Why did Galileo use multiplication instead of addition in his gravity equation?

[Naveen3456]

I was reading about Galileo's gravity and there I came across the formula.

F= -mg.

Galileo said that Force is proportional to mass but in the opposite direction. I don't want to discuss whether he was right or wrong.

I just want to ask that when he replaced the proportionality sign and used a 'constant' 'g', why did he use the multiplication sign (this is the standard practice till today).

In other words, whenever there is an interaction why only multiplication sign is used and not the plus (+)sign.

Plus sign can also mean some kind of 'interaction'.

Doesn't F= - (m+g) show some kind of interaction. I mean doesn't the + sign show interaction between two entities in nature. ( BTW, I know this formula is wrong)

Forgive me, if it seems too childish a question to ask.

 

[HallsofIvy]

You say you have no problem with "Force proportional to m" but ask why it is "multiplication"? Do you not understand that that is what "proportion" means? "A is proportional to B" means that A is equal to some constant times B: A= kB for some number B.

When we say that Galileo determined that "force is proportional to m", we mean that he measured the gravitational force on many different masses and found that ratio of force to mass was always a constant: F/m= k so that F= km. Of course, now we call that constant -g: "g" for "gravity". And it is negative because gravitational force acts downward.

The fact fact that it is a multiplication and not an addition is "experimental result".

 

[Naveen3456]

You say you have no problem with "Force proportional to m" but ask why it is "multiplication"? Do you not understand that that is what "proportion" means? "A is proportional to B" means that A is equal to some constant times B: A= kB for some number B.

When we say that Galileo determined that "force is proportional to m", we mean that he measured the gravitational force on many different masses and found that ratio of force to mass was always a constant: F/m= k so that F= km. Of course, now we call that constant -g: "g" for "gravity". And it is negative because gravitational force acts downward.

The fact fact that it is a multiplication and not an addition is "experimental result".

Let's consider an experiment.

I push a small block (less weight) of wood on the floor of my room and I get some noise (sound waves/energy).

I push a bigger box sound energy is more.

If I keep on increasing the size of box, sound energy keeps on increasing.

1. Can I now say that size/weight of the box is directly proportional to the sound that I get. can I ever get a proportionality constant in this case, though these two things are proportional?

2. Can anybody tell me an example where interaction between two bodies is represented by a + sign?

 

[nasu]

You don't know they are proportional until you actualy measure some parameter of the noise (level, intensity) and find out that the ratio between this parameter and the mass is constant.
The fact that the noise increases with mass does not automatically means proportional.

Of course, this is an imaginary experiment. In reality the noise can decrease with mass and it depends on many other parameters. Try to pull an empty can over the floor versus a heavy block of aluminum. Which one makes more noise?

The second question does not make sense to me.
The interaction is not represented by signs (plus or minus).
The interaction force can be attractive or repulsive but the sign depends on the axes you choose.

 

[DrewD]

2. Can anybody tell me an example where interaction between two bodies is represented by a + sign?

In reality, there are many additive terms in most forces. For example, a falling object feels the force of gravity, $-mg$ plus a drag force from the air, $c\rho Av^2$, where $\rho$ is the density of the air and $c$ is a constant that needs to be determined and $A$ is the area of the object that is perpendicular to the motion. So the two forces add and the total force is
$F=-mg+c\rho Av^2$

But, imagine if we had the force of gravity where $F_g=-(m+g)$. In this world, an object near the Earth would accelerate as $a=-\frac{m+g}{m}$. If that were the case, very very light objects would accelerate extremely quickly. If the interaction were additive instead of proportional, the extreme case would be very weird.

I'm tempted to keep rambling on, but when you get down to it, physics is an experimental science. The theories are the way they are because that's the way they are.

 

[russ_watters]

You slipped on the "entities" vs "interaction" thing there, naveen: that equation is describing one INTERACTION, not one entity. One force. A plus sign would indicate considering multiple forces simultaneously, as drew said.

