# Multiplication by different units

1. ### thedy

75
Hi,

What does N*m mean?
What does N/m mean?

What kWh mean?
What kW/h mean?

What does F=m*a mean?

I read maybe all articles on web about concepts of multiplication,units,differences.But respondents only starts with some question(You know,what is dividing,but you do not know what multiplication is?),but they are not giving the clear answer.

All my life,I thought that every problem could be sketched on paper,visually explained.I was trying to imagine for example F=ma on paper by making x line like mass and y line like acceleration and many others ways.

For me,multiplication gives me sense only when I have some unit,that is per something and that unit is multiplied.For example: s/t*t,BUT s*t is totally something super highly transcedence paranormal for me.What is distancesecond?It attracts me to think about it like it is distance per some time.Or time per distance.But what is then s/t?

What is apples times apples? Hour times hour?Why is easy to imagine square metre or cubic metre,but rest non length units are difficult to imagine.And if I do not know to imagine,How can I count them and know,what I am doing with them?

### Staff: Mentor

The first is probably Newton-meters; the second is probably Newtons per meter.
Kilowatt-hours and kilowatts per hour, respectively.
The force F on an object is equal to the mass m of the object times the acceleration of the object. On the right side the units are <mass> and <dist/time2>. The units of F on the left side are <mass> * <dist/time2>. This way, the units on both sides of the equation are the same.
Some units are very simple, such as length (meters), time (seconds or minutes or the like), but other units are more complex, being built up of other units. Acceleration is typically given in units of length/(time2). Force is another one whose units are built from simpler units.
For the most part, these are just useful notations. Seconds time seconds doesn't make sense from a physical standpoint, but it might be helpful for you to understand where this came from. Velocity is distance divided by elapsed time, or d/t, typically in units such as m/sec, ft/sec, km/hr, or mi/hr. Any unit that is <something> per <something>, the unit will involve division; e.g., miles per hour = mi/hr, and so on.

Acceleration is the time rate of change of velocity, as in meters per second per second. This could be written as
$$\frac{\frac{m}{sec}}{sec}$$
or in simpler form as ##\frac{m}{sec^2}##.

You have asked this before, as I recall, so it must be something you're still struggling with. My advice to you would be to just accept it as notation, and not worry so much about it.

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3. ### MrAnchovy

550
In a slightly different order:

It is easy to imagine square metres because this is a measure of area, and similarly cubic metres is a measure of volume. There is no such thing as a "square apple", the concept is meaningless. Similarly there is no such thing as a "square hour", I don't think you will ever come across this in physics so don't worry about it.

Are you using s as distance? distance x time does not have any physical significance, it is not something you need to calculate or worry about. distance divided by time on the other hand is very important - for instance if you travel a distance of 3.4 kilometres in 10 seconds you have travelled at an average speed of 340 metres per second, which is about the speed of sound. Speed = distance / time.

This makes more sense if you put force along the x axis and acceleration on the y axis. If you measure the acceleration of an object subject to different forces and plot them on this chart you should get a straight line with a gradient or slope equal to acceleration / force i.e. mass.

We only use the * symbol for multiplication in computer code, so we would normally write ## F=ma ##. It is read as "force equals mass times acceleration".

It is usually read "kilowatt hour". A kilowatt is the measure of the rate of consumption of (usually electrical) power used by a domestic appliance - for example a electric heater may use 3 kW. If you have the heater on for an hour, you have used 3 kilowatt hours. In physics we don't usually use kWh (we use Joules to measure energy instead: 1 kW is 1000 Joules per second so 1 kW hour is 1000 x 60 x 60 = 3,600,000 Joules).

It doesn't mean anything useful.

You've picked a tought one there. Newtons x metres is used in two different contexts. 1 Nm is a force of one Newton applied at the end of a lever 1 metre long. Alternatively it is the amount of energy used in applying a forece of one Newton to move an object a distance of 1 metre: this is equivalent to 1 Joule.

4. ### Stephen Tashi

4,330
As far as physical reality goes, units like "newton meters", "kilowatt hours", "meters per second" don't have a particular meaning.

You can get the impression that they mean something in particular because there are equations in physics that are very common. For example, the motion of an object is often studied and in that context "meters per second" refers to the velocity of an object. But there can be equations where "meters per second" has a different physical meaning.

To make an artificial example, suppose we have an complicated machine. You press down a button on the machine and hold it for t seconds. This causes the machine to remain stationary for 8 seconds and then move forward for x meters and stop.. By doing experiments on the machine you develop an equation to predict its behavior by measuring various values of x and t. The equation might involve the quantity x/t, but the meaning of x/t in "meters per second" would not be the velocity of an object.

