Work done = force x displacement?

[Ayesha_Sadiq]

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.

 

[UltrafastPED]

The vector dot product is the projection (like a shadow) of the one vector onto the other.

If you started with two unit vectors, that would be the end of it. To include general vectors we then multiply this "projection" by the magnitudes of the two vectors.

See http://mathworld.wolfram.com/DotProduct.html
And http://betterexplained.com/articles/vector-calculus-understanding-the-dot-product/

 

[Ayesha_Sadiq]

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.

 

[A.T.]

But I still couldn't get what is implied when we multiply two physical quantities.

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

Then wouldn't a 5N times 3m be like 15 Newtons total force applied in 3 unit distances of 1m each.

No, because it's 5N * 3m not 5N/m * 3m.
5 meters * 3 meters = 15 square meters
5 Newton * 3 meters = 15 Newton metersFor a nice visal introduction into these concepts I recommend this book:
Thinking Physics by Lewis Carroll Epstein

 

[Phy_enthusiast]

In physics there is no thing as multiplication,there is only dot product or cross product.

 

[Ayesha_Sadiq]

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?

In physics there is no thing as multiplication,there is only dot product or cross product.

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.

 

[A.T.]

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.

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

Not of two physical quantities, would it?

Velocity and time are both physical quantities.

Oh. But then what would division be in physics?

100 candies / 50 candies per packet = 2 packets
100 miles / 50 miles per hour = 2 hours

 

[Ayesha_Sadiq]

Hmm. I think I see it. This is how it works then. Thank you. =)

 

[sophiecentaur]

In physics there is no thing as multiplication,there is only dot product or cross product.

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.

 

[Ayesha_Sadiq]

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.

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.

 

[sophiecentaur]

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.

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.

 

[BruceW]

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.

 

[A.T.]

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.

Everything about integers is more intuitive than continuous numbers. But multiplication of continuous numbers is not specific to physics.

 

[BruceW]

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.

But multiplication of continuous numbers is not specific to physics.

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.

 

[A.T.]

I didn't really mention physics because I thought that the OP's issue was more with the mathematical intuition.

Maybe, but she asks specifically about multiplication of physical quantities in the physics forum

 

[BruceW]

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.

 

[Ayesha_Sadiq]

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...?

 

[sophiecentaur]

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.

 

[BruceW]

So then, real, continuous numbers are multiplied that way, and it doesn't really mean what multiplication means generally.

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).

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...?

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.

 

[sophiecentaur]

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.

 

[Ayesha_Sadiq]

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...
.

Yes. Physical quantities do take real values, but we can only measure them to a specific degree of precision. Got no tools or instruments for that. =P No division on an instrument could be the smallest, you could go on making the smallest divisions smaller and smaller, and increasing it's precision. So then, a real value like that, would be a continuous number? And we don't perceive multiplications of continuous numbers like we do for integers?

 

[Ayesha_Sadiq]

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.

Indivisible pebble, eh? =P
Well, the quantisation of energy for electrons certainly amazed people.
I am still in college, and working towards the day I can tell to a youngie that I study the complicated working details of things all the time too. But the working of algebra in a physics context, I did not understand, and I asked.

 

[Phy_enthusiast]

Your question is similar to a intuitive question as why 1+1=2,these numerals since their invention have been useful both mathematically and physically,you can lead us to new system by inventing something better than this.

 

[BruceW]

Yes. Physical quantities do take real values, but we can only measure them to a specific degree of precision. Got no tools or instruments for that. =P No division on an instrument could be the smallest, you could go on making the smallest divisions smaller and smaller, and increasing it's precision. So then, a real value like that, would be a continuous number?

when I said 'real number' I mean the mathematical definition 'real number'. They have certain nice properties. In a sense, they 'fill up' the number line very tightly. So, for example, the set of all possible fractions is also infinite, but they don't fill up the number line as tightly as the real numbers do. So anyway, the 'real numbers' is the strict mathematical name for what we would usually call 'continuous numbers' in everyday talk.

And we don't perceive multiplications of continuous numbers like we do for integers?

I agree, we don't perceive multiplication of continuous numbers like we do for integers. For integer multiplication, we have the 'candy intuition' that you mentioned. But for continuous numbers, I don't think we have any kind of intuition like that.