Meaning of Multiplication and Division in Physics

[mremadahmed]

Mathematics is all around sciences.It is also considered a language.In language everything should have meaning.

I know addition and subtraction meaning.Same things[quantities ,etc.] can add or subtract . It cannot be done on different things.

What is the meaning of multiplication and division in sciences[Physics,.. etc].I know what it means in real life.Multiplication is higher form of addition.Division is higher form of subtraction.

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

Why any quantities can multiply or divide,contrary to adding and subtracting.

Can anyone answer this question,or provide a book to read about this subject.

 

[Drakkith]

I know addition and subtraction meaning.Same things[quantities ,etc.] can add or subtract . It cannot be done on different things.

What do you mean "different things"?

What is the meaning of multiplication and division in sciences[Physics,.. etc].I know what it means in real life.Multiplication is higher form of addition.Division is higher form of subtraction.

It is the same as it is in math.

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

I could easily add or subtract or add M and A. I just wouldn't be using the F=ma equation. That is only useful in certain contexts. The letters themselves are just placeholders for numbers, and the equation itself is only useful for real life applications when the numbers you use come from the force applied, the mass of the object, and the acceleration. Mathematically F=ma is identical to X=cd if you plug the same numbers in.

On the other hand, the reason WHY m times a is equal to the force applied in real life is a mystery. That's just the way it works. In some other universe it may be the case that F=2ma, or F=ma². But not in our own universe.

Why any quantities can multiply or divide,contrary to adding and subtracting.

Because otherwise it wouldn't be useful to us. I can take the numbers I plug into F=ma and do a million different things with them and it would all be mathematically correct. They just wouldn't mean anything in real life unless I use them in the appropriate equation.

 

[dipole]

You've completely misunderstood his question, Drakkith. He's not asking why any specific physical law has a specific form, he's asking what is the physical meaning of multiplying or dividing physical quantities.

What do you mean "different things"?

He means different physical quantities which have different units.

It is the same as it is in math.

No I don't think so. In math, the statement ab = a+a+a+... b number of times is valid and makes perfect sense when dealing with pure numbers, but in physics the statement ma = m+m+m+m... is nonsense, because a is not a number, but a physical quantity with a magnitude which is represented by a number. In physics when we write down a multiplication, we're really talking about a different kind of multiplication than the kind we use in normal arithmetic. However, we represent certain aspects of the world by real numbers, to which we can apply the normal rules of arithmetic.

I could easily add or subtract or add M and A. I just wouldn't be using the F=ma equation. That is only useful in certain contexts. The letters themselves are just placeholders for numbers...

No, you can't. As I said above it makes no sense to write down an equation like F = m + a because m and a have different units.

So, as far as the OP's original question goes... I think it really depends on the situation. What it means in a physical way to multiply something like mass and acceleration may be different than what it means to multiply something like force and distance. However, just because we can represent many quantities by numbers, especially in one dimension, doesn't mean they are numbers.

I have no good general response to the question, and I think it's a rather deep one and one I've thought about myself before. I hope some other more enlightened persons can weigh in.

 

[Drakkith]

You've completely misunderstood his question, Drakkith. He's not asking why any specific physical law has a specific form, he's asking what is the physical meaning of multiplying or dividing physical quantities.

Huh. Well then ignore everything I said.

 

[jedishrfu]

we create new physical quantities from multiplication and division operations:

length * width = area with units of measure of meters^2

or

distance/time = average velocity with units of measure of meters/second

or

F=ma --> we've defined a new quantity called force with units of measure Newtons (kg*meters/second^2)

We add and subtract the same kinds of quantities but create new ones when we multiply or divide.

Differentiating or integrating operations are also used to create new quantities that aid us in understanding a physical system.

The key is that the equations must be internally consistent in units of measure.

 

[A.T.]

What it means in a physical way to multiply something like mass and acceleration may be different than what it means to multiply something like force and distance.

And multiplying force and distance itself can have two different meanings (work vs torque).

 

[technician]

Huh. Well then ignore everything I said.

What do you mean by this ?

