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Work done = force x displacement?

  1. Jul 12, 2014 #1
    When two people have three candies each, the total number of candies is six, 2x3=6. So what does multiplication imply when we say work done = force x displacement? I know it's a dot product and the force is in the direction of the displacement.
     
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  3. Jul 12, 2014 #2

    UltrafastPED

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  4. Jul 12, 2014 #3
    Thank you, those links were very helpful. But I still couldn't get what is implied when we multiply two physical quantities. Like 2 packets have 50 candies, then the total candies are 100. We multiply the numbers. Or we could add 50 two times, like 50+50, it's the same thing. But when we are multiplying a force by a distance then does that mean the total force in those unit distances (example, 5m is made up of five 1m units of distances, but I know it's arbitrary, the choice of the smallest division). Then wouldn't a 5N times 3m be like 15 newtons total force applied in 3 unit distances of 1m each. That would be the total force exerted in that distance, not the work done, would it? Or what would it mean when we multiply two masses like in gravitational force's calculation.
     
  5. Jul 13, 2014 #4

    A.T.

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    Start with something simple like: velocity * time = distance
    50 candies per packet * 2 packets = 100 candies
    50 miles per hour * 2 hours = 100 miles

    No, because it's 5N * 3m not 5N/m * 3m.
    5 meters * 3 meters = 15 square meters
    5 Newton * 3 meters = 15 Newton meters


    For a nice visal introduction into these concepts I recommend this book:
    Thinking Physics by Lewis Carroll Epstein
     
  6. Jul 13, 2014 #5
    In physics there is no thing as multiplication,there is only dot product or cross product.
     
  7. Jul 13, 2014 #6
    Thank you for the reference of the book. I'll try to get my hands on it as soon as possible. But then, in this case, wouldn't it be a somewhat 'rate' multiplication. Like, I am covering 2 metres per second, so how many metres I cover in 10 seconds would be a sort of 'rate' multiplication. Not of two physical quantities, would it?


    Oh. But then what would division be in physics? I am guessing this is how relationships are developed between physical quantities. How physical quantities are sort of quantified relative to one another, and many factors considered at once. And it just all somehow works out.
     
  8. Jul 13, 2014 #7

    A.T.

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    Your example with candy is also a "rate" multiplication: candies per person, candies per package.

    Velocity and time are both physical quantities.

    100 candies / 50 candies per packet = 2 packets
    100 miles / 50 miles per hour = 2 hours
     
  9. Jul 13, 2014 #8
    Hmm. I think I see it. This is how it works then. Thank you. =)
     
  10. Jul 13, 2014 #9

    sophiecentaur

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    That is a bit of a sweeping statement. If you multiply heat capacity by temperature change, to get heat input, you are just multiplying scalars and there is no 'dot' or 'cross' involved.
    The original concept of Multiplication would have been repeated integer addition. e.g. ten sheep with four legs gives you forty legs altogether. sheep plus sheep plus sheep etc.
    Once the mechanics of this was established, it was found that you got the right answer if non-integers were involved. The history of Maths is full of instances of taking a basic operation and then finding it works in complicated situations.
    The use of Maths to model the real world is an interesting point of philosophical discussion.
     
  11. Jul 13, 2014 #10



    Thank you, I think this is what I was looking for. So, it's just Mathematics can describe relationships between physical quantities too, without it necessarily being a 'multiplication' of some sort.
     
  12. Jul 13, 2014 #11

    sophiecentaur

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    There seems (to me) to be a dichotomy between discrete (integer) operations and continuous operations. Take the exponential operator, for instance. You can square, cube etc an integer number and it is easy to grasp what goes on. One step further - into fractional indices and logarithms - and, despite the fact that the same rules apply, it is hard to believe it's the same thing.
     
  13. Jul 13, 2014 #12

    BruceW

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    Ayesha - in your first post, (2 people with 3 candies each), you can think of this in terms of sets and elements. Consider each candy to be a unique element (they are all different from each other). Then each person represents a disjoint set that contains 3 elements each. So the union of these two sets contains 6 elements. So your interesting 'intuitive' definition of multiplication can be closely related to the concept of the cardinality of sets. (cardinality means how many elements).

