### Mass, weight, kilos, pounds, Newtons

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I have a problem. I have an exam in my science class and realize I don't understand a basic concept as put forth by my professor. It has to do with Newtons, mass, kilograms and pounds. She maintains that a pound is a unit of weight while a kilo is a unit of mass. I understand what mass is at least in theory. The space station has mass, but no weight. But if I step on a scale calibrated in pounds it’s measuring my weight, but if I now calibrate it in kilos, how is it now measuring my mass? She pulled out a couple of Newton meters the other night and they looked exactly like the scales my butcher uses to weigh a leg of lamb. Leg of lamb weight: 2.2 pounds. Calibrate the scale in kilos and it's now measuring 1 kilo but that's not its weight, that's its mass? How? Now put it on the Newton meter and it's measuring force? I always thought that a kilogram was the weight of a liter of water. Now I discover that it’s really its mass. I think I'm doomed.

Please help.

-- Frank Kinney (fnkinney@hotmail.com), October 28, 2001

### Answers

She maintains that a pound is a unit of weight while a kilo is a unit of mass.

Correct. The unit of mass in the English system is a "slug". (Really)

I understand what mass is at least in theory. The space station has mass, but no weight. But if I step on a scale calibrated in pounds its measuring my weight, but if I now calibrate it in kilos, how is it now measuring my mass?

Both scales are measuring your weight, which is the force gravity exerts on your body. Were metric scales properly labeled, they would register Newtons, which is the metric unit of force.

It is possible to interpret weight on a scale as a measurement of mass because things with the same mass have the same weight on the surface of the earth. So you can calibrate a scale in kg and get away with it.

She pulled out a couple of Newton meters the other night and they looked exactly like the scales my butcher uses to weigh a leg of lamb. Leg of lamb weight: 2.2 pounds. Calibrate the scale in kilos and it's now measuring 1 kilo but that's not its weight, that’s its mass? How?

It is measuring the weight and converting it to mass by assuming you're using it in earth gravity. If you used the scale on the moon, it would correctly measure the force (in Newtons) but would get the mass (in kilograms) wrong.

Stop here if you want to be less confused rather than more.

Mathematically, it works like this: Newton's second law relates the force on an object to its acceleration:

F = m a (1)

That is, a one Newton force on a 1 kg object results in an acceleration of 1 m/s^2. By definition, mass is an object's resistance to acceleration. Now, weight is the force exerted by gravity. On the surface of the earth, the weight is:

W = m g (2)

where g is a constant, g = 10 m/s^2. Notice the similarity between (1) and (2). In (2), the force exerted by gravity is proportional to the mass. In (1), the acceleration given a particular force is proportional to the mass. The fact that it's the mass in both cases is really kind of an accident, because the two laws are completely unrelated. There's no reason a priori that the force of gravity couldn't be proportional to say the mass squared:

W = m^2 g

There's no reason this couldn't be so, but the world doesn't work that way. In fact, the fact that the mass enters identically into (1) and (2), which is the source of your confusion, is a very deep thing, known as the "equivlence principle": the response of a body to gravity is the same as its resistance to acceleration. This is the basic postulate of Einstein's General Theory of Relativity, and why it is so is one of the single most important questions in physics.

Cheers! Hope at least a little of this makes sense.

Will Kinney
Research Scientist, Columbia University
(And brother of Frank Kinney)

-- William H. Kinney (kinney@physics.columbia.edu), October 28, 2001.

Hi Frank (and Will):

Part of your confusion arises from the fact that in everyday usage, the kilogram (or gram) is the standard, metric unit of weight. So if you travel in Europe, for example, all "weights" are recorded in grams/Kg. As your brother and your professor correctly (obviously!) point out, however, the technically correct unit for weight in the metric system is the actually the Newton, which is your mass (kg) times gravitational acceleration (9.8 meters per second per second on earth, roughly). That unit (the Newton) is only used in the scientific community as far as I know. Everyone else uses the kilo/gram and calls it "weight." I can actually send you some small spring scales that are calibrated in Newtons and Dynes (Not sure what a dyne is -- Will?).

I think that the key here is that on the surface of earth your "weight" as expressed on a standard bathroom metric scale (Kg) is, for all practical purposes, the same as your mass in Kg. As has already been mentioned, that same scale (assuming it is a spring-type scale) used on the moon would NOT tell you your mass on the moon, because that scale depends on the force of gravity acting on your mass, and the gravity on the moon is different than the gravity on earth. So if the scale reads 100kg on earth, on the moon it would read about 17kg -- you will have "lost" 83kg weight. Your mass would be the same as on earth but the scale would give you an incorrect reading for mass. (Correct mass = 100kg. Scale reads 17kg).

This next point will either clarify or confuse the issue: Your doctor probably uses a balance rather than a spring scale. The balance works like a see-saw, so it would give you the same measurement on the moon as on earth. (Think of two people on a see-saw. If they are balanced in earth's gravity, they would likewise be balanced in the moon's gravity). A balance scale works by balancing the mass of an object on one side with the mass of a known quantity on the other side. If the masses are balanced in earth's gravity, they are balanced in any gravity. Play around with a spring scale and a triple beam balance at school and this will make more sense.

(Will: Correct me if I'm wrong!)

-- Michael Gatton (mg143@aol.com), October 28, 2001.

(Not sure what a dyne is -- Will?)