If you have formulated the equation correctly as a "physical law" then an experimenter who used different units of measurements (such as minutes and feet) would confirm your equation provided he obeyed the conventioins for converting one system of measurement to another. This implies the left and right hand sides of the equation must have the same physical units. For example, if the equation 5 = x/t predicts the behavior of the machine when x is in meters and t is in seconds then we must specify that the constant 5 has units "meters per second".

(Using visual representations to understand things is useful, but making it your only method of understanding will be a handicap.)

Last edited: Aug 22, 2014
5. ### thelema418

109
This is an issue I'm having with some of the recommendations to use non-standard units for math education.

Let's say that a student has to measure the area of a rectangle. The non-standard measuring device is apples. The students find that they can cover the rectangle with an array of dimensions 3 apples × 4 apples for a total of 12 apples covering the rectangle.

The complexity in this problem is that number of apples in a row (apples.x) and the number of apples in a column (apples.y) decomposes apples into linear units; but it is also an area unit and could also be a volume unit. Not to sound snarky, but is it any wonder that we don't use apples for measurement?

This complexity comes out more when an oblong non-standard unit is used, like a paperclip. Most people will use the longest side of a paperclip as their measure of both the x and y component of the rectangle. Others will use the long side to measure the y component and the short side to measure the x component. These methods produce different results. Yet here, I would need to say ##\mathrm{paperclip}^2## for the first situation and just ##\mathrm{paperclip}## for the measure in the second situation.

6. ### MrAnchovy

550
Where is "here"? Whoever is responsible for that idea should have nothing to do with mathematical education.

7. ### thedy

75
Thanks for answers. Everyone of you are right. Well to be brief,I just would like to know,why all physical laws,relations are described by multiplication and division.I just would like to know what is concept or description of multiplication and division,because I can see,that definitely,it is not repeated addition of substraction.

Like F=ma. It seems that this equation is solved in many forums on net,because it is clear typical schoolbook problem. Maybe I want to know,what kind of relations(relationship) show us multiplication and division.

Like addition show us adding of something,multiplication shows us something too.Seems like multiplication for example is some kind of ,glue" that make from stuffs stuff.In textbook F=ma is described like direct proportion.If I have twice more mass I have twice more F.But it is not definition of relation that is used in physical laws,equations that describe some relation between two or more quantities.What is it?

F=ma does not show us repeated mass.Because repeted mass it always just repeated mass and not Force.But in some cases, multiplication is defines like repeated addition,but in F=ma case I can not aplly it.Well what kind of relation,definition of math operation is used in Newton law?If multiplication has many faces,which one is used in F=ma?

I do not want start problem with:,,Why we can not add two different units?" but seems like this one is too in place.

And in the end,I know that t*t or apples times pears are meanless but we should know describe,at least in our minds what are squared apples and what are not.If 4 times apple are 4 apples,and it could mean everything,not just fruit,we could maybe be able exaplain what is squared apples.

Sorry for my long reply,but I need to give you more analogies to make appropriate picture,what I mean.

8. ### jbriggs444

1,538

The area of a rectangle is proportional to its length. It is also proportional to its height. And as Stephen Tashi's insightful post suggests, the numeric measure of the area is inversely proportional to the size of the units you use to express the measurement.

Proportionality is intimately related to multiplication and division.

9. ### thedy

75
Could you explain this a little more.I do not understand.:(

And why is addition of two different units nonsense and multiplying not. Like kW hour....We mix two different units as well.....

Last edited: Aug 29, 2014
10. ### Stephen Tashi

4,330
Suppose someone describes a physical process by the equation W = x meters + y minutes.

This implies that there are different pairs physical inputs that produce the same outcome as far as the measurement of W goes. For example:

W, x meters, y minutes
4, 1, 3
4, 2, 2
4, 3, 1

If the equation is a correct description of a physical situation then an experimenter using different uinits of measurement will get results that agree. Suppose the experimenter uses centimeters to measure distance and seconds to measure time. In those units, the equation says: W = 100 x cm + 60 y seconds. However, this equation does not agree with the above table of data. The values of distance and time that produce the same measurement of W in the table do not produce the same value of W when different units of measurement are used.

W , x cm, y sec
280, 100 cm, 180 sec
320, 200 cm, 120 sec
360, 300 cm, 60 sec

11. ### thelema418

109
Things are meaningless when they have no purpose to the individual. Even when an individual finds meaning, the thing may be meaningless to society because it provides no general purpose or greatly conflicts with established meanings.

I would say that a squared apple is a mathematical object. The meaning depends on context, if a meaning exists. And this might all depend on what "*" means.

