Junal Estandarte: I want to ask something about the basic arithmetic and its fusion with the physical quantities in physics. Like for example the formula F=ma, in this equation mass and acceleration are multiplied. What I want to know is why are they multiplied?! I mean you can add them like this F=m+a.

I know that the result will be different from when you multiplied them but why multiply them?! Instead of adding them. When can we multiply and add physical quantities? And what do we exactly mean, by the way, by the product of the number and the sum of the number? I’ve been to many math lessons in high school but so far my teachers haven’t made it clear yet.

Vitali Ko

Math is the language of physics. If you want to say in English “cow” “, you say “cow” and not “dog”. Newton’s law which you mentioned is an axiomatic statement of physical reality in the language of math. If he tried to say anything else like F=m+a, he`d be easily debunked as it wouldn’t hold experimentally and then you’d never hear of Newton.

Alon Ben Israel

To further clarify the point, what you’re actually asking is “why are the laws of nature the way they are? Why does F=M*A make a consistent model, but F=M+A does not?”

It’s a very, very good question – but not one that physicists actually deal with. It’s more in the realm of philosophy. Physicists usually accept the fact that we don’t know why the laws are the way they are. Like Vitally said, the reason Newton’s laws have been so immensely successful is because they can be proven experimentally (y’know, up ’till they start falling apart). The same goes for every physical concept.

Many times in history people have, in fact, tried to formulate different physical laws which were entirely mathematically consistent. You couldn’t find anything wrong with them just by following the math. The reason they didn’t work is that they failed experimentally; the universe simply doesn’t work that way. A physical theory cannot be considered true until it has been shown to predict experimental results (within a certain margin of error).

By the way, just to further elaborate – you actually can’t write F=M+A because it is not mathematically consistent. Every physical quantity has a unit or dimensions. F is force (Newton’s), M is mass (kg), A is acceleration (meter/second^2).

One basic principle is that you cannot add together two quantities with different dimensions. Quite simply because it would not make sense. If someone tells you that a quantity is in the units of meter^2, you know what he means – it’s surface. If someone tells you a quantity is in joule/meter^2/sec, you also know what he means – it’s energy per surface unit per amount of time (an energy flux). However units such as “meters + kg” don’t make any physical sense.

As a rule in physics which always holds true, if two terms in an equation are separated by addition or subtraction, they must always have the same units. Exponents must also always be dimensionally neutral, because units of, say, e^kg just wouldn’t make sense.

This is called dimensional analysis, and is a powerful tool in formulating general solutions to problems and coming up with new theories.

Kevin Pretorius

Hi Junal Estandarte I don’t quite know how to approach your question, as it seems your problem is actually about understanding maths at a very basic level.

It’s intuitively obvious to me that you can’t add a mass to acceleration and expect a force to result. Mass, Force and Acceleration are different things. You can only add things which are measured in the same units.

For example you can add masses together and you’ll get a total mass out. You can add accelerations together and get another acceleration out.

But when you multiply mass time acceleration, you’ll get something which is neither a mass nor acceleration. The fact that it’s a force is something (after a fashion) Newton discovered.

Sorry Alon – we seem to have been writing about the same points at the same time.

Alon Ben Israel

The more, the merrier. : P

Kel Van Der Meel

@Kevin and Alan…..I had a conversation like this concerning the definition of heat, and heat transfer with him yesterday. And it was the same issue. A lack of basic understanding. I admire his bravery; it can’t be easy asking basic questions, knowing that someone might inadvertently make him feel silly for asking.

But the reality is, don’t ask questions if you are unsure you will be able to comprehend the answers. he has homework to do. :)~

Junal Estandarte

Please never feel sorry, Kevin. I really appreciate your help. I guess my conclusion is that I actually use the philosophical way of understanding it instead of the scientific way. I see now why I really find it hard to understand this question. Thank you guys. But if you guys won’t mind, Could you give me a brief story on how Newton conducted his experiment. I want to know specifically how he formulated the law of universal gravitation formula. In this formula you can actually notice that he multiplied two physical quantities of the same kind. Mass1 times mass 2. Now since they are the same, you can mistakenly add them actually, Can you?! m1 + m2. So why multiply it and not add them. if you guys won’t mind off course because I can actually learn this on my own if I study it with observation and experiment.

