# Defining inertial mass scale

I am following the approach to define mass scale as given in the book 'Newtonian Mechanics' by A.P.French (page 171).

Take different objects labeled 1, 2 and 3, pull them one at a time with the same force and measure individual accelerations: $a_1,a_2,a_3$. Proportionality factors $m_1,m_2,m_3$ are unknown yet. Then to define an inertial mass scale, we use these experimental results by putting $F= m_1a_1=m_2a_2=m_3a_3$. From this, we get: $\dfrac{m_2}{m_1}=\dfrac{a_1}{a_2}, \dfrac{m_3}{m_1}=\dfrac{a_1}{a_3}$.

Now, choose $m_1$ arbitrary as 1 kg. A quantitative measure of other objects can be obtained as multiple of this standard object.

But to Prove the fact that the inertial mass obtained this way is truly a property of an object itself and independent of force used, following argument is used (book uses argument in words only, I have put symbols):

Repeating the above procedure in a similar way for a different force $F^{'}$, we get: $F^{'}= m^{'}_1a{'}_1=m{'}_2a{'}_2=m{'}_3a{'}_3$ and henceforth, the ratios: $\dfrac{m{'}_2}{m{'}_1}=\dfrac{a{'}_1}{a{'}_2}, \dfrac{m{'}_3}{m{'}_1}=\dfrac{a{'}_1}{a{'}_3}$. We find experimentally that $\dfrac{m{'}_2}{m{'}_1} = \dfrac{m_2}{m_1}, \dfrac{m{'}_3}{m{'}_1} = \dfrac{m_3}{m_1}$.

Now, from this, how to conclude that $m^{'}_1 = m_1$? (As only then, we can conclude the value of the mass does not change with the force used in the experiment).

Edit 1: After looking at the answer below, I have now doubt over my interpretation of the concluding remark in the book, quoting the remark: "The fact that these experimentally determined mass ratios are independent of the magnitude of the force establishes the inertial mass as a characteristic property of the object."

What would be the right interpretation of it?

## 1 Answer

Now, from this, how to conclude that $m^{'}_1 = m_1$? (As only then, we can conclude the value of the mass does not change with the force used in the experiment).

You applied a certain procedure using force $F$ to assign the three masses, and then you repeated the procedure using force $F'$. The procedure involves arbitrarily assigning a mass of 1 kg to $m_1$, so as written, this does not prove that $m_1=m_1'$. You've simply chosen $m_1$ as your 1 kg standard, and then made the same choice again.

To establish that the mass scale doesn't vary as the force is varied, you need to do something else. For example, if you assume that forces are additive, then you have a way of doubling a force by applying two forces of the same size. You can then test whether doubling the force really does produce twice the acceleration.

• Why the downvote? – Ben Crowell Jan 15 '18 at 20:49
• I had edited my question after reading your answer. I did not downvote. Hopefully someone will justify. – user127249 Jan 25 '18 at 19:40