 

[DiracPool]

we create new physical quantities from multiplication and division operations

Yeah, I like that explanation. We are creating new relationships between otherwise isolated phenomena in order to label and quantify some emergent property of a system wherein those hitherto isolated phenomena/physical quantities interact. This is the utility of the multiplication and division operators in physics. As noted above you can't do that with the addition and subtraction operators because they must act on like units.

Force=kg*meters/second^2. I always thought the balance of units of all of these physical terms (such as force, work, etc.) was downright goofy and confusing. But I think I see their utility more now. Interesting.

 

[mremadahmed]

Thanks guyss ... I got the concept ...

 

[technician]

On the other hand, the reason WHY m times a is equal to the force applied in real life is a mystery. That's just the way it works. In some other universe it may be the case that F=2ma, or F=ma2. But not in our own universe.

I would disagree with the statement F = 2ma indicates a 'different universe'
The relationship between physical quantities is best given as F ~ ma (proportional)
The numerical factor indicates the units used in measurement.
If you had written F = 2.2ma it would be a fairly good indication of Newton's law in this universe if m was measured in lbs, and a in m/s²
F = ma² could be another universe but that is science fiction (as far as I know)

 

[AlexS]

Interesting conversation, I think that the criterion with unit is very important.I actually think that after you have this fundamental defining formulas such as average speed=total distance/total time, you can obtain other formulas such as Galileo's formula(v square=v0 square+2ad) just doing the math,without any trouble. But,in my view, the problem is still with this fundamental equations. For instance, why the speed is distance per time and not distance multiplied with time? I see the proportionality between time and distance (if speed is constant), but I still cannot perfectly understand this problem.

 

[willem2]

The distributive law seems to be logically required for F, m and a.
Suppose you accelerate a mass, that's composed of two smaller masses m₁ and m₂, since the masses are attached to each other a, is the same for both masses and for the total mas, so you have m_tot * a = F_tot = F₁ + F₂ = m₁ a + m₂ a. Also if a = 0, then F=0, another multiplication axiom m*0 = 0.

 

[sophiecentaur]

You've completely misunderstood his question, Drakkith

That's a bit harsh, ain't it? Any given mathematical operation can be fitted to a whole set of different physical situations with different quantities. There are Logical Rules that limit which operations can be applied to which relationships between physical quantities; some operations are commutative or distributive and some aren't. How far do you want to take this discussion and how deep into Maths Analysis do you want to go?
When we learn Maths we start with discrete quantities and slide gently into applying the same rules to continuous variables. When we derive any Physics equation from basics, we start with discrete steps and extend into a continuum. Calculus is a great example of this when the results always involve "limit as x approaches zero" statements.

I must say though, it gives me a weird feeling that Maths seems to have sprung up, out of nowhere, in the heads of humans who have constantly been trying to describe and predict the world around them. Just why should it be common to everything we experience?

 

[OldYat47]

The basic issue is dimensional analysis and the definitions of various physical quantities. For example, F=ma could be in English or SI units as long as they agree. Force is defined as mass times acceleration, so both sides of the equation have the same dimensions. The meaning of, for example, F = (20 kg) X (2 m/s^2), F = [(40)(kgXm/s^2)], is that the numbers give the magnitude, the dimensions give the system of measurements used. You could not, for example, say F = (20 slugs) X (2 m/s^2) because the systems of measurement are not the same. Force is not defined as (slugs X m/s^2) or (kg X ft/s^2). You have to be consistent. The dimensions are handled algebraically.

And it pays to be careful. You may see lbm (pounds mass) instead of slugs. lbf (pounds force) or just lb (pounds) as weight, and kg (kilograms) is commonly used as a measure of weight (actually 1/9.81 kg mass measured by a scale on the surface of earth, which should be Newtons to be accurate).

 

[pixel]

I must say though, it gives me a weird feeling that Maths seems to have sprung up, out of nowhere, in the heads of humans who have constantly been trying to describe and predict the world around them. Just why should it be common to everything we experience?