Kevin Pretorius

I’m sure I don’t know what experiments Newton did. To my understanding he mostly had an incredible flair for analysis, noting that the motions of astronomical bodies were consistent with the idea of an attractive force pulling things downward on earth.

These two sections of the Wiki article on Newton make some references to this.

Kel Van Der Meel

Yes! Good place to start Junal.

Kevin Pretorius

As for the second part of your question, again you are missing the nature of what is going on. When I explained previously that you CAN’T add different types of quantities like mass and acceleration, I didn’t imply that if they’re the same that you MUST add them. That’s not correct.

When you have the lengths of two sides of a field, you MULTIPLY them to calculate an area, and ADD them to get the (semi) perimeter. Different mathematical operations to achieve different types of results.

And so it is with gravitation. The formula involves the PRODUCT of the two masses, not the SUM.

Junal Estandarte

So is it all about “area” then?

Dimensions, you should say….

Kevin Pretorius

“Is it about ‘area’ then?”

Sorta/kinda in a very rough analogy, but there’s no such concept as a square mass.

Junal Estandarte

For mathematicians why did they call the result of multiplying two variables, PRODUCT? And for addition, Sum.

Mazikeen Morningstar

Why would anyone mention philosophy here? Another way to look at it is adding quantities does nothing? The units don’t cancel. They need to cancel or else you’re going to have a result that is wonky. It won’t be commensurate to experiment.

Kevin Pretorius

Hi Junal “Why are products and sums so called?” I don’t think there’s a specific reason, any more than we have ratios and differences for division and subtraction.

Also (but less familiarly) one adds an addend to an augends to calculate a sum, and take a subtrahend away from a minuend to calculate a difference.

They’re just words.

Junal Estandarte

They only thing I know is that multiplication is actually adding in an easier way. Or making addition easy. Or making it easy to add many numbers or variables. Thanks anyway Kev… I appreciate it…

So the word is “dimensional analysis”. Thanks to Alon Ben Israel for introducing me this word.

James Maxwell

My favorite problem, almost solved by dimensional analysis alone is period of a swinging pendulum, mass M on the end of a string of length L, with the earth’s gravitational acceleration g.

To get time out of this, where ^ means to the power of:
T = M^a * L^b * g^c

Just looking at the units:
(sec) = (kg)^a * (m)^b * (m/(sec sec))^c

There are no (kg) on the left, so a=0. There are no (m) on the left so b+c = 0. There are (sec) to the power 1 on the left and to the power -2c on the right, so c = -1/2 and b = 1/2. The period is then:

T = (L/g)^(1/2)
The period is the square root of the length of the pendulum divided by g=9.81 m/(sec)(sec). This is almost right, for small oscillations we are missing a factor of 2 Pi on the right.

All this from dimensional analysis.

Adam Boudreau

Junal Estandarte that is not true. Multiplication is NOT just making addition easier. They are fundamentally different things. Multiplication is only related to addition in the case of positive integers. For example: 2 x 3 = 3 + 3, because I added 3 two TIMES.

However, if I have Sqrt(3) * 2, how do I add 2 square root of 3 times?

It makes no sense.

Prateek Prasad

Dude, F is not equal to MA…..
But F=dP/dt……

DP/dt=d (mv)/DT=mdv/dt+vdm/dt

James Maxwell

Prateek Prasad, quite right about F=dP/dt, but if, and that is a BIG IF, mass is constant, then F = ma

Prateek Prasad

Ya, that’s true…..

Junal Estandarte, now u will be asking why P=mv and not m+v

Junal Estandarte, u can also understand by applying the concept of vectors……as F=m+v would be absolute non sense when we will talk in vectors……

James Maxwell

To balance two children one of mass M1 there other of mass M2 on a teeter-totter, you will find that when one is distance L1 and the other distance L2 from the fulcrum, they balance if the products are equal:
M1 L1 = M2 L2

You don’t add their mass and distance from the fulcrum, you multiply. Momentum is similarly useful concept and you multiply the mass by the velocity, that is just how it is … useful.

