# Operational definitions in Newtonian Physics

Operational definitions are constructed from the observations we make in nature. Experiments show us that two objects $m_1$ and $m_2$ in a local inertial frame, isolated from the rest of the universe, interact in a way such that the ratio of their accelerations $a_1$ and $a_2$ is a constant. Also from observations, an object's acceleration depends on a property (which is observed to be an inherent property) termed mass. The property Mass is then defined as, $$m_2 \equiv m_1 \frac{a_1}{a_2}$$

We could define some other property, call it property $Z$, such that, $Z_2 \equiv Z_1 (\frac{a_1}{a_2})^2$. This property is not useful when compared with mass.

• How is it evident that the definition of Mass and not property $Z$ (or any other property) is useful?

• What is the need of defining mass when it is known that mass is the amount of matter in a substance?

The operational definition of Force sidesteps the question, "What is a Force?"

• How does the definition of Force avoid the question?

• Are there any operational definitions of physical quantities that tell us what physical quantities are or do these definitions only exist because they help us do physics?

• Related: physics.stackexchange.com/q/70186/2451 and links therein. – Qmechanic Nov 22 '17 at 15:01
• Your quotes are from Kleppner and Kolenkow. Please don't cut and paste random stuff on the internet without attribution. It's rude. – Ben Crowell Jan 28 at 17:09
• @BenCrowell I agree re: attribution, but perhaps it would be less antagonizing and more conducive to maintaining a convivial internet learning community to call people out in a slightly less harsh way? I share the desire to promote certain values, and I understand that when one feels someone has done something wrong, one might feel justified in himself being blunt or even a bit harsh, but I wonder if it's less productive. Just some food for thought. – joshphysics Feb 27 at 22:20

How is it evident that the definition of Mass and not property Z (or any other property) is useful?

Mass is additive in Newtonian mechanics (and approximately additive in the Newtonian limit of relativity). Z is not additive.

What is the need of defining mass when it is known that mass is the amount of matter in a substance?

"Amount of matter" is not a useful definition. It doesn't define a number. It's also incorrect, since we know that a big chunk of the mass of ordinary matter comes from the kinetic energies of the quarks, via $$E=mc^2$$.

'The operational definition of Force sidesteps the question, "What is a Force?"' How does the definition of Force avoid the question?

It provides an operational definition, which allows us to deal with forces without even needing to supply a conceptual or dictionary definition.

Are there any operational definitions of physical quantities that tell us what physical quantities are or do these definitions only exist because they help us do physics?

Operational definitions can be interpreted in order to gain conceptual insight, but they are not themselves dictionary/conceptual definitions. Their advantage over conceptual definitions is precision.

For the first question, because we have uses for the property we term mass. If there is a usefulness for property Z, it hasn't been discovered yet, or at least I am unaware of it. There is an infinite class of properties that can be derived that way that are all semi-related to mass, by changing the function.

For your second question, it is a useful relationship; it can be used to calculate masses given accelerations, which are easier to measure for certain classes of objects (such as orbiting objects).