There's a paper by Wigner titled "The unreasonable effectiveness of mathematics in the natural sciences" that may provide some answers: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

 

[OldYat47]

pixel has referenced an excellent article. On thing that the author discussed in an interview I heard was that mathematics (or at least the way we understand mathematics) doesn't always work (weather prediction was an example he used). It's an interesting question whether it's our mathematical inability to explain certain phenomena because we haven't figured out the math yet, or if the phenomena are truly outside the realm of mathematics. I tend to believe the former, just because.

 

[rcgldr]

One mathematical model for dealing with different units is to consider the area under a curve (or between two curves) where x and y-axis have different units. The area can be considered to be the limit of a bunch of squares with sides Δx and Δy as the size of the squares approaches zero and the number of squares approaches infinity. For simple cases with constant values, then the area is just the product of the sides of a rectangle x by y, with x and y having different units, such x = mass and y = acceleration, where the area would be force.

Other examples of multiplication in physics: torque = force x radius, power = force x speed, ... .

 

[David Lewis]

What does F=ma means? m is in kg. a in ms-2. How can both multiply? They cannot add or subtract.

When mass and acceleration are multiplied, it just means if you double the mass then you double the force. If you triple the mass, you triple the force, etc. And, if you double the mass and triple the acceleration, you will get 6 times the force.

Multiplication started out as shorthand for this abstract generalized relationship between numbers. The original application was counting. It was subsequently discovered some disparate physical quantities obey the same relationship.

 

[AlexS]

Another document about the mathematical operations in physics- http://www.physics.ohio-state.edu/~aubrecht/OSAPSdivision.pdf

 

[arul_k]

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

We also need to look at how we define certain quantities in science. Force is defined in terms of the accleration produced on a certain mass.

So the equation F=ma indicates that if you double the force you would get twice the accleration, mass remaining constant and experiment proves this to be correct.

 

[gleem]

We are introduced to addition as a method of counting objects in groups. We are introduced to multiplication as a method of adding things that are grouped in equal quantities as a dozen of eggs in a carton packed 20 to a box. So we can find the total quantity of eggs by multiplying 12 by 20 instead of having to remove the eggs and count them. Addition expedites counting and multiplication expedites addition. Addition requires that we add only similar objects that is when you count you count something particular, cows, bulls, black sheep or herbivores, They can be different in actuality but must have a common characteristic to which the addition is to apply and the counting is appropriate answering the question "How many of this type do we have?"
Now in the use of multiplication to "non counting situations" as in mass x acceleration one can see that this is still in fact a counting or addition situation. The total force for example on a body is the sum of the forces on a multitude of elemental masses. There is the force acting on each real elemental mass but this force is the sum of the basic specification unit of the force i.e, Newton. Each elemental mass will have a delimited number of Newtons associated with it. So we have an array of force units associated with a number of masses. analogous to the total number eggs in a box containing a number of cartons associated with a dozen eggs count in each carton.

We are led to notice that units can be treated as mathematical quantities e.g., meter/meter =1 or meter⋅meters = meters² which leads to dimensional analysis and of course the adoption of names for certain combinations of these quantities.

 

[OldYat47]

Think about it this way: F = m*a is algebraically the same on both sides of the equal sign. Force is defined as m*a so the equation reads (expanding force into its components) m*a = m*a. That's dimensional analysis, making sure the various quantities make sense algebraically. The magnitude of the equation is in the numbers. So for example, 10*m*a = 5*m*2*a. This makes perfect sense even if the definition of force doesn't exist. Similarly we could work equations with (550 ft-lbf/sec) without the "abbreviation" of horsepower.

All the quantities and defined units must follow the rules of algebra and calculus, so you can't subtract or add mass from or to acceleration.

Even though multiplication is often introduced as a form of addition for simplification, in reality it is not.