12. ### thedy

75

And it inspired me say opinion,why is creating units for me so un-intuitive. I still think,and make visual analogy like multiplication is a repeated addition.It is difficult to see what it means,when I multiply unit by unit.One unit by one unit to get totally new one.Because I do not understand how qualitative terms can be multiplied and therefore,why is multiplication a function, concept, which seems to be as repeated addition and this repeated addition which is quantitative process can make new qualitative concept.

13. ### mikeph

Addition of two different units isn't necessarily wrong. 1 kg plus 1 ounce is a valid quantity because the resulting number will represent the total mass of the collection. 5 kg can be written in units of pounds, ounces, tons, miligrams or any other unit of mass without ambiguity.

Addition of quantities with different dimension is what doesn't make sense. 4 metres plus 7 kilograms is nonsensical becuase the two dimensions are, in some sense, fundamentally irreconcilable. There is no accepted conversion factor between quantities expressed in dimensions of metres and kilograms because they refer two two fundamentally different things.

14. ### HallsofIvy

40,370
Staff Emeritus
A "Newton" is a unit of force equal to the force necessary to move an object of mass 1 kg at an acceleration of one meter per second per second. "N*m" is "Newton meter" which is a unit of energy (also called a "Joule"). It is the energy used in applying a force of 1 Newton to move an object one meter. It is also the kinetic energy of an object of mass 2 kg moving at 1 meter per second: kinetic energy= $\frac{1}{2}mv^2$= (1/2) (2 kg)(1 m/sec)^2= 1 kg m^2/sec^2= (1 kg m/sec^2)(1 m)

It would be read "Newton's per meter" but I don't believe there is any useful physic quantity that has those units. Since a Newton is "1 kg m/sec^2" dividing by meters would give "1 kg/sec^2" and, as I say, I don't believe there is a physical quantity that is described by that.

kW is a measure of the rate of production of energy (or use) of energy- it is 1000 Joules per second. (A "kilo" watt is 1000 Watts and a Watt is one Joule per second). Multiplying by "h" (hours or 3600 seconds) that gives 3600000 Joules, measuring the amount of energy produced (or used) in a given time.

"KiloWatt per hour" would be a rate of production per hour or "Joules per hour per hour. That could be thought of as the rate at which the production of energy is changing but I don't believe that is an important quantity in physics.

"Force equals mass times acceleration", a basic law of physics (some people think of that as a definition of force).

You have an idea what "speed" is, m/sec, miles/hour, or km/hour don't you?

But not every combination of basic units, such as "apples times apples" or "hours times hours" has a useful meaning (though "miles per (hour times hour) = acceleration which is useful)

15. ### DrClaude

2,260
Spring constant, stiffness, surface tension, tear strength, tensile impact strength, and force gradient all have units of N/m.

16. ### Stephen Tashi

4,330
This illustrates the fact that products and ratios of fundamental units don't have a unique meaning. (This includes commonly encountered combinations like m/sec , kg m/ sec^2 , kilowatt hour etc.) There are equations in physics that are commonly studied and this leads to the impression that the meaning of such units in the context of those equations is their only possible interpretation.

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17. ### thedy

75
Hi,thanks for answer,but why does multiplication creates new units,although sometimes there are meaningless, like square apple.But in context of pure math they are correct despite square-apple does not exist.What is the ability of multiplication that make it possible,but by addition not.
But addition of different units are wrong mathematicaly and they are meaningless as well.
Why is 6kWh unit of energy and not unit of time.We can also say that,more kW we have,more progress in time is.
THanks

18. ### Stephen Tashi

4,330
That is a hard question to answer in a precise mathematical manner.

An imprecise answer is: If you look at the operations needed to convert experimental results from one system of units to another, they are most conveniently done by multiplying by conversion factors. If you convert the units in another (correct) way, it always amounts to using conversion factors.

For example, suppose a friend does a statistical study. He does a linear reqression between the age and weight of students at a university and fits the equation W = 5.3 T + 101.6 to the data. Where W is weight in pounds and T is age in years. Suppose you decide to verify this equation by doing an independent study, but you measure weight in newtons and age in days. Naturally, your linear regression produces an equation with different constants. You can figure out how to compare your equation to your friend's equation in various ways. If you do the conversion correctly it will amount to using conversion factors. Not being a physicist, your friend didn't put any dimensions on his constants 5.3 and 101.6. To use conversion factors, you would to assign dimensions to them so that the terms 5.3 T and 101.6 and W all have the same units of measure.

The field of study of "Dimensional Analysis" should deal with your question. However, the sources I've seen (e.g. http://web.mit.edu/2.25/www/pdf/DA_unified.pdf ) simply assume that unlike units cannot be added. They don't make more basic assumptions and then prove that unlike units cannot be added.

19. ### HallsofIvy

40,370
Staff Emeritus
Thank you. "spring constant", at least, I should have known!