Meaning of Multiplication and Division in Physics

[valenumr]

What @ergospherical discussed is also Newtonian mechanics, just a bit advanced

Fluid mechanics is a bit beyond what Newton had in mind.

 

[meekerdb]

I agree with the explanations by David Lewis and gleem. But I think they missed a point that acceleration is not only dimensionally different, it's also a vector, and so is F. So F=(m1+m2)a makes sense but F=m(a1+a2) doesn't.

I also noticed that no one suggested a book on this subject. There is an excellent one by Hart "Multidimensional Analysis" which will probably show you that it's more complicated than you thought and is commonly abused (especially by engineers).

 

[Mark44]

But I think they missed a point that acceleration is not only dimensionally different, it's also a vector, and so is F. So F=(m1+m2)a makes sense but F=m(a1+a2) doesn't.

I don't see why the latter form doesn't make sense. For example, you could have two separate forces being applied to an object of mass m -- $F_1$ and $F_2$. We could have $F = F_1 + F_2 = ma_1 + ma_2 = m(a_1 + a_2)$.

 

[sophiecentaur]

F=m(a1+a2) doesn't.

Can you help me with why it doesn't when it appears to me to be fine? Or is there an 'always' that needs to be added somewhere.

 

[meekerdb]

First thing to notice is that force and acceleration are vectors, mass is a scalar. So the multiplication is in vector space. Multiplying a vector by a scalar is quite different from multiplying two vectors which may be a dot product or a cross product. But you can't add things with different units and sometimes you can't add things with the same units, e.g. torque+energy.

Second it's a way of defining force. There's an excellent book by Hart "Multidimensional Analysis" which shows how dimensions are used (and abused) in physics and engineering.

 

[Dale]

Multiplication is higher form of addition.Division is higher form of subtraction.

What does F=ma means. m is in kg. a in ms-2.How can both multiply.They cannot add or subtract.

Why any quantities can multiply or divide,contrary to adding and subtracting.

It is not generally correct that multiplication is simply a higher form of addition. When you were in elementary school this is indeed how the topic is introduced, but that is simply because elementary school students only know addition and are not mentally prepared for an axiomatic approach to math. In the axiomatic approach, multiplication is not constructed from addition, but a set is proposed and multiplication is defined abstractly as an operation on elements of that set with certain axiomatically imposed properties. It is those axioms that define multiplication, not the elementary-school construction.

The reason for that is to allow the extension of the concept of multiplication to other sets besides the real numbers. For example, a mathematical field (not to be confused with a physical field) is a set that has addition, subtraction, multiplication, and division that all work just like real numbers. So, of course, the real numbers are a field, but so are the rational numbers and the complex numbers. It doesn't make sense to think of adding $5+2i$ to itself $3-i$ times, but for complex numbers $(5+2i)(3-i)$ is a perfectly valid operation defined axiomatically. Other more exotic sets include algebraic functions, where it likewise makes sense to consider multiplication between two functions, but it does not make sense to consider it as adding one function to itself a function number of times.

In the case of dimensional analysis the appropriate abstract mathematical formalism is that of vectors, which is described in section 2 here: https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/. Vectors have their own scalar multiplication and vector addition axioms.

So, specifically for your question we have a one-dimensional real vector space $V^M$ of all possible masses, so $5 \text{ kg}$ is the vector in the one-dimensional mass space which is the scalar product of $5$ times the named vector $\text{kg}$. Similarly we have one-dimensional real vector spaces $V^L$ and $V^T$ for all possible lengths and times. Then, we can use a standard tensor product to construct a new vector space $V^{LT^{-2}}=V^L \otimes V^{T^{-1}}\otimes V^{T^{-1}}$ which is the space of all possible (1D) accelerations, and $V^{MLT^{-2}}=V^M \otimes V^{LT^{-2}}$ which is the space of all possible (1D) forces. That is what is implied by something like $F=ma$.

So then the issue with addition of dimensionally inconsistent quantities is that you are trying to add vectors of different vector spaces, so that doesn't make any sense because no such operation is defined. Whereas multiplication of dimensionally inconsistent quantities makes sense because you use it to create a new vector space using the standard tensor product.

 

[Mark44]

First thing to notice is that force and acceleration are vectors, mass is a scalar. So the multiplication is in vector space.

Yes, and so what? One of the axioms of any vector space has to do with the multiplication of a vector by a scalar.

Multiplying a vector by a scalar is quite different from multiplying two vectors which may be a dot product or a cross product.

The vector space axioms don't even define the multiplication of two vectors. The only operations defined in the vector space axioms are addition of vectors and multiplication by a scalar.

But you can't add things with different units and sometimes you can't add things with the same units, e.g. torque+energy.

This has nothing to do with your earlier statement in post #53 that $F = m(a_1 + a_2)$ doesn't make sense. I gave you a scenario in which it does make sense.