 

[ogg]

I may have missed it, but in case it hasn't been made explicit. "Addition" (or "multiplication" or "subtraction" or "division" or "integration" or "differentiation" or "taking the limit" or...) does NOT have a single unique meaning in Science/Physics. Adding groups, adding vectors, adding areas, adding lines, and adding scalars are DIFFERENT types of addition. Please look up the history of mathematics: counting (accounting) was considered almost completely different than geometry (areas of surfaces, volumes,...). The rules we use in the math we apply DEPEND on the need (the situation, the context, the problem). Sometimes a+b = b+a but sex and a dinner is a different date than dinner and then sex. (seriously, the "normal" properties of arithmetic (commutative property, distributive property, etc.) are NOT always correct in given contexts. I haven't looked into the philosophy, but I suspect there is no "bright line" distinguishing when it's meaningful to attach units to ratios and when its not. Since much of physics is ratios, I expect that interpretation (dimensional analysis) at some point requires a pragmatic (it works, so it's correct) approach.

 

[David Lewis]

Please look up the history of mathematics: counting (accounting) was considered almost completely different than geometry (areas of surfaces, volumes,...).

Geometry was originally used to count the area units of Egyptian farmland so the pharaoh knew how much property tax was due.

 

[gleem]

I can't seem to think of a quantity in physics than cannot be counted, included, accumulated, added, or summed in the simple strategy of arithmetic

 

[OldYat47]

gleem, that may be true working with individual units, but not when working with combined defined units. Mass and acceleration, for example, can be multiplied to give a resulting force, but they cannot be added or subtracted from one another. So algebra always enters early on.

 

[jbriggs444]

I can't seem to think of a quantity in physics than cannot be counted, included, accumulated, added, or summed in the simple strategy of arithmetic

Adding the Celsius temperatures of all the students in a freshman physics class is of questionable value, even though they all have the same units.

 

[gleem]

I can't seem to think of a quantity in physics than cannot be counted, included, accumulated, added, or summed in the simple strategy of arithmetic

I guess I should added where permitted or sensible for adding temperatures is usually of dubious value as is adding densities and you know even adding masses together can be of dubious value unless there is a relationship or unifying connection between them, the same for example, resistance or battery voltages.

The OP questioned the meaning of multiplication in scientific applications. My post was to point out that I see multiplication basically has the same meaning as counting equal size groups where the groups size is precounted (summed) and usually interpreted as a rate or ratio. An note if the group size varies then this counting processes can be appreciated as a primitive form of integration.

Multiplication begets division as addition begets subtraction. I think when you look at a ratio as an indicated division and not stemming from a sum of things in a group you what is happening. A moving object for example put the number of miles traveled in a group with defined characteristic e.g.. one hour intervals. I guess I just been concentrating too much on my odometer counting of the miles while my watch is counting off the hours.

 

[OldYat47]

Again, I respectfully disagree. Multiplication and addition are fundamentally different as applied in common physics problems. And again, we can define certain physical properties and fundamental concepts with algebra where we could not do the same with arithmetic. And indeed a ratio is a division operation not stemming from a sum of things in a group.

 

[sophiecentaur]

It's easy to forget that Maths is only just another model of real life. You write down an equation or formula to represent what happens and you immediately make assumptions about the accuracy of that formula. Solving a simple quadratic equation for a computation of the motion of an object assumes that the variables are all continuous. Quantum Mechanics applies everywhere, so that implies that the energy changes are actually not continuous but discrete. Fact is that it makes not difference in most cases but you have to break away from that simple formula when the number of energy states becomes limited. The Maths model breaks down - no problem, unless you let it bother you. The 'faith' involved in using Maths should be just based on experience that multiplication, in most cases, produces results that can be verified (within the limits of experimental error).
3 (discrete) boxes of 12 (discrete) cakes gives you (correctly) 3X12 = 36 cakes. But you can only have complete boxes and complete cakes. 10s of acceleration of a car at 2m/s² will only give an approximate answer of 10X2 = 20ms², whatever accuracy you may claim to have measured the time and the acceleration.
If you want to introduce the concept of Chaos into the argument, the correspondence between Maths and Measurement becomes even more dodgy. The results for infinitesimal differences in input conditions for a chaotic model can be vastly different. The Butterfly can be absolutely anywhere around the flower but the Maths prediction assumes a discrete input of position - so many decimal places. Increase the number of decimal places by one and the possible outcomes can involve flat calm or a hurricane around your house.

 

[valenumr]

We also need to look at how we define certain quantities in science. Force is defined in terms of the accleration produced on a certain mass.

