What does the multiplication between two units mean?

In summary: 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.
  • #1
Ehden
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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?
 
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  • #2
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".
 
  • #3
Ehden said:
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.
 
  • #4
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.
 
  • #5
Khashishi said:
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.
 
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  • #6
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.
 
  • #7
Windadct said:
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.
 

1. What is the purpose of multiplying two units together?

The purpose of multiplying two units together is to determine the combined value of the two units. It is a way to express the relationship between the two units and how they contribute to a larger quantity.

2. How do you multiply two units together?

To multiply two units together, you simply multiply the numerical values of the units. For example, if you are multiplying 5 meters by 3 seconds, the result would be 15 meters-seconds.

3. What are the rules for multiplying units?

The rules for multiplying units are as follows:

  • When multiplying like units (e.g. meters x meters), simply multiply the numerical values and keep the unit the same.
  • When multiplying unlike units (e.g. meters x seconds), convert the units to a single unit before multiplying.
  • When multiplying by a unitless number, the unit remains the same.

4. Does the order of the units matter when multiplying?

Yes, the order of the units does matter when multiplying. When multiplying unlike units, the order in which they are written determines which unit will be in the numerator and which will be in the denominator of the final unit.

5. What is the resulting unit when multiplying two units together?

The resulting unit when multiplying two units together will depend on the specific units being multiplied. It is important to pay attention to the units and their order when multiplying to ensure the correct unit is used in the result.

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