 

[dipole]

You slipped on the "entities" vs "interaction" thing there, naveen: that equation is describing one INTERACTION, not one entity. One force. A plus sign would indicate considering multiple forces simultaneously, as drew said.

Furthermore, it makes no sense to add mass and acceleration - they have different units! Mathematically we can write interactions by summing things which are the same kind of thing, but it doesn't make sense to say something like "mass and acceleration interact". Objects, which have mass, interact via forces. It doesn't make sense any other way.

 

[russ_watters]

Good point.

 

[Naveen3456]

Furthermore, it makes no sense to add mass and acceleration - they have different units! Mathematically we can write interactions by summing things which are the same kind of thing, but it doesn't make sense to say something like "mass and acceleration interact". Objects, which have mass, interact via forces. It doesn't make sense any other way.

If speed and mass can 'interact' ( more the speed more the mass), why can't mass and acceleration interact.

Now, please don't tell me that mass and speed don't interact via forces but via some yet undiscovered 'mechanism'.

I am talking about 'interaction' not about how it takes place, whether via forces or some other mechanism.

 

[Naveen3456]

In reality, there are many additive terms in most forces. For example, a falling object feels the force of gravity, $-mg$ plus a drag force from the air, $c\rho Av^2$, where $\rho$ is the density of the air and $c$ is a constant that needs to be determined and $A$ is the area of the object that is perpendicular to the motion. So the two forces add and the total force is
$F=-mg+c\rho Av^2$

These two forces are acting 'simultaneously' on the body. This is a very complex situation. Mere addition of these forces should not give the real picture, IMHO.

Plz don't add gravity is not a force, I know it. I am posing this question in the context of what is quoted.

 

[Aero_UoP]

These two forces are acting 'simultaneously' on the body. This is a very complex situation. Mere addition of these forces should not give the real picture, IMHO.

Plz don't add gravity is not a force, I know it. I am posing this question in the context of what is quoted.

Actually this is a quite simple situation. A complex situation is for example the shock wave/boundary layer interraction :P

To me, there is no point in asking "why multiplication and not addition" since multiplication IS addition... 4*5 = 5+5+5+5 or 4+4+4+4+4, the result is the same.

 

[dipole]

If speed and mass can 'interact' ( more the speed more the mass), why can't mass and acceleration interact.

Now, please don't tell me that mass and speed don't interact via forces but via some yet undiscovered 'mechanism'.

I am talking about 'interaction' not about how it takes place, whether via forces or some other mechanism.

Yes well words have specific meanings, and "interaction" has a specific meaning, whether you want it to or not. It is nonsense to say "speed and mass interact" because both speed and mass are properties of some object. Objects interact, quantities do not.

It's like saying that blue and round interact when describing a blueberry. Blueberries, however, can interact, very weakly through gravitation or by contact forces. Fields can also interact, but fields are physical things just like a blueberry is. This interaction, classically, is described in terms of forces. That's what physics is.

 

[Dale]

If speed and mass can 'interact' ( more the speed more the mass), why can't mass and acceleration interact.

I think that you mean that relativistic mass (a deprecated concept) is a function of speed. That is a definition of the term "relativistic mass", where that term is defined as a function of speed. I cannot think of a definition of mass where acceleration is part of the definition.

The "interaction" as you call it, with speed and relativistic mass is simply a matter of definition and other terms are not defined the same way. It makes no sense to insist that the result of one definition apply to other definitions. Not only does acceleration not "interact" with speed, but other definitions of mass do not "interact" with speed. For instance, the invariant mass is completely independent of speed.

 

[DrewD]

These two forces are acting 'simultaneously' on the body.

That is absolutely correct, but you don't need the quotations.

This is a very complex situation.

This is a simplified description, but it is quite good in air at moderate speeds. Physics doesn't get much simpler than this, so I'm confused about why you consider this a complex situation. Maybe I am misunderstanding what you consider complex.