So the equation F=ma indicates that if you double the force you would get twice the accleration, mass remaining constant and experiment proves this to be correct.

Actually, the formal definition of force in Newtonian mechanics is change of momentum with respect to time. Varying mass isn't a problem.

 

[sysprog]

Adding the Celsius temperatures of all the students in a freshman physics class is of questionable value, even though they all have the same units.

You can of course divide the sum by the number of students and get the arithmetic mean.

 

[ergospherical]

Actually, the formal definition of force in Newtonian mechanics is change of momentum with respect to time. Varying mass isn't a problem.

To the contrary, variable mass systems are way more subtle than you think. The expression $\boldsymbol{F} = \dot{\boldsymbol{p}}$ in fact only holds for systems of constant mass.

I won't go into details, but Spivak's Physics for Mathematicians: Mechanics I has some nice discussion on this issue, for example.

And a footnote, completely inappropriate for this thread so start a new thread for any followup:

Spoiler

In general treatments, the derivative of the momentum $\boldsymbol{p} = \int_{\mathcal{D}(t)} \rho \boldsymbol{v} dV$ of the medium within some domain $\mathcal{D}(t) = g^t \mathcal{D}_0$ must be evaluated using the Reynold's transport theorem of vector calculus, and the result involves a term $-\int_{\partial \mathcal{D}(t)} \rho\boldsymbol{v}(\boldsymbol{v}-\boldsymbol{w}) \cdot d\boldsymbol{S}$ in addition to the total force on the body. [Here $\boldsymbol{w}$ is the velocity of the boundary and $\boldsymbol{v}$ the velocity of the medium.]

 

[bhobba]

To the contrary, variable mass systems are way more subtle than you think.

Indeed they are.

Regarding forces in a generalised sense, I think the Lagrangian approach is needed:
http://www.physics.arizona.edu/~varnes/Teaching/321Fall2004/Notes/Lecture13.pdf

Question: What happens to the equations if m happens to depend on position?
https://www.scielo.br/j/jbsmse/a/fGPcfL8tYJSCpBJVMdcVmSh/?lang=en

Thanks
Bill

 

[sophiecentaur]

Regarding forces in a generalised sense, I think the Lagrangian approach is needed:

Isn't that more 'nuts and bolts' than the OP question needs, though? Whatever Maths you use is just a model of a particular accuracy and the general comments will apply.

 

[bhobba]

Isn't that more 'nuts and bolts' than the OP question needs, though? Whatever Maths you use is just a model of a particular accuracy and the general comments will apply.

Well yes. It's a personal thing. When considering general questions, I tend to fall back on the most fundamental way I think of something. I guess I am a bit too influenced by Landau - Mechanics I always recommend.

The interesting part is the second link, where mass depends on the position, which used Lagrangian formalism.

Thanks
Bill

 

[jrsintel]

One case where multiplication comes up is attenuation. If you have a filter that passes half the light I that enters it, then the light that emerges is 0.5 * I. If you added a second filter, then the light coming out would be 0.5 * 0.5 * I. Attenuation is one example where multiplication models the effect. Units don't really come up because the attenuation coefficient is unitless.

 

[Robert Stenton]

Mathematics is all around sciences.It is also considered a language.In language everything should have meaning.

I know addition and subtraction meaning.Same things[quantities ,etc.] can add or subtract . It cannot be done on different things.

What is the meaning of multiplication and division in sciences[Physics,.. etc].I know what it means in real life.Multiplication is higher form of addition.Division is higher form of subtraction.

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

Why any quantities can multiply or divide,contrary to adding and subtracting.

Can anyone answer this question,or provide a book to read about this subject.

F= ma is Newton's most famous equation but he never used it! Instead he used m=F/a to define mass. F/a is a ratio (fraction) so you have to use multiplication or division. Suppose you buy a pie and cut it into 8 slices. Being a pig you ate 3 of the pieces then decided to be a hog and have another piece. Since the pie is in units of 8, you can't simply add or subtract. Suppose you decide to be a hog and have another piece. Now you ate 4/8th of the pie which you can simplify by dividing by 2. I hope this was your question and that I answered it.

