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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?

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    $\begingroup$ Well, mass can also be energy. $\endgroup$ – Leo Liu Jan 11 at 2:08
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    $\begingroup$ They amount to the same thing, surely? The more "stuff" there is in an object, the more resistant it is to acceleration. $\endgroup$ – Harry Johnston Jan 11 at 3:10
  • $\begingroup$ Before ever hearing of mass in the classroom, I heard it's resistance to acceleration. Then in the classroom I heard it's "amount of matter". Only the resistance-to-acceleration version enabled me to understand it. I wonder why anyone would feel satisfied with the amount-of-matter version. $\endgroup$ – Michael Hardy Jan 11 at 23:06
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    $\begingroup$ @LeoLiu What's true is that $E^2 = m^2 + p^2$. Photons have no mass, so $E^2 = p^2$, i.e. $|E| = |p|$. A photon has both momentum and energy all while being massless. What you're referring to as "the effects of having mass" are really the effects of having momentum and energy, which massive and massless particles can both possess. There are observable differences between massive and massless particles, so it's important not to conflate having mass and having momentum and energy. $\endgroup$ – Charles Hudgins Jan 13 at 1:48
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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.

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  • $\begingroup$ For a second I thought you were going to say that inertia mass could be different from gravitational mass due to the difference between rest mass and relativistic. $\endgroup$ – Michael Jan 11 at 1:24
  • $\begingroup$ For a second I thought you meant "we use cams to measure ...". $\endgroup$ – user21820 Jan 12 at 2:30
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    $\begingroup$ This does not answer the OP's question. You're explaining the difference between inertial and gravitational mass, but the question is only about the concept of inertial mass and how it's related to "how much stuff" there is in an object. You answered the question you wish the OP had asked. $\endgroup$ – Wood Jan 12 at 22:37
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    $\begingroup$ @tfb: No sorry needed, I was just amused. =) $\endgroup$ – user21820 Jan 13 at 1:47
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    $\begingroup$ @Wood: my take is that gravitational mass is how much stuff there is: that's why I answered it the way I did. $\endgroup$ – tfb Jan 13 at 10:06
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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.

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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.

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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

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  • $\begingroup$ The formula $\sum \vec{F}=\frac{\mathrm{d} \vec{p}}{\mathrm{d} t}$ does not respect Galilean Invariance. For variable mass systems check this Wikipedia. $\endgroup$ – Antonios Sarikas Jul 16 at 12:13

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