with some mathematical rigor. There is also an equivalence principle.

I would like to know if the concept of mass in Newton's "Principia" is not expressed in a much simpler manner.

edit:Newton on Mass and Force. By Ed Dellian, Bogenstr. 5, D-14169 Berlin.

What does Newton say about “mass”? Let us read the “Definitio 1” which opens the Prin- cipia: “Quantitas materiae est mensura eiusdem orta ex illius densitate et magnitudine conjunctim.” That is: The quantity of matter is that measure of it which arises from its density and volume conjointly.

So Newton's mass is simply "quantity of matter" obtained through density x volume (density => weight concept). Had we defined our unit mass with a fixed volume of water, then any object's mass could easily be measured using a scale against a certain volume of water - the volume of water would give the mass. This manner of definition could be taken as a fundamental definition of mass; it may seem to be just gravitational mass.

With a definition of mass done, then force is defined as simply: mass x acceleration. We may not need any discussion of "inertial mass".

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    $\begingroup$ The very first paragraph of Principia is about inertial and gravitational mass, i.e., the equivalence principle... $\endgroup$ – AccidentalFourierTransform Mar 4 '16 at 9:53
  • $\begingroup$ Most texts on classical mechanics that I have seen do not "express that need" and basically use the equivalence principle (however implicitly) by simply ignoring the theoretical difference after pointing out the problem. You can't define anything in physics with "mathematical rigor". Physical quantities have to have one (or multiple) experimental implementations. That's what Newton's second law is: it "defines" how to measure (inertial) mass and force at the same time. $\endgroup$ – CuriousOne Mar 4 '16 at 9:53
  • $\begingroup$ There is no "equivalence principle" in Principia. Newton defines mass = "quantity of matter" = density x volumn. Concept of mass, like time, is only an abstract. There is but only one way to fill "m" with a physical value. From Newton (mass = lb-weight) till the present SI unit kilogram, it is "quantity of matter" through use of the weighing balance. $\endgroup$ – thinking Mar 6 '16 at 8:45

I am a freshman in physics so i had the same confusion a some time ago. What i understood is the following.

There are four interactions in nature that are fundamental. In physics we don't care so much for pointing out the source of the interaction as to describe the very interaction itself. Therefore we describe them by giving to space specific properties with wich we can detrmine the forces that are acted upon particles/bodies that enter it. That space of course is called a force field. In order for a particle/body to interact with this force field, it has to have a specific property according to the kind of the interaction.

For example: In order to interact with an electromagnetic field, the particle has to have a property that we call charge.

Accordingly, a particle that enters a gravitational field has to have a specific property that we call gravitational mass.

Now as for the inertial mass, there is a phenomenon in nature that is called inertia and that is the tendency of bodies/particles to resist to any change of their kinetic state.

So inertial mass is the a numerical value of the enertia that a body has, meaning that the more inertial mass a body has the "harder" it changes its kinetic state and vice versa. Of course with the word harder we mean that the force that has to act upon it has to be larger. This definition occurs from the second axiom of Newton.

$$\vec{F}=\frac{\mathrm d\vec{p}}{\mathrm dt}$$

Let's examine an isolated system of tow particles.

According to the third axiom of Newton, the forces that are acted upon them are opposite.


So the net force of the system, since it is isolated it is zero.

$$ \sum\vec{F}=0 \quad\Longrightarrow\quad \frac{\mathrm d\vec{p}}{\mathrm dt}=0 $$

Therefore the momentum of the system is conserved.

$$ m_1 \vec{v}_\mathrm{initial} + m_2 \vec{u}_\mathrm{initial}= m_1 \vec{v}_\mathrm{final} + m_2 \vec{u}_\mathrm{final} $$

$$ m_1 \Delta \vec{v} = -m_2\Delta \vec{u} \quad\Longrightarrow\quad \frac{m_2}{m_1}= \frac{\left|\Delta \vec{v}\right|}{\left|\Delta \vec{u}\right|} $$

If we define $m_1$ as standard inertial mass then

$$ m_2= \frac{\left|\Delta \vec{v}\right|}{\left|\Delta \vec{u}\right|} $$

So $m_2$ is inversly proporional to the change of kinetic state of the body. Which means that the greater the inertial mass is, the "harder" the velocity changes.

As all experiments show till today, inertial and gravitational mass have the same value.

  • $\begingroup$ .Goggle: Newton on Mass and Force Ed Dellian pdf. Ed Dellian's discuss Newton's original concepts - helps me greatly. Newton's definition of mass is exactly = quantity of matter = density x volume, (density = weight force per vol) >Max Jammer rightly states on p. 91 (COM 2000) that Henri Poincaré in 1908 was one of the first “to use explicitly a term to denote gravitational mass”. Defining inertial mass m = F/a is not valid unless force has an unambiguous definition in itself; but there are four force type! $\endgroup$ – thinking Mar 4 '16 at 15:21
  • $\begingroup$ I did not define inertial mass as m=F/a. I simplly said that the defenition occurs from the second axiom. The definition that i gave was that the inertial mass is the numerical value of inertia the a body has and gravitaional is the property that a body has to have in order to interact in a gravitational field. $\endgroup$ – George Smyridis Mar 4 '16 at 15:35
  • $\begingroup$ But a physical definition of a quantity has to be have a means to quantitatively measure its value. $\endgroup$ – thinking Mar 4 '16 at 15:48
  • $\begingroup$ I made some edits. Check them and make your objections. $\endgroup$ – George Smyridis Mar 4 '16 at 16:05
  • $\begingroup$ please note that of the four fources, only the long range ones really fit the pattern. The strong and the weak are short range ones and need quantum field theory for any mathematical formulation. $\endgroup$ – anna v Mar 4 '16 at 17:19

General relativity showed that there is no such thing as gravitational mass, only inertial mass.

Force is something that can be measured with a spring or transducer. Inertial mass can then be measured as the (constant) ratio of the net force on an object to its acceleration.


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