 

[Philip Koeck]

I don't know whether this has been brought up already.
In physics many quantities are extensive, which means they are proportional to some amount.
For example mass and energy. If you add two amounts of matter with a certain mass each, the masses add (ignoring relativistic effects).
Adding and subtracting makes sense for extensive quantities.

Then there are intensive quantities such as temperature. They are not proportional to the amount of substance. You don't usually add these quantities unless you calculate an average.
Subtracting them only makes sense for example as a temperature difference. You're not actually removing one temperature from another.

I'm speculating quite a bit here but anyway:
Many multiplications in physics involve an extensive and an intensive quantity so that the result is extensive.
Divisions of extensive by extensive quantities give intensive quantities, such as some kind of density.
Intensive quantities can be freely divided and multiplied in many ways and the results are always intensive.

A multiplication of two extensive quantities doesn't make sense to me. The result would be proportional to the square of an amount.

I think these considerations limit which algebraic operations make sense when applied to physics.

Again: This is a lot of speculation.

 

[Physhead]

The intellectual abstraction vexed me until the light bulb came on in my head: IT IS ALL ABOUT ENERGY AND REDISTRIBUTION THEREOF and nothing more. Mass is energy. Acceleration is energy... EVERYTHING is about energy and states thereof, which necessitates vector analysis.

The confusion comes from the way that physics is taught, (mostly because schools are largely interested in churning out "working scientists and engineers" rather than in producing great thinkers) in that we are introduced to "practical" problems with Newtonian "solutions" (yes, I used quotes, and real physicists know why) first. If we were to teach (1) ENERGY first, (2) then explain that every quantity we deal with is a form thereof, (3) constantly reiterate how every problem is about energy (4) then teach how SCALE dictates the level of necessary precision for practical results (i.e. you don't really need general relativity to figure out how much energy it takes to kick a football 30 meters to clear a goal post that is X meters high... but you could, if you were effing masochistic, and that you can't "do lasers" using Newton), the cognitive dissonance would be greatly diminished.

UNIT analysis is what confused our original inquirer mremadahmed . All units distill to energy, because that is all that there is, and we are merely adding and subtracting.

All of this is best contemplated over a pint of Guinness or a nice scotch, but not a pint of scotch, which is counter productive in all but the rarest of circumstances.

 

[bhobba]

F= ma is Newton's most famous equation but he never used it! Instead he used m=F/a to define mass.

It's a definition and has no physical content. It assumes something called mass that also appears in Newton's law of gravitation. People can use gravitation to give mass a value via, say, a mass balance. The status of F=ma as a law is as a paradigm that says - get thee to the forces in analysing mechanics. That makes it a bit subtle, which is why I prefer the Lagrangian formulation. However, you would have to have rocks in your head to start with that in studying mechanics. Physics sometimes is like that. You start with a basic concept that later has to be scrapped and replaced with something more advanced. Even math is sometimes like that. In calculus, you begin with an intuitive idea of the limit, then replace it with the standard epsilon definition later.

Thanks
Bill

 

[bhobba]

The intellectual abstraction vexed me until the light bulb came on in my head: IT IS ALL ABOUT ENERGY AND REDISTRIBUTION THEREOF and nothing more. Mass is energy. Acceleration is energy... EVERYTHING is about energy and states thereof, which necessitates vector analysis.

There is a lot of confusion here. My suggestion is to study Noether's Theorem:
https://math.ucr.edu/home/baez/noether.html

Thanks
Bill

 

[Physhead]

There is a lot of confusion here. My suggestion is to study Noether's Theorem:
https://math.ucr.edu/home/baez/noether.html

Thanks
Bill

I am terribly familiar with Em, and YES! Everyone needs to know and love her work!
I am confusioned about what you think there is a lot of confusion about. Please help deconfuse me as to my confusion over your allegation of confusion...or was this a reference to my statement about the original poster's confusion. I am so sorry. English is way difficulter than it looks.

 

[italicus]

I will address some considerations to the OP original request, because I’ve the sensation that no one has ever clarified him that Mathematics is NOT Physics. Math is only an instrument , a way to put physical ideas and experimental findings in formulae, in order to give us the faculty to manage them, do calculations and obtain results, that then are to be compared with the experimental results.

