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Please explain the sign

  1. Jul 11, 2013 #1
    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.
  2. jcsd
  3. Jul 11, 2013 #2


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    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".
  4. Jul 11, 2013 #3
    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?
  5. Jul 11, 2013 #4
    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.
  6. Jul 11, 2013 #5
    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.
  7. Jul 11, 2013 #6


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    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.
  8. Jul 11, 2013 #7
    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.
  9. Jul 11, 2013 #8


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    Good point.
  10. Jul 12, 2013 #9
    If speed and mass can 'interact' ( more the speed more the mass), why can't mass and acceleration interact.

    Now, plz 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.
  11. Jul 12, 2013 #10
    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.
  12. Jul 12, 2013 #11
    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.
  13. Jul 12, 2013 #12
    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.
  14. Jul 12, 2013 #13


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    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.
  15. Jul 12, 2013 #14
    That is absolutely correct, but you don't need the quotations.
    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.

    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.

    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.
  16. Jul 13, 2013 #15
    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?
  17. Jul 13, 2013 #16
    Thanks. That's what I meant by complex. So, it means that the formula is an approximation but not a complete picture of things.

    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:approve:) 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.
  18. Jul 13, 2013 #17
    I don't get irritated. It just seems as if you're arguing just for the sake of arguement :p (and it's not just you, I've noticed that many users do so and I can't really understand why...)
  19. Jul 13, 2013 #18
    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
  20. Jul 13, 2013 #19


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    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. :wink:

    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.
  21. Jul 13, 2013 #20
    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
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