Exploring Defined Quantities in Physics: Momentum, Velocity, and Force

[pmb_phy]

I'm curious about something - What quantities in physics do you think are "defined" quantities, i.e. quantities which are defined in terms of other physical quantities and therefore are not required? e.g. momentum is defined as p = mv, velocity is defined in terms of space and time, i.e. v = dr/dt. Force is defined as F = dp/dt, etc.

Thanks for your opinions.

Pete

 

[jcsd]

I'd think of such a quantity as a quantity that has an SI derived unit as opposed to an SI base unit, of course that doesn't cover every one of these quantities, but it's a guide:

http://en.wikipedia.org/wiki/SI_derived_unit

 

[HallsofIvy]

There is nothing "inherent" about defining one quantity in terms of another. We tend to think of "distance" and "time" as more fundamental and then define "speed" as distance divided by time but that's just because we find it easier to measure distance and time.

From a purely natural point of view, it might make more sense to take "speed" as fundamental- because "c" is given by nature. similarly, the gravitational constant (the G in (GmM)/r²) appears to be a universal constant as well as Plank's constant (a measure of "action"). In some texts, units are take so that those "universal constants" are all equal to 1 and other units are defined in terms of them. For example, distance is defined in terms of speed and "action". They aren't very handy for everyday use: if I remember correctly, the unit of distance is the "radius" of an electron and the unit of time is the time it take light to cross the radius of an electron!

 

[jcsd]

Yes that's true, but on the other hand it tells which quantites that we assume in some way exist in our theories rather than defining them via mathematical relationships using other quantities.

 

[Garth]

Could not one define time as multiples of the Planck time and distance as c times this? The 'magic number' of the age and size of the observable universe is then OOM 10^60 Planck units.

The physics experiment that I found most significant was in high school, the measurement of coefficient of expansion of various liquids. We tested water, alcohol, paraffin wax and mercury. The graphs of volume against temperature all showed slight deviations from a straight line except mercury, which was exactly straight. We were asked why mercury should be unique in this respect and we guessed: it was a metal, it was very dense, it conducted heat well etc. etc. It was some time before the penny dropped, we were using mercury thermometers!

What quantities in physics do you think are "defined" quantities

It depends on how you measure them.
Garth

 

[pmb_phy]

I'd think of such a quantity as a quantity that has an SI derived unit as opposed to an SI base unit, of course that doesn't cover every one of these quantities, but it's a guide:

http://en.wikipedia.org/wiki/SI_derived_unit

In my opinion if that were true then there'd be no ambiguity in the definition whereas in relatity there is.

There is nothing "inherent" about defining one quantity in terms of another. We tend to think of "distance" and "time" as more fundamental and then define "speed" as distance divided by time but that's just because we find it easier to measure distance and time.

If that were true then consider this: Suppose you take speed and time as basic quantities and distance a derived quantity. How would you measure speed?

For example: Suppose I take the distance between two events and time interval between those two events as basic quantities, where the events are the locations of the same freely moving particle. Then the time interval and the distance are directly measurable. Speed is then calculated. But how would you do it with speed and time as the basic quanties? I.e. how would you measure speed and time without measuring a distance?

Pete

 

[Rob Woodside]

Suppose you take speed and time as basic quantities and distance a derived quantity. How would you measure speed?

I.e. how would you measure speed and time without measuring a distance?

Pete

Reply:
I'd use doppler shift and a clock.

 

[jcsd]

In my opinion if that were true then there'd be no ambiguity in the definition whereas in relatity there is.

The SI units are not specific to anyone heory, rather they reflect the assumptions of physics in general. In mist physical theories concepts such as 'time' and 'postion' are primitive.

 

[pmb_phy]

Reply:
I'd use doppler shift and a clock.

How would you measure the doppler shift of an electron? A car? The wind?

You're relying on quantum principles to make classical measurements. Are you saying that you can do this so long as you have quantum mechanics available to you?

Pete

 

[robphy]

Suppose you take speed and time as basic quantities and distance a derived quantity. How would you measure speed?

I.e. how would you measure speed and time without measuring a distance?

Pete

Reply:
I'd use doppler shift and a clock.

How would you measure the doppler shift of an electron? A car? The wind?

You're relying on quantum principles to make classical measurements. Are you saying that you can do this so long as you have quantum mechanics available to you?

Pete

Actually Rob Woodside's proposal is what is advocated by Bondi's k-calculus.
There's nothing especially "quantum" about it. It's classical special relativity emphasizing the Doppler Effect.

A similar measurement of [local] spacetime intervals without using distances is described in Geroch's General Relativity from A to B. The square-interval between two nearby events (one on your worldline and one possibly distant one) is the product of two time-intervals (measured by your clock) marking certain light transmission and reception events.

In both formulations, the "wristwatch" time and the constancy of the speed of light lead to derived concepts like "[apparent] elapsed time between two events", "[apparent] distance between two events", "[apparent] speed [of a timelike object]".

 

[pervect]

It's easier to look at the base units than derived units, IMO - there are less of them. Si base units are a good starting point as jcsd mentioned.

Wikipedia lists them as

meter, kilogram, second, ampere, kelvin, mole, candela

You can start paring base units down from there.

If you go down to Planck units, you can get rid of _all_ the base units. This may be going too far, I would suggest keeping the second as a base unit. A succinct way of saying this is that I'm a fan of geometric units (which shouldn't be a huge surprise to anyone who has read a lot of my posts).

This keeps Plancks's constant as an experimentally measured constant. The meter is already defined in terms of the second, so it's not really fundamental anymore. Defining mass in terms of seconds requires good values for c (which we have, it's already a defined quantity rather than a measured one in SI), and G (which we don't have which is why it isn't done already). Setting G=c=1 defines the geometric units for the most part (except for charge, which we get to next).

The charge of the electron would be sufficient to get rid of the ampere as a fundamental unit. There's another route to getting rid of the ampere/coulomb as a fundamental unit as well, that's to set the electric permitivity of free space to one - that's the approach used in geometric units. Either way, charge can be eliminated as a unit, and an ampere is just a columb/second. Kelvin is just another name for energy. Moles are just a convenient number. The candela is (I think), an artifact of the human visual system as opposed to being a fundamental physical unit. (I don't use candela's much, I'm fairly sure that's what they are there for). That takes care of all of the SI base units :-)

So we are left with one unit, the second, to represent (as Weyl says), the scale dependence of the universe - or, we can get rid of that by making hbar =1, and then not have any units at all.

The downside of not having units is the same as the downside of not having "types" in a programming language - it's a lot easier to make errors in an untyped programming language, it's a lot easier to make errors in a unitless physical system.