Definition of mass

Question

In high school physics, I was taught that mass was just how much "stuff" or matter there is in an object. However, now that I am learning physics again in college, I am taught that mass of an object (inertial mass) is how resistant an object is to acceleration. Which one's the correct definition of what mass is?

Answer 1

This is a deep question. There are (at least) two definitions of mass:

• gravitational mass is how something is influenced by gravity, which is the $m$ in $F = Gm_1m_2/r^2$, and is more-or-less 'how much stuff there is';
• inertial mass is how resistant to acceleration something is, and it's the $m$ in $F = ma$.

If we call these two versions of mass $m_G$ and $m_I$, then we can conduct experiments which ask whether they are the same (or to be more precise, whether there is a constant ratio between them, which ratio can be absorbed into $G$).

It's easy to see how to set up such experiments in principle. Given the two equations above we can equate the forces to get

$$m_{I,1} a = \frac{Gm_{G,1}m_{G,2}}{r^2}$$

or

$$\frac{m_{I,1}}{m_{G,1}} = \frac{Gm_{G,2}}{a r^2}$$

Well we can measure all the quantities on the right hand side of this, and we expect the left-hand-side always to be $1$ if the two definitions of mass are equivalent. Even if we can't measure $G$ or $m_{G,2}$ very well, we can repeat the experiment with lots of objects on the left-hand-side and we should always get the same answer.

The weak equivalence principle (WEP) says that they are the same, and experiment has so far borne this out.

There are various stronger equivalence principles which matter in General Relativity in addition. I won't go into them here as I am always confused about exactly which is which. However it's fairly easy to see that if we want a theory of gravity which states that gravity is about the geometry of spacetime, then we really must have only one definition of mass, and so we need to claim rather strongly that all definitions of mass are equivalent.

Answer 2

Which one's the correct definition of what mass is?

In a way, both.

Mass is a fundamental measure of the amount of matter in an object, or as you say, a measure of the amount of "stuff" in an object. At the same time it is a numerical measure of its inertia. Because mass is a fundamental property, definitions of mass can tend to be circular, as a fundamental property is difficult to define in terms of something else.

Newton's first law, that objects will remain in their state of motion unless acted upon by a net force, is a statement about the inertia of objects. Newtons's second law says that an object of mass $m$ will experience an acceleration $a$ when subjected to a net force $F_\text{net}$, or $F_\text{net}=ma$. If the net force is zero, the acceleration is zero and the object will remain in its current state of motion. Based on that, the first law can be considered as a special case of the second law.

Hope this helps.

Answer 3

Making use of Newton's second law, the two definitions are indirectly equivalent, see:

The net force applied to an object is proportional to its acceleration.

The constant of proportionality is itself the mass, the matter measurement of an object. $$ F = ma $$ Therefore, for a given net force, mass and acceleration are inversely proportional, i.e., if you have a higher mass you will have less acceleration. That can be interpreted as resistance to acceleration.

Answer 4

Strictly speaking, neither is correct. First, as you will learn, the more rigorous and general formulation of Newton's second law is not $\sum \vec{F}=m\vec{a}$, but $\sum \vec{F}=\frac{\mathrm{d} \vec{p}}{\mathrm{d} t}$, where $\vec{p}$ is the momentum. This is because the former is valid only for non-relativistic constant mass particles, while the latter is valid for variable mass systems and relativistic particles. Also, the mass as inertia definition is more philosophical than physical since does not let us determine the mass of a particle or a system of particles, so it is bogus. Regarding the mass as amount of matter definition, it is a tautology, since when you look for the definition of matter it says that is "anything that has mass and volume". You can further research by reading the following paper: http://www.physicsland.com/physics10_files/mass.pdf