The second law of Newtonian dynamics : F=kma ( where k is a constant, that can be assumed equal to 1 by convention, with an opportune choice of units of measure: see later) is nothing else that the way to put mathematically the following experimental findings :

1) I suppose the OP knows the first law of Mechanics : a body (imagine a small rigid body, don’t want to be extremely detailed here...) remains "at rest” or in rectilinear uniform motion, if is not acted upon by any external action that changes this state ( a lot of contours should be better précised here, but I’ll leave them).

2) Let me also suppose that the OP knows what the acceleration is : change of speed with time. But to define it quantitatively I need a unit of measure; I decide that meters and seconds are good for measuring space and time, and assume that speed is measured in m/s : every quantity in physics has a physical significance only when measured , with conventional units. If you don’t measure, no one will appreciate you. Any physical quantity is made of a number and a unit of measure , the number alone doesn’t mean anything.
Therefore, acceleration being defined as the change of speed wrt time , I’ll find that, for dimensional reasons of coherence, the unit of measure of acceleration must be : m/s².

2) So far so good. Now take a ball , kick it. Repeat this several times, with different “intensity” of the kick. You will notice that, the greater the intensity, the greater the acceleration taken by the ball. Suppose to repeat this very many times, and to also have invented a system for determining with precision the “intensity” of you kick: the experimental result is that , when you double the intensity, the acceleration doubles...and so on. There is a direct proportionality between the intensity of the kick and the acceleration assumed by the ball.
Now , it’s a matter of definitions and conventions.
The previous ideas suggest that there is, for the body ( better: material point; for systems, a lot of more precise concepts should be given) , a defined quantity, given by the intensity of the kick and the acceleration taken by the body. This quantity is called “inertial mass” of the body; the intensity of your kick is called “force” . So, To express the proportionality between force and acceleration, found experimentally, we simply write :

$$F = ma$$

mass “m” is given a proper unit, for example “kg” . Therefore “ma” is expressed in kg*m/s². The LHS is given the name of force, and the unit takes the name of Newton N.

That is, it is needed the force of a N to give the mass of 1 kg the acceleration of 1 m/s² .

This is the end of story , and the beginning of mechanics. As you can see, all quantities have units, conventionally chosen of course. All quantities can be someway measured, all observers must agree on the results.

I Hope I have been able to give you at least an idea on how things work, when speaking of physical relations. F=ma doesn’t mean that you take m factors equal to a ! . They are not simple numbers, are physical entities.

 

[bhobba]

IPlease help deconfuse me as to my confusion over your allegation of confusion...or was this a reference to my statement about the original poster's confusion. I am so sorry. English is way difficulter than it looks.

Ah - English is not your primary language. That likely explains my 'confusion' over what you wrote. Acceleration, for example, is not energy. That was the sort of thing that was leading me to think you needed to investigate what things like energy are further. Since you know them already I will be more careful in reading your responses in future.

BTW welcome to Physics Forums. I see you just joined yesterday. It is always a joy to have new members.

Thanks
Bill

 

[bhobba]

Suppose to repeat this very many times, and to also have invented a system for determining with precision the “intensity” of you kick

That's the bit I do not quite understand. What would be an example of a system of determining the intensity of the kick, other than of course defining it as ma? Would you use for example stretched springs? But that relies on Hooks law which requires for its statement the concept of force. You could take it as a given. But that would mean Hooks law is true by fiat - and we know it is true for only small displacements.

My personal view is that Feynman had the right approach:
https://www.feynmanlectures.caltech.edu/I_12.html

It is more than just a definition because it allows a precise statement of the third law and other laws like gravitation. Basically, that is the true content of F=ma. It is, as I said, the correct paradigm to analyse mechanical phenomena from experience. It is a law, but not of the usual type such as the third law which is a testable statement about nature. Rather it is saying when analysing problems in classical mechanics look at the forces. That is a statement about nature.

