B How and why can multiplication combine physical quantities?

[Haris045]

TL;DR Summary

how and why can multiplication combine physical quantities to form a new physical quantity, to make an equation?

I am on a journey to not just understand how to manipulate physics equations but to understand why they work , and how they describe physical phenomena.

I understand how division combines physical quantities. I have this much physical quantity 'per' this much physical quantity. It puts 2 physical quantities in a ratio to describe a physical phenome e.g. velocity=distance/time, It makes sense.

I understand what multiplication does numerically. But I don't understand how it can combine physical quantities to form a new one.

Intuitively I see that multiplication means 'applied over' e.g. W=F*d a 'force applied over a distance' or d=s*t 'a speed applied over a certain time'. mass = density*volume 'a density applied over a volume'. But why

1)So my first question is why does multiplication have this property of combining physical quantities and forming a new one, what is the multiplication doing to these physical quantities.

2)How does the multiplication relate back to the physical phenomena (how can it combine 2 physical quantities to describe a physical phenomena using an equation)

3)what does it show in an equation (about how it is combing the physical quantises to form a new on) , what does it mean in the equation , what is it doing to the physical quantises to form a new one?

4)so I can see that multiplication means 'applied over' but why does it mean this?

Thank you for the help and sorry if I am being ignorant.

 

[berkeman]

For whole numbers, it means how many "times" you have a quantity. So if I count a quantity of "A things" "6 times", I get 6*A of the things. Think about a 2-d rectangle made up of a whole number of smaller rectangles. There are "A" rectangles laid out in a row, and "6" rows of those A rectangles...

EDIT -- similar to images like this one:

https://mathisvisual.com/wp-content...ompts.129-7-groups-of-5-squares-1024x576.jpeg

 

[kuruman]

Summary: how and why can multiplication combine physical quantities to form a new physical quantity, to make an equation?

I understand how division combines physical quantities. I have this much physical quantity 'per' this much physical quantity. It puts 2 physical quantities in a ratio to describe a physical phenome e.g. velocity=distance/time, It makes sense.

To continue your example, if velocity is distance per unit time, then the distance covered in a time T would be velocity times T. That should be obvious to anyone who has traveled in a vehicle. It should also be obvious to you that if your pay per hour is $15.00, your pay after working 10 hours would be $150.00. Note that hourly pay is a different physical quantity from total pay with different units.

Carrying that one step farther, if acceleration is net force per unit mass, then the net force on a 10-kg mass with acceleration 15 m/s² would be 150 N. $$F_{\text{net}}=ma \Leftrightarrow a=\frac{F_{\text{net}}}{m}.$$You can do this with all products, so if you understand division, you should be able to understand multiplication.

 

[Baluncore]

e.g. velocity=distance/time, It makes sense.

Therefore, velocity * time = distance. It makes sense.

https://en.wikipedia.org/wiki/Dimensional_analysis

 

[pbuk]

1)So my first question is why does multiplication have this property of combining physical quantities and forming a new one

It doesn't. The mathematical relationships we write describe relationships we observe between physical quantities, nothing more and nothing less.

what is the multiplication doing to these physical quantities.

Multiplication is an abstract operation, it cannot "do" anything in the real world.

2)How does the multiplication relate back to the physical phenomena

It describes the relationship between them.

3)what does it show in an equation (about how it is combing the physical quantises to form a new on) , what does it mean in the equation , what is it doing to the physical quantises to form a new one?

Again these questions are not meaningful: multiplication is just multiplication. Similarly division is just division

4)so I can see that multiplication means 'applied over' but why does it mean this?

No it doesn't, it means multiplication. Examples where the words "applied over" don't fit:

• Force = mass times acceleration $(F = m a )$
• Potential difference = current times resistance $(V = IR)$
• Area = width times height

Also, division just means division. Examples where the concept of a rate "per" somthing doesn't fit:

• Newtonian gravity $\frac{GMm}{r^2}$
• Current = potential difference divided by resistance $\left ( I = \frac V R \right )$
• Frequency = wave speed divided by wavelength $\left ( f = \frac v \lambda \right )$

 

[Andy Resnick]

Summary: how and why can multiplication combine physical quantities to form a new physical quantity, to make an equation?

I am on a journey to not just understand how to manipulate physics equations but to understand why they work , and how they describe physical phenomena.

Many students struggle with compound units for the very reason you mention- units seem to obey rules of multiplication and division, but while 3/5 is readily understood to be a number, feet/year is clearly not a number. And it gets worse- for example, the units of magnetic field are [mass]/[time* coulomb], whatever that means.

I tell my students ( college & univ. Intro physics) that units are a principal difference between science and math. Unit analysis helps in problem-solving strategies (all forms of energy have units ML²/T², for example).

Does that help?

 

[DaveC426913]

Multiplication is an abstract operation, it cannot "do" anything in the real world.

Try saying that over in the Biology forum...

 

[kuruman]

Try saying that over in the Biology forum...

Be fruitful and multiply?

 

[robphy]

Summary: how and why can multiplication combine physical quantities to form a new physical quantity, to make an equation?

I am on a journey to not just understand how to manipulate physics equations but to understand why they work , and how they describe physical phenomena.

I understand how division combines physical quantities.
[snip]

I understand what multiplication does numerically. But I don't understand how it can combine physical quantities to form a new one.
[snip]

I think one should not think of these operations as "forming new quantities."
Instead,
one should understand (for example)

• the "work done by a constant force along a straight path" $W$ is a physical quantity
  that is proportional to the force along the path $\vec F_x$
  and proportional to the displacement along the path $\vec \Delta x$: thus, $W=(F_x)(\Delta x)$.
• Similarly, a "constant velocity"
  is proportional to the displacement traveled $\Delta x$
  and inversely-proportional to the elapsed-time $\Delta t$: thus, $v=(\Delta x)\left(\frac{1}{\Delta t}\right)$.

(While one can perform the mathematical operation $F_x (\Delta x)^3$,
it has no immediate physically-meaningful interpretation.)

4)so I can see that multiplication means 'applied over' but why does it mean this?

In some situations,
where the quantity is an integral (a sum of products where one factor may be varying),
one uses the colloquial description "applied over".
For example, if (in the example above) the work done is due to a variable force,
one is trying to describe the force (a dependent variable) as the position (an independent variable) varies.
But, as mentioned above by @pbuk, there are some situations where that is not the case.
(However, area could be described as a variable height along locations on the base.)