Mere addition of these forces should not give the real picture, IMHO.

If it is a rigid body that is symmetric so that there is no torque from the air resistance, addition of the vectors is sufficient.

Plz don't add gravity is not a force, I know it. I am posing this question in the context of what is quoted.

It is my understanding that this conversation concerns classical forces, so gravity is a force.

If speed and mass can 'interact' ( more the speed more the mass), why can't mass and acceleration interact.

dipole was saying that adding mass and acceleration does not make sense because the units do not match. There would have to be some sort of factor in front of each of them so that they add to be a force. It would be like saying,
"I walked one mile and then started to run at 6 mph. How far did I travel?"
It is not a well formed question.

 

[Naveen3456]

To me, there is no point in asking "why multiplication and not addition" since multiplication IS addition... 4*5 = 5+5+5+5 or 4+4+4+4+4, the result is the same.

Plz don't get irritated. It's not for the sake of argument, but I want to really understand things.

As per you, Mass and acceleration can be added again and again ( as multiplication is continued addition) but they can't be added once i.e. F=m+a is wrong.

'Why' is this so?

 

[Naveen3456]

If it is a rigid body that is symmetric so that there is no torque from the air resistance, addition of the vectors is sufficient.

Thanks. That's what I meant by complex. So, it means that the formula is an approximation but not a complete picture of things.

It is my understanding that this conversation concerns classical forces, so gravity is a force.



dipole was saying that adding mass and acceleration does not make sense because the units do not match. There would have to be some sort of factor in front of each of them so that they add to be a force. It would be like saying,
"I walked one mile and then started to run at 6 mph. How far did I travel?"
It is not a well formed question.

Not arguing uselessly, I have heard that in quantum mechanics, interactions are more important than entities. In fact entities/objects are not considered to exist, until and unless they interact with the surroundings. It may be due to extremely small scale of things, but quantum mechanics experts say that it is infact the very truth of nature and size has not to do anything with it.

I know, we are considering a classical example, but still the concept of entities, interactions and forces seems to be somewhat arbitrary and even overlapping to my fragile brain.

For example, Dalespam said that invariant mass has nothing to do with speed, buy how would it even exist without the 'speed' of its constituent parts (like electrons, nucleons etc.) So, does it mean there are two kinds of 'speeds' for an object. One that deals with its internal structure and the other with its 'externality'( perhaps, I coined a new word) as a whole.

If we consider water instead of ball the 'internal' and 'external' speed seem to be dependent on/related to each other even if by a small amount.

BTW, this a bad example. Hope someone provides a better example.

 

[Aero_UoP]

Plz don't get irritated. It's not for the sake of argument, but I want to really understand things.

As per you, Mass and acceleration can be added again and again ( as multiplication is continued addition) but they can't be added once i.e. F=m+a is wrong.

'Why' is this so?

I don't get irritated. It just seems as if you're arguing just for the sake of argument :p (and it's not just you, I've noticed that many users do so and I can't really understand why...)

 

[Aero_UoP]

Yes well words have specific meanings, and "interaction" has a specific meaning, whether you want it to or not. It is nonsense to say "speed and mass interact" because both speed and mass are properties of some object. Objects interact, quantities do not.

It's like saying that blue and round interact when describing a blueberry. Blueberries, however, can interact, very weakly through gravitation or by contact forces. Fields can also interact, but fields are physical things just like a blueberry is. This interaction, classically, is described in terms of forces. That's what physics is.

I think I agree with dipole...
I have another example too: black and round interact when describing... a blackberry! and if you put blue into the equation you have a blue blackberry, naveen! (at an excellent price too) :p lol
just kidding :p

 

[sophiecentaur]

Plz don't get irritated. It's not for the sake of argument, but I want to really understand things.

