## What is mass ?

In our everyday life the concept of mass is pretty straightforward and seems so deceivingly simple. We don't make any distinction between weight and mass, they both mean the same thing to us. But in Physics things look a'bit different, we need to be more precise if we are going to discover better approximation of reality. What are those differences ? To show you the distinction I will start with two approaches we can use to measure the "massivity" of objects. Using a scale (image on the left) or with a spring (image on the right).

Why would I trouble you with those examples ? The reason is a subtle difference of the resulting measurement when we use those two methods. Here is it ... the scale will give you the same value no matter where you are, but the spring balance will give you different results on different places around the world. The reason for it is that spring balance measurement is dependent on the strength of the gravity at the particular place you are doing the measurment and as we know gravity althought approximately the same across the globe is not exactly the same, it varies from place to place. So what we are measuring is the weight of the object i.e. the force acting on the the object. On the other hand the normal scale will give us a consistent value at any place, because it uses different principle of measurement ... i.e. comparison of one object to another which is ratio and this is constant no matter what the exact gravitational attraction is. Because both the standard weights and the object are afffected in the same way.
So to summarize the scale measures the mass - "quantity of matter" in the object, the spring balance on the other hand measures the weight i.e. force of gravitation acting on the object of specific mass. For theoretical purposes mass (aka "inertial mass") is defined as the amount of force needed to produce given acceleratation i.e. the more massive the body is the greater the force have to be to produce change in its motion.
Pfuuu ... nice going but you wont go off that easy, there is more complications ahead. The suprising thing is that as we go from macroscopic to microscopic world things change a "little" ... it is observed that when radioactive bodies emit beta particles (electrons), the faster their speed is the harder is to bend their trajeory (by electomagnetic fields) i.e. the more massive they look for stationary observer. But we know that electrons apart from this effect of motion have the same mass, hmmm....yuck.
With the invention of the theory of special relativity this effect could now be explained i.e. the electrons have "proper mass" which is the same for all electrons and can be "measured" (i.e. inferred) by observer who moves alongside the particle. And the difference from "proper mass" is due to realtivistic effects which a stationary observer will observe.
Finally we can sleep easy, we got to the bottom of it .... not so fast :).
There is this curious fact that even after we do the mentioned corrections we still do not get the same "proper mass" for the same object. If the object receives energy (i.e. becomes "hotter") there is slight increace of mass and vice versa.
This is very appearent in one of the basic reaction happening inside the core of the Sun i.e. the combining of the 4 hydrogen into a one helium atom. The resulting helium atom is little bitty less massive than the combined hydrogen atoms !!! Where did the mass go ? As any fifth grade kid will tell you, it became energy : E = m*c^2, need I say more.

We still have to go one step further though, if we are to take General relativity into account ... where the things become even more blury.. ... ultimately our goal was to find a definition of mass that do not change under different curcumstances i.e is invariant, that is general trust in every natural phenomena exploaration, right. ... todo

### More theoretical approach to mass

We discussed the general meaning of mass from layman perspective, Let's now try a little bit more Physics approach. Those interpretation are based on different laws which were discoverd as we understood more the nature of motion.
• Intertial mass - this is the measure of intertia of an object i.e. the amount of resistance of an object to accelerate when force is applied. f.e. push,pull ). This definition comes from Newton second law of motion : m = F/a
• Gravitational weight - this is defined by law of Gravitation i.e. the measure with which object responds to a gravitational force (falls), F = G*(m_1*m_2)/r^2
All experiments point that graviataional weight and intertial mass are the same. That is why the acceleration due to gravity is independednt of the mass of the object itself. The bigger the mass of object the higher the gravitational attraction, but also the higher the resistence to acceleration.
General relativity equivalence principle states that "inertial" and "gravitational" force are the same i.e. acceleration and gravity are the same thing ( m*a = m*g => a = g ).
• Relativistic mass - Is the intertial mass as defined by and observer with respect to whom the object is at motion.
• Proper mass (rest mass, invariant mass) - Is the intertial mass as defined by and observer with respect to whom the object is at rest.
The last two definitions come from special realitivity and their relation is as follows :

m_r = m_p / sqrt(1-(v^2/c^2) or m_r ~~ m_p + (m_p*v^2)/(2*c^2)

where m_r is relativistic mass and m_p the proper mass, v is the velocity of the object and c the speed of light.