The interesting thing of course, and Feynman initially resisted this, is that the Lagrangian formulation is more like the usual testable statements about nature we call laws and often makes more difficult problems easier to solve. Evidently, Feynman would insist on using forces when the Lagrangian approach would be easier. The real twist of course is when he developed his path integral approach to QM the Lagrangian formulation popped out immediately. Just one of the many curious things in the life of that curious character.

Thanks
Bill

 

[italicus]

@bhobba

Feynman is a master scientist, but in the linked chapter he repeats several times that if one defines force as the mass times the acceleration, he has found out nothing. This is an abstract:

If we have discovered a fundamental law, which asserts that the force is equal to the mass times the acceleration, and then define the force to be the mass times the acceleration, we have found out nothing. We could also define force to mean that a moving object with no force acting on it continues to move with constant velocity in a straight line. If we then observe an object not moving in a straight line with a constant velocity, we might say that there is a force on it. Now such things certainly cannot be the content of physics, because they are definitions going in a circle. The Newtonian statement above, however, seems to be a most precise definition of force, and one that appeals to the mathematician; nevertheless, it is completely useless, because no prediction whatsoever can be made from a definition.

Furthermore, in one of the initial lines of his lesson he says that one has, more or less, an intuitive idea of what is “mass” of a body. Then I'd ask, if possible : what is mass, beyond intuition ? When it comes to definitions, some problems arise.

On another side, let’s take into consideration what Benjamin Crowell, estimated member of PF , says in his book on Mechanics :

0.6 The Newton, the metric unit of force
A force is a push or a pull, or more generally anything that can change an object’s speed or direction of motion. A force is required to start a car moving, to slow down a baseball player sliding into home base, or to make an airplane turn. (Forces may fail to change an object’s motion if they are canceled by other forces, e.g., the force of gravity pulling you down right now is being canceled by the force of the chair pushing up on you.) The metric unit of force is the Newton, defined as the force which, if applied for one second, will cause a 1-kilogram object starting from rest to reach a speed of 1 m/s. Later chapters will discuss the force concept in more detail. In fact, this entire book is about the relationship between force and motion.

May be this is more intuitive than the concept of mass, and IMO it gives a hint towards an operational definition of force.

As far as the Lagrange equations are concerned, one has first to define Energy, another not-easy task (look for Feynman chapter on energy) , and then apply other not so immediate concepts of the calculus of variations. I find it difficult to introduce this to students of a first course of Physics.

Thanks.

 

[bhobba]

As far as the Lagrange equations are concerned, one has first to define Energy, another not-easy task (look for Feynman chapter on energy) , and then apply other not so immediate concepts of the calculus of variations. I find it difficult to introduce this to students of a first course of Physics.

I agree it is not for the first physics course. But once you move onto Lagrangians, Noether comes into play and what energy is and why it is conserved is a snap.

Thanks
Bill

 

[valenumr]

To the contrary, variable mass systems are way more subtle than you think. The expression $\boldsymbol{F} = \dot{\boldsymbol{p}}$ in fact only holds for systems of constant mass.

I won't go into details, but Spivak's Physics for Mathematicians: Mechanics I has some nice discussion on this issue, for example.

And a footnote, completely inappropriate for this thread so start a new thread for any followup:

Spoiler

In general treatments, the derivative of the momentum $\boldsymbol{p} = \int_{\mathcal{D}(t)} \rho \boldsymbol{v} dV$ of the medium within some domain $\mathcal{D}(t) = g^t \mathcal{D}_0$ must be evaluated using the Reynold's transport theorem of vector calculus, and the result involves a term $-\int_{\partial \mathcal{D}(t)} \rho\boldsymbol{v}(\boldsymbol{v}-\boldsymbol{w}) \cdot d\boldsymbol{S}$ in addition to the total force on the body. [Here $\boldsymbol{w}$ is the velocity of the boundary and $\boldsymbol{v}$ the velocity of the medium.]

Well, to be fair, I explicitly said "Newtonian mechanics" which allows us to figure out how to send rockets to the moon and such.

 

[weirdoguy]

Well, to be fair, I explicitly said "Newtonian mechanics

What @ergospherical discussed is also Newtonian mechanics, just a bit advanced