As per you, Mass and acceleration can be added again and again ( as multiplication is continued addition) but they can't be added once i.e. F=m+a is wrong.

'Why' is this so?

The reason that the two (adding and multiplying) are not the same in your example is because of the dimensions involved. Apples and pears do not add together because they are different entities. However, as part of the set 'fruit' (they both have the property of being a fruit), their numbers can quite reasonably be added together. BUT mass and the second time-differential of displacement have nothing in common (no property) and adding them together means nothing.

If you were interested in a quantity "applepears" then you could usefully multiply the number of apples by the number of pears ( as with man-hours and foot-pounds). You might have a hard time justifying this to anyone else, though.

Maths can only be used validly in Physics when it is a meaningful representation of a Physical situation. It is all too easy to write down nonsense expressions which are perfectly OK, Algebraically.

 

[Aero_UoP]

The reason that the two (adding and multiplying) are not the same in your example is because of the dimensions involved. Apples and pears do not add together because they are different entities. However, as part of the set 'fruit' (they both have the property of being a fruit), their numbers can quite reasonably be added together. BUT mass and the second time-differential of displacement have nothing in common (no property) and adding them together means nothing.

If you were interested in a quantity "applepears" then you could usefully multiply the number of apples by the number of pears ( as with man-hours and foot-pounds). You might have a hard time justifying this to anyone else, though.

Maths can only be used validly in Physics when it is a meaningful representation of a Physical situation. It is all too easy to write down nonsense expressions which are perfectly OK, Algebraically.

Earlier, I tried to write a similar example (but with appricots and watermelons :P) explaining what you just said but at some point I found out that my English vocabulary needs some refreshing so I quit trying :P

 

[DrewD]

Thanks. That's what I meant by complex. So, it means that the formula is an approximation but not a complete picture of things.

All of physics is an approximation, even QFT.

Not arguing uselessly, I have heard that in quantum mechanics, interactions are more important than entities. In fact entities/objects are not considered to exist, until and unless they interact with the surroundings. It may be due to extremely small scale of things, but quantum mechanics experts say that it is infact the very truth of nature and size has not to do anything with it.

Stop reading Michio Kaku

I know, we are considering a classical example, but still the concept of entities, interactions and forces seems to be somewhat arbitrary and even overlapping to my fragile brain.

Which is why physics uses more well defined concepts. Force is an interaction that causes a mass to accelerate. I don't think I have ever read "entities" in a physics book, definitely not in a technical sense.

For example, Dalespam said that invariant mass has nothing to do with speed, buy how would it even exist without the 'speed' of its constituent parts (like electrons, nucleons etc.) So, does it mean there are two kinds of 'speeds' for an object. One that deals with its internal structure and the other with its 'externality'( perhaps, I coined a new word) as a whole.


If we consider water instead of ball the 'internal' and 'external' speed seem to be dependent on/related to each other even if by a small amount.

BTW, this a bad example. Hope someone provides a better example.

In some ways there is internal and external speeds. The internal motion does not effect the external. Technically, the internal energy does change the mass a negligible amount, but the difference is so minor that it not only can, but should, be ignored.

However, I don't think any of this has anything to do with your original question which was answered. The reason $F=ma$ and not $F=m+a$ is
1) The units don't match if you try to add
2) That's the way it works because experiments say so. We could make up a new law of motion, but that would be stupid because what we have works and $F=m+a$ does not.

 

[sophiecentaur]

As per you, Mass and acceleration can be added again and again ( as multiplication is continued addition) but they can't be added once i.e. F=m+a is wrong.

/QUOTE]

Not correct. Multiplication of m by a is adding mass to itself a number of times, governed by the numerical value of acceleration. That's not adding mass to acceleration.

 

[Naveen3456]

Not correct. Multiplication of m by a is adding mass to itself a number of times, governed by the numerical value of acceleration. That's not adding mass to acceleration.

Now, as is logical to assume, an equation that is correct always gives the correct answer.