 

[Mark44]

But you can't add things with different units and sometimes you can't add things with the same units, e.g. torque+energy.

Torque and energy have different units. And if you meant force and torque, they have different units, as well.

 

[ergospherical]

torque and energy don't have the same units

They do

 

[jbriggs444]

They do

They have the same dimensionality. Not the same units.

 

[ergospherical]

They have the same dimensionality. Not the same units.

Having equal dimensionality implies they can be expressed in the same units…

It is usually the Nm (Newton metre)

 

[jbriggs444]

Having equal dimensionality implies they can be expressed in the same units…

I distinguish between units and dimensions.

 

[ergospherical]

I distinguish between units and dimensions.

Indeed, because they are different things… (loosely, dimensionality is associated with an [infinite] set of possible units…)

Nonetheless torque and energy share the same set of units

dW = F . dr
t = r x F

 

[jbriggs444]

Indeed, because they are different things… (loosely, dimensionality is associated with an [infinite] set of possible units…)

Nonetheless torque and energy share the same set of units

dW = F . dr
t = r x F

With radians (arguably a dimensionless unit) as a conversion factor between torque applied and energy expended through a unit rotation angle).

 

[hutchphd]

This discussion has given me a headache. But it did remind me of a favorite quote (seems apropos):

"Under capitalism, man exploits man. Under communism, it's just the opposite."

John Kenneth Galbraith

.

 

[dromarand]

Interesting conversation, I think that the criterion with unit is very important.I actually think that after you have this fundamental defining formulas such as average speed=total distance/total time, you can obtain other formulas such as Galileo's formula(v square=v0 square+2ad) just doing the math,without any trouble. But,in my view, the problem is still with this fundamental equations. For instance, why the speed is distance per time and not distance multiplied with time? I see the proportionality between time and distance (if speed is constant), but I still cannot perfectly understand this problem.

The ideal model of the discussed experiment assumes that the first falling snowflakes (constant vertical precipitation velocity, constant density) touch the ground at the moment of starting the measurement of the dozer movement (horizontal, uniform rectilinear motion).
Which dozer, under the same conditions, will pick up more snow; the one who traveled 40 m in 30 s, or the one who traveled 50 m in 20 s?
The entire experience (with two dozers) repeated at a other snowfall speed will not change which of them will sweep more snow.
Imagine a right triangle against the falling snow. The triangle moves vertically down with the snow. One cathetus is the distance s, the other cathetus - height is proportional to the time t. The movement of the point of intersection of the hypotenuse with the ground line is the movement of the dozer. The precipitation pushing capacity x=st/2 could be a physical quantity, the measurement unit of which would be a meter-second.

 

[sophiecentaur]

I distinguish between units and dimensions.

Different animals entirely; I agree
'They' had to invent dimensional analysis to take care of this and other problems. As far as I know, the US has to accept the same convention for dimensions as we use in UK, despite their continuing love affair with feet.

 

[jbriggs444]

Different animals entirely; I agree
'They' had to invent dimensional analysis to take care of this and other problems. As far as I know, the US has to accept the same convention for dimensions as we use in UK, despite their continuing love affair with feet.

Feet go up to twelve. That's two more.

With apologies to Spinal Tap.

 

[sophiecentaur]

Feet go up to twelve. That's two more.

With apologies to Spinal Tap.

I wonder where that movie would be now without the volume control gag.

One afternoon, I watched it with my teenage son in a cinema in Brighton (UK). There were just three of us in the audience; Tom and I and a little old man on the other side of the gangway. Without the '11' gag, he would probably not have been there.

 

[Herman Trivilino]

Why any quantities can multiply or divide,contrary to adding and subtracting.

Let's take a simple equation from rectilinear motion: $\Delta x=vt$. In other words, displacement equals the product of the velocity and the time. You know that multiplication is a series of additions. So if you have a velocity of $2 \ \mathrm{m/s}$ for 3 seconds you travel 2 meters in the first second, 2 meters in the next second, and finally 2 meters in the third second. $2 \ \mathrm{m}$+$2 \ \mathrm{m}$+$2 \ \mathrm{m}$=$6 \ \mathrm{m}$.

Note that you never add two quantities with different units.

 

[sophiecentaur]

A question that gets my mind going in ever decreasing circles is "what is it about three that allow it to be applied to so many quantities?" . There's a 'threeness' that is outside the physical world. We do our three times table but when we explain it to a child we talk about three apples and four boxes of three apples etc.

 

[jbriggs444]

A question that gets my mind going in ever decreasing circles is "what is it about three that allow it to be applied to so many quantities?" . There's a 'threeness' that is outside the physical world. We do our three times table but when we explain it to a child we talk about three apples and four boxes of three apples etc.

There are only four many numbers. One, two, three and many.

 

[hmmm27]

There are only four many numbers. One, two, three and many.

I haven't the reference material on hand, but isn't it "One, Two, Many, Lots" ?