I'm starting first year in the Autumn and I'm doing some pre-university maths and physics. One of the questions I had a go at doing was this one, from Classical Mechanics by Kibble:

Discuss the possibility of using force rather than mass as the basic quantity, taking for example a standard weight (at a given latitude) as the unit of force. How should one then define and measure the mass of a body?

Answer: an object's gravitational mass (not the inertial mass) could be calculated by using $$ m_g = \frac{F}{\omega^2 r}. $$

For small objects like an apple, using this equation would be straightforward. However, if one was interested in the mass of a plane, for example, that would entail having to measure the plane's weight, which could be difficult depending on a variety of factors.

My method assumes the Earth's mass and Newton's gravitational constant are unknown. If they were known, you could use $F=GMmr^{-2}$.

What do people think of my answer?

I'm not too familiar with how university-style questions usually work, but there was no indication of how many marks this question was out of, but it was marked with a * in the textbook, meaning it is "somewhat harder".

  • $\begingroup$ This question is unclear. You write that the gravitational mass of some object can be calculated and you present a formula with a variable $\omega$. $\omega$ is commonly used to denote angular velocity. You need to specify what measurements need to be made in order to populate the formula. You need to specify the procedure you have in mind. $\endgroup$
    – Cleonis
    Jul 27 '21 at 12:46
  • $\begingroup$ @Cleonis Yes, I was using $\omega$ to denote angular velocity, $2 \pi / T$, where $T$ is the period of an object undergoing circular motion. $r$ would be the distance from the Earth's centre of mass to the centre of mass of the object in question. $F$ is the object's weight. As to the procedure, you could determine weight through the use of scales or a spring and $\omega$ is easy to calculate given the length of a day in seconds. The Earth's radius can be determined using a method similar to the one used by Eratosthenes. $\endgroup$ Jul 27 '21 at 19:53

If you have a spring scale with (k) defined in terms of a known standard force, then just hang the unknown mass on the scale to get the gravitational mass from F = mg (where g is the measured acceleration of a freely falling object). For the inertial mass, let it oscillate and solve from $ω^2$ = k/m. A standard force could be chosen arbitrarily, as were the original standards for length and mass, but these days we like to redefine the standards from reproducible measurements (like the wavelength of light from, or the number of atoms of a particular isotope). Where in nature do you find a measurable force that is always the same?

  • $\begingroup$ That makes sense. I did think about using springs although I wasn't sure. As for finding a measurable force that is constant, you could use the force between two charged particles held 1 metre apart, so a standard force could be defined as $$ F_{\text{standard}}= k q_e q_e. $$ $\endgroup$ Jul 27 '21 at 20:00
  • $\begingroup$ But then, how do you determine the magnitude of each of the charges? Actually if force were taken as a basic quantity, one could define both mass and charge in terms of force. $\endgroup$
    – R.W. Bird
    Jul 28 '21 at 13:31
  • $\begingroup$ Would you not be able to use an existing definition such as $Q=E/V$? $\endgroup$ Jul 28 '21 at 15:41

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