    So, is it fairly easy to extend this to multiplication of continuous numbers? No, I don't think so. And so maybe this is the reason why multiplication of continuous numbers is much less intuitive than multiplication of integers, which can be thought of in terms of unions of sets.

    edit: I should also have said, the reason why this works is because each person has the same number of candies. But I'm guessing you know that already. So the main idea is that the cardinality of the union of disjoint sets of the same size is equivalent to the cardinality of each set, multiplied by the number of sets. Which is a very intuitive concept.
     
    Last edited: Jul 13, 2014
  14. Jul 13, 2014 #13

    A.T.

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    Everything about integers is more intuitive than continuous numbers. But multiplication of continuous numbers is not specific to physics.
     
  15. Jul 13, 2014 #14

    BruceW

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    the integers make a ring, the reals make a field. In some ways a field is more intuitive than a ring, I think. Specifically, the multiplicative inverse. But yeah, in a lot more ways I think the integers are more intuitive.

    neither is multiplication of the integers... I don't understand what you mean here. I didn't really mention physics because I thought that the OP's issue was more with the mathematical intuition.
     
  16. Jul 13, 2014 #15

    A.T.

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    Maybe, but she asks specifically about multiplication of physical quantities in the physics forum
     
  17. Jul 13, 2014 #16

    BruceW

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    hmm. good point. Well, i'd say that using the 'candy intuition' gives us a good intuition for multiplication of integers generally, even if our application is mathematical or physical. But I guess it's true that the 'candy intuition' can only be directly used to solve our specific problem when the problem is effectively a counting problem.
     
  18. Jul 13, 2014 #17
    It was a bit of both, but I actually wanted to know with relation to physical quantities. But this is good insight about continuous numbers. So then, real, continuous numbers are multiplied that way, and it doesn't really mean what multiplication means generally. And I think physical quantities do take a real value, as we cannot measure them more than a specific degree of precision. And their actual value is really not what we measure, only a very, very close approximation, which works. I hope I am getting this right...?
     
  19. Jul 13, 2014 #18

    sophiecentaur

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    We don't 'measure anything' with total accuracy as there is always system noise of one form or another. This mostly gives us the ability to predict what will happen - unless we are dealing with a chaotic system.
    There are some things which we can 'count', however and that can give complete accuracy.
     
  20. Jul 14, 2014 #19

    BruceW

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    I agree that multiplication of the reals does not have a direct link to your 'candy intuition'. I think most mathematicians would say the 'candy intuition' is one specific example of multiplication, and multiplication of continuous numbers is another specific example. I think 'multiplication' in maths is used as a very loose term. For example, multiplication of square matrices would be another specific example of a kind of multiplication.

    So the word 'multiplication' can mean a lot of things. I think the most general definition of 'multiplication' is the definition that you find when people talk about a ring. So we have a set with some 'addition' and 'multiplication' defined. The set is a monoid under multiplication and multiplication distributes over addition. This defines the most general kind of multiplication (I think).

    Surely our ability to measure to arbitrary precision is a different issue? It looks like you are implying that if we could measure to arbitrary precision, then physical quantities would not take real value...

    edit: I think what SophieCentaur said is right. If we have a physical quantity that takes on real value, and we use some arbitrary units, then we will generally get a number with an arbitrarily large number of decimal places. So we can't even write down our measurement, because we have only a finite amount of pen and paper.
     
    Last edited: Jul 14, 2014
  21. Jul 14, 2014 #20

    sophiecentaur

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    This problem with Multiplication is only the tip of the iceberg; the whole thing gives cause for concern. Everywhere we involve Calculus (Greek of an indivisible pebble, btw) we just accept the implication that the variable we are dealing with is continuous and often we further assume it is differentiable over its whole range etc. etc. Imagine someone came up with proof about the granularity of the Universe - we could still 'assume' continuity for most of our work.
    I think the best idea is to separate the thing into two areas of worry. First of all, get to grip with the Maths and 'make it work' for you. Then, when you have time to spare, get into Mathematical Analysis - or whatever they call it these days and see how everything that (legit) Mathematicians do is justifiable, going back to simple stuff like 0+1 = 0 and 1X1 = 1. Trying to tie those two things together at every step can really spoil your day.
    PS You are quite happy to sit and watch TV and use your computer and you probably have very little idea of the working details of those things. We do it all the time. Chill.
     
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