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let's look at the second concept mass.
the definition of mass in Principia says that -
if you compress or expand or twist and do whatever you wants to do with the object, the mass within that object will never change.
or either you could take it into the space, still the mass within that object will remain same.
if you take a one kilogram bar and expand it to the size of the earth
or compress it to a point mass, still the mass within that object remain the same.
that's what the definition says.
let's look at how we measure the mass and what exactly it represents within an object.
we normally measure the mass using the balance scale.
take balance scale and put one kilogram bar on one side and then the other object
(for) which we wants to measure the mass, put it on the other side..
and make the balance parallel to the surface of the earth,
then we say that both the objects are having the same amount of mass.
what exactly.. when you say - same amount of matter within this..
let's say this is cotton and this is iron bar, what exactly is same within these two objects.
is the size of the objects same? no;
is the number of particles within these two objects same? basic particles like protons, neutrons and electrons, are same? no.
is the mass of these basic particles, combined mass of these basic particles in iron and then in cotton the same? no.
so, what exactly.. when you say these two objects are same in mass, what it exactly represents?
it (doesn't) represent anything related to these objects.
what it is representing is -
when you balance a scale, here, the gravity to the earth..
here, the gravity to the earth for this cotton and then for this iron bar will be equal.
earth is pulling the iron bar with certain amount of gravity
and the same earth is pulling down the cotton with certain amount of gravity.
when these strengths.. both these strengths are equal in amount,
and then the balance will be parallel to the surface of the earth.
so, we defined the mass as something unique to an object, and which doesn't depend on anything else.. is actually..
completely dependent on the gravity of the earth to these objects.
that's what.. actually, when we say mass (or) equal mass, we are actually measuring the gravity of these objects to the earth.
that is what we are making equal.
and if, by some reason, if the gravity between these two objects
(always) wouldn't remain same, then mass is not (the) same.
so the definition of mass is totally dependent on the gravity of the earth on these two individual objects.