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In R. Shankar’s “Fundamentals of Physics : Vol 1” while discussing Newton’s Second Law of Motion, Prof. Shankar raises the question : how do we know Newton is right? I quote from the book :

Take yourself back to the seventeenth century, when Newton was inventing these laws. You have an intuitive definition of force: when somebody pushes or pulls an object we say a force is acting on it. Suddenly, you are told there is a law $F = ma$. Are you better off in any way? Can you do anything with this law? What does it help you predict? Can you even tell if it’s true? Here’s a body that’s moving. Is Newton right? How are we going to check? Well, you want to measure the left-hand side and you want to measure the right-hand side. If they’re equal, you will say the law is working. What can you measure in this equation?

He then proceeds to explain how we can measure acceleration. I easily understood that, but trouble started when he came to mass. Essentially, Prof. Shankar defines a standard quantity to be 1 kg. Then he puts the defined 1 kg mass on a spring, stretches the spring to some displacement x, and measures the acceleration to be some a. He then takes the unknown mass m’, puts it again on the spring and stretches it to the same displacement x, and measures the resultant acceleration a’.

Now he reasons: equal displacements in a spring should produce forces of equal magnitude. And thus he equates

$$ 1.a = m’.a’$$

And then he gives $m’$ as $a/a’$. My question is, we want to measure the mass of a particle, and its acceleration, to find out whether mass x acceleration equals the force applied on said particle, i.e, we want to experimentally verify Newton’s Second Law.

But in measuring mass, we invoke the second law itself (where Prof. Shankar equates $1.a$ to $m’.a’$ ), so doesn’t this result in some sort of circular reasoning? I am also confused as to how this way of measuring mass is supposed to verify Newton’s Law, as in the process itself we are assuming the law is true.

And is there a way to measure mass, force and acceleration without explicitly assuming Newton’s Law?

Qmechanic
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NiceGuy
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  • Historically, we (crudely) determine the mass of something by balancing a scale of (x copies of) the object we want to measure against (y copies of) the prototype of the kilogram. This assumes that weight specifically is proportional to mass, but not the second law in general (someone correct me if this is wrong). – Marius Ladegård Meyer Mar 09 '22 at 14:21
  • Related: https://physics.stackexchange.com/q/70186/2451 and links therein. – Qmechanic Mar 09 '22 at 14:58

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Keep in mind that until the advent of special relativity, mass was a conserved quantity. It was equaled to weight as measured on scales against fixed weights for all the years of human civilizations, the basis of money in gold , after all, and exchange of money for produce.

The only variable to see that Newton's law holds in the proof above, is acceleration.

After the gravitational theory was established it was necessary to separate the weight variable from the mass of the object.

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anna v
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