What does the multiplication between two units mean?

[Ehden]

It seems like division between two units is a simple intuitive concept to grasp, such as velocity, for every interval of time, a particle travels a certain distance. However, I've always had trouble trying to find an intuitive sense for multiplication between two units, e.g. what exactly does kg*m in the unit Newton intuitively mean?

 

[fresh_42]

It's $kg \cdot \frac{m}{s^2}$ and means an accelerated $1 \, kg$ mass.
There isn't always an intuitive concept, e.g. units of some constants are sometimes pretty non-intuitive.
On the other hand $m^2$ or $m^3$ can be grasped naturally.

As you've said, division means "per". Multiplication perhaps can be thought of "applied to".

 

[cnh1995]

It seems like division between two units is a simple intuitive concept to grasp, such as velocity, for every interval of time, a particle travels a certain distance.

Division of two physical quantities is a mathematical description of rate of change of one quantity w.r.t. the other. For example, m/s gives rate of change of displacement with respect to time, which we call as velocity. Division of physical quantities can be described using the derivative form i.e. v=dx/dt or a=dv/dt etc.
Multiplication of two physical quantities can be described using the integral form. It gives the area under the graph relating the two quantities.
e.g. s=∫v⋅dt or for constant v, s=vt. This gives the area bounded by v-t graph in a particular time interval.

 

[Khashishi]

If we observe that one quantity increases proportionally to quantity A and increases proportionally to quantity B, then we might make a model which looks like:
C = A * B
If A is in kg and B is in m, then it's natural to assign C the units of kg*m.

 

[jbriggs444]

If A is in kg and B is in m, then it's natural to assign C the units of kg*m.

To make this concrete, consider a railroad in the shipping business. They charge $0.04 per ton mile

If you want to ship one ton 1000 miles, it will cost you $40.00
If you want to ship 100 tons 100 miles, it will cost you $400.00

To quote dollars per kilogram meter instead of dollars per ton mile they have to convert.

A meter is about 0.00062 miles. A kg is about 0.0011 U.S. short tons. A kilogram meter is about 0.00062 * 0.0011 = 0.000000682 ton miles.

Accordingly, the price quoted by this railroad should be about $0.000000027 per kilogram meter.

 

[Windadct]

Also - torque is a good example, the units are Force * Distance.

So a 10KG weight at the end of a 1M lever generates the same torque as 1 KG at the end of a 10M lever. Like seaSaws and balances.

Actually - look at area or volume. we prefer to use the same units, Like M*M - but you can say M * inches ( and upset some people) but it is still an area.

 

[Nugatory]

Actually - look at area or volume. we prefer to use the same units, Like M*M - but you can say M * inches ...

And there are even a few situations in which it is more convenient not to use the same units. For example, in America commercial volumes of water are often measured in acre-feet - one acre-foot is the volume of water that will cover one acre one foot deep, or one-half-acre two feet deep, or ...
If you're considering how much water you need to irrigate a field of a given size, the acre-foot is a the most convenient unit.