So, why has this answer that force is 'adding mass to itself a number of times, governed by the numerical value of acceleration' emerged from this equation.

Does force mean 'this' also in the' physical sense' for a body(i.e. Force acting on a body = adding mass of the body to itself a number of times, governed by the numerical value of acceleration for that body). If we add mass of a body to itself say 3 times won't it become triple the mass, i.e. will the body really have 3 times its previous mass? This seems out of this world.



I

 

[Aero_UoP]

The mass of the body will be the same because the material of which the body is made of is still the same! THE FORCE ACTING ON THE BODY WILL BE (say) 3 TIMES THE MASS OF THE BODY!What is that you don't get?

EXAMPLE: W=mg
W= weight in N (force)
m= mass in Kg (property)
g= gravitational acceleration (constant)

What does this tell us? That the force acting on the body due to Earth's gravity (the weight of the body) is g times its mass. That is, multiply the body's mass by 9.81 (or if you are a masochist with low IQ just add the body's mass to itself 9.81 times) and you have the FORCE acting on the body. The body's mass will be the same either on Earth or on the moon or on saturn or wherever!

 

[sophiecentaur]

Now, as is logical to assume, an equation that is correct always gives the correct answer.

So, why has this answer that force is 'adding mass to itself a number of times, governed by the numerical value of acceleration' emerged from this equation.

Does force mean 'this' also in the' physical sense' for a body(i.e. Force acting on a body = adding mass of the body to itself a number of times, governed by the numerical value of acceleration for that body). If we add mass of a body to itself say 3 times won't it become triple the mass, i.e. will the body really have 3 times its previous mass? This seems out of this world.
I

If you want a conversation about how Maths relates to the real world then it is only fair to stick to the rules of Maths. You seem to have deliberately interpreted my 'verbal' description of the process of Multiplication in an unfortunate way. Of course, adding three times the mass, to itself, will produce a final result of Four times the mass. That's why I used the words "governed by" and not "proportional to", because I was being deliberately unspecific (was that not obvious?).
But the fact is that the axioms of Arithmetic are pretty well established and the way that Areas, Weights, Energy, Volts and all the others fit in with the Multiplication operation is hardly in question. Nor is the the fact that similar quantities can be added together.
I really can't be sure why you are arguing about this. You just need to go through some of the School definitions in Science (and Finance) and see how the two operations work in different ways. Start by assuming it has to be OK and work towards that, rather than trying, fruitlessly, to show we all got it wrong. Maths (mine included, in this case) is very very consistent.

 

[D H]

To me, there is no point in asking "why multiplication and not addition" since multiplication IS addition... 4*5 = 5+5+5+5 or 4+4+4+4+4, the result is the same.

This is a side track, and it isn't true in general. How do you add pi copies of √2?

See http://www.maa.org/devlin/devlin_06_08.html and the follow-up article http://www.maa.org/devlin/devlin_0708_08.html .

 

[Aero_UoP]

How does our personal computer perform multiplication? :)

 

[Dale]

Dalespam said that invariant mass has nothing to do with speed, buy how would it even exist without the 'speed' of its constituent parts (like electrons, nucleons etc.)

This is irrelevant. If a quantity X is defined to depend only on a set of quantities Y then it does not depend on Z. Even if all objects for which X is defined also have Z the value of Z is irrelevant to determining X, which is fully determined by Y.

I simply do not know of any definition of mass which depends on acceleration. It is useless to ask "why"; the answer is simply "by definition" and there is no deeper reason. if you wish to define the "Naveen mass" and make it dependent on acceleration then you can, but "Naveen mass" would be a new concept of dubious utility.

 

[D H]

How does our personal computer perform multiplication? :)

Not by repeated addition. This is an O(2ⁿ) algorithm, where n is the number of bits in the multiplier. That would be ridiculously inefficient. The first computers used the binary equivalent of long multiplication. This is an O(n²) algorithm. In 1950, Andrew Booth developed such an O(n²) for multiplying a pair numbers represented using two's complement.