Prateek Prasad

James Maxwell, that’s a good example of similar behavior but i think it does not clearly explains as to why that multiplication is needed

James Maxwell

A force is that which causes an object with mass to change its velocity. Mass is the measure of a resistance to the force. The greater the mass of something the less it changes its velocity when under the same force.

So these are intuitive ideas about force. F = ma is the simplest equation that has the relationship that the greater the force the greater the change in velocity (i.e. the greater “a” is) and the greater the mass “m” is the smaller its acceleration is when under the same force.

The equation has proved useful. One can try other equations, like F = m a a, but then then don’t lead to the beautiful simple results that Newton got.

Suleiman Bashir Adamu

The product of the number and the sum of the number? what it means is that product stands for “multiplication eg, the product of A and B = A*B”, while sum of a number is used for “Addition” eg, the sum of a number 2 and 5 is ? Then 2+5=7…. For f=ma, they are multiplied because m is quantity of weight in the body or object while acceleration a, is the rate at which the body or object is moving.

Nina Astro-Nut Logan

The one creating the equation can make it anything that they want it to be. Also they can explain the results any way they want to as well. However, once the equation is adopted, validated and used by others and interpreted to mean a certain accepted truth, it must stay that way. He could of made F=m+a, but he didn’t.

Adam Boudreau

“He could of made F=m+a, but he didn’t.” Completely untrue, this would not have made sense with the data.

Kevin Pretorius

Hi Nina Astro-Nut Logan – I’m with Adam on this one. Isaac Newton didn’t just make up F=ma. He’s trying to describe the behavior of nature, and F=ma is exactly how nature behaves.

Nina Astro-Nut Logan

Nature did not give the plus “+” sign and the multiplication “*” sign meaning… man gave them meaning. Just switch the values or meanings of those two signs and you have the same formula. By trying to explain nature mathematically you must concur on all symbols. Make the “+” sign to mean multiply and it can still be written F = m+a. That is why I am correct in my statement.

Kevin Pretorius

Hi Nina, OK, so if Isaac Newton had chosen to break with the mathematical conventions of the day, and have + mean multiply then yes you would be correct in your statement.

Of course when I say “yes you would be correct in your statement” I am breaking with the conventions of my day, and intend it to mean something else.

James Maxwell

Sheesh, F = m + a can only mean that you add the mass and the acceleration to get the force, but … but the units are all screwed up, how do you add kilograms with meters per second squared and come up with units for force. Its like adding 4 dogs and 5 cats … what have you got there, 9 what?

Adam Boudreau

No, Nina you’re still wrong. Even if you switched the meaning of the statements one would still be a product and the other sum. You don’t get to choose how that operation “really works” with respect to the data. All we get to do is name the damn thing, which is a meaningless point, because I can easily say, for things that are called two, I name them three. And everything would still work; the fundamental concept behind operations doesn’t change if you switch the definition of plus and times. It’s an inherently meaningless exercise, apparently so you can say “this is why I’m right”… but in the end you’re still dumb and don’t understand.

Sophie Grillet

Because multiplication is the zipper that holds the universe together

Nina Astro-Nut Logan

Sophie that might be the answer to his question. How is multiplication the zipper that holds the universe together?

Kel Van Der Meel

Multiplication by division. Cells divide to become two, chromosomes divide to become two. For one example. With other forms of matter, two H atoms fuse to become one He atom. It is a perpetual symphony of calculations. Each form of matter attracting to form larger masses and more complex systems. Each system comprising of lower order scale systems.

To us it seems chaotic, but it really is perfection in motion. When we look into the cosmos all we see is the result of this perfect process. So we try to define what we see as an end result initially. but it is the process that binds the Math, us, and the universe together. What we see as a whole doesn’t define the universe as a whole. To me this is why quantum theory and practicality are of the up most importance to our eventual understanding.

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