That is not what computers use nowadays. A decade after Booth developed his algorithm, Anatolii Karatsuba developed an even faster O(n^log₂3) method. Karatsuba's algorithm has been further refined, with the latest refinement being embodied in US patent 7,930,337, "Multiplying two numbers".

 

[sophiecentaur]

This is a side track, and it isn't true in general. How do you add pi copies of √2?

See http://www.maa.org/devlin/devlin_06_08.html and the follow-up article http://www.maa.org/devlin/devlin_0708_08.html .

I think there is more of a parallel than you suggest - if you think in terms of the convergent series, which use only integers and fractions but which yield irrational and trancendental numbers in their limit. Also, an indefinite integral has no direct connection with 'adding up' but we are quite happy to turn the handle and get answers with computers when we use those numbers.
But this thread is too loose to get anywhere further, I think. It introduced another of those "why" questions which is guaranteed to spoil your day.

 

[Dale]

In physics the units don't work out for multiplication to be repeated addition. It is true that 5*3 is 5+5+5, but that requires 3 to be dimensionless. It makes sense to "do something 3 times", but it doesn't make sense to "do something 3 kg times".

 

[Aero_UoP]

Not by repeated addition. This is an O(2ⁿ) algorithm, where n is the number of bits in the multiplier. That would be ridiculously inefficient. The first computers used the binary equivalent of long multiplication. This is an O(n²) algorithm. In 1950, Andrew Booth developed such an O(n²) for multiplying a pair numbers represented using two's complement.

That is not what computers use nowadays. A decade after Booth developed his algorithm, Anatolii Karatsuba developed an even faster O(n^log₂3) method. Karatsuba's algorithm has been further refined, with the latest refinement being embodied in US patent 7,930,337, "Multiplying two numbers".

I don't know this guy Karatsuba, it's out of my field of interest and/or expertise but I was taught that the only mathematical operation a CPU can perform is the addition... maybe it's wrong, maybe I got it wrong. The problem here though, it's neither me nor computers. You're missing the point trying to prove me wrong instead of actually answering the initial question which I noticed you ignored completely...
Come on then, answer the initial question, tell us your opinion about the subject and don't focus on what I say...

 

[sophiecentaur]

In physics the units don't work out for multiplication to be repeated addition. It is true that 5*3 is 5+5+5, but that requires 3 to be dimensionless. It makes sense to "do something 3 times", but it doesn't make sense to "do something 3 kg times".

But does it make any more sense to multiply the length of a carpet '2 metres times', to find the area? I reckon this is mainly a matter of familiarity. We seem to be quite happy, these days, with the 'twoness' or 'threeness' of two or three ducks, apples or kg, these days but I believe the numbering system started off quite irregular. We still have a brace of pheasants, a pair of shoes and a few others. The whole thing about connecting Maths with real life is very 'deep'.

 

[dipole]

As per you, Mass and acceleration can be added again and again ( as multiplication is continued addition) but they can't be added once i.e. F=m+a is wrong.

/QUOTE]

Not correct. Multiplication of m by a is adding mass to itself a number of times, governed by the numerical value of acceleration. That's not adding mass to acceleration.

No, because then force would still have the units of mass. That method of multiplication is only an arithmetic trick, and it's not the real definition of multiplication.

 

[jbriggs444]

No, because then force would still have the units of mass. That method of multiplication is only an arithmetic trick, and it's not the real definition of multiplication.

If you are adding mass a number of times where the number of times is given by the numerical value of acceleration then you need a conversion factor from acceleration to number of times. That's where the units of acceleration come into the picture. That's why the result has units of acceleration in it.

[Or at least that's how I'd strain to preserve the intuition of multiplication as repeated addition in the realm of real-valued multiplication of quantities with units]