A Newton is the unit of force in "MKS" ("meter-kilogram-second") units. A dyne is the unit of force in "cgs" units ("centimeter-gram- second"). Just like a Newton is the amount of force necessary to accelerate one kilogram at 1 meter/second^2, a dyne is the unit of force necessary to accelerate 1 gram at 1 centimeter/second^2.

There is more than one system of metric units. MKS is more widely used.

This next point will either clarify or confuse the issue: Your doctor probably uses a balance rather than a spring scale. The balance works like a see-saw, so it would give you the same measurement on the moon as on earth.
[...]
If the masses are balanced in earth's gravity, they are balanced in any gravity.

Yes. A balance scale measures the difference in weight between two masses, while a spring scale measures the weight of an object compared to the force exerted on it by a spring. In this sense, neither directly measures the mass, although a balance scale would measure 1 kg on both the earth and the moon, whereas a spring scale would measure 1kg on earth and 1/6 kg on the moon.

Neither one would work in orbit, however, and the mass would still be 1 kg!

This is one of those things that takes a bit of thinking to understand, then takes a lot more thinking to realize you don't actually understand it at all ;-).

Will

-- William H. Kinney (kinney@physics.columbia.edu), October 28, 2001.

I think you do not know wat ur talkin bout. WIEHT IS POUNDS AND THAT IS THAT!!!

-- Christopher Skalaway (shawty31055@hotmail.com), November 01, 2003.

Not to get totally off center, but it is interesting to make the comparison of the "weights" a person would have on the earth vs the moon. Gravity might initially be considered the consequence of interaction of a force field on a mass. In reality gravity is an "acceleration" field resulting in identical motion of two objects of differing masses. Consistant with F=MA.

-- Bob Steding (bsteding@hotmail.com), November 12, 2003.

The original poster may never see this answer but I will reply for future people that may read here. As an expert in the field of mass measurement, I would like to add the following:

The mistakes made in some of the previous posts are common. The main source of the mistakes is the improper use of the term pound. When the meter convention was signed in 1875 the US Avoirdupois pound was redefined relative to the kilogram. One Avoirdupois pound (US customary unit) was defined to be 0.45359237 kilograms the United Kingdom's National Physical Laboratory also defines the pound with this value. This makes the pound like the kilogram a unit of mass. The units are traceable to the International Prototype Kilogram maintained at the International Bureau of Standards near Paris France. This is the only unit of measurement which continues to be referenced to a physical artifact.

The confusion arises in that both the Kilogram and pound are also sometimes used as units of force. The Pound Force which is the correct way to refer to the force unit is defined as the force acting on a one-pound mass in a gravitational field for which the for which the acceleration of free fall is 9.80665 meters per second squared. The Pound Force is a unit derived from the mass unit.

For a full description of this subject you can see the Journal of Research of the National Institute of Standards and Technology Volume 106, Number 1, January – February 2001. The article published by Dr. Zeina Jabbour covers this topic very well.

-- Mark Fritz (mark31rp@yahoo.com), December 11, 2003.

Thanks to Mark for the information on mass & weight. The National Institute of Standards also has a vast collection of resources on measurement standards (as their name might suggest). You can request a class set of metric measurement tools (rulers, metric conversion cards, activities, etc.) by downloading a form here:

http://ts.nist.gov/ts/htdocs/200/202/orderfrm.htm

or by simply e-mailing your request:

"The FREE Teacher's Metric Resource Kit can also be ordered via email at TheSI@nist.gov. Please indicate that you wish to order the Teacher's Metric Resource Kit and include your name, mailing address, phone number, fax number, and email address."

Mike

-- Michael Gatton (mwgatton@hotmail.com), December 16, 2003.

100Kilo what would it be in pounds

-- Beatrice Doucet (beat53@sympatico.ca), April 06, 2004.

Earl Carlyon taught me a failsafe* way of converting between different units of measurement. It’s called “the funny looking one” (FLO) method.

Any two equal quantities expressed as a fraction equals one (x/y = 1 where x = y). The following conversion factor will be used to create our funny looking 1s:

1 kilogram = 2.2 pounds (actually 2.20462262 lbs)

(Click for a list of other conversion factors.)

Therefore we can make the following funny looking 1:

1 kilogram
2.2 pounds

and also

2.2 pounds
1 kilogram

Both of these expressions are equal to 1.

We further know that any number multiplied by 1 equals that number.

i.e.,

100 kilos x 1 = 100 kilos

So we can take our funny looking 1 and substitute it for “1” and multiply it by the number that we want to convert and get a number that looks different.However the value of the quantity we get will be equal to the value of the original quantity:

 100 kilos X 2.2 pounds 1 kilogram = 220 lbs

100 kilos has the same value as 220 pounds, the mass of the object doesn't change.

Notice that the kilogram units in the numerator of the original number and the denominator of the funny looking 1 cancel out and we are left with pounds as our unit. If we were converting from pounds to kilograms, we would simply invert the funny looking 1 so that the pounds would cancel out:

 220 lbs  X 1 kilogram 2.2 lbs = 100 kilograms

Again, the pound units cancel out, leaving kilograms as the unit. The decision as to which of the two funny looking 1s to use is entirely dependent on which units you want to get rid of in the conversion process.

This took care of what always confused me about conversions, namely whether to multiply or divide. In this method ALWAYS multiply by a funny looking 1.

*This may seem at first like a long and complicated way to figure out how many pounds equal 100 kilograms, but I promise you that if you learn this method now you will be glad you did so when you get into basic chemistry or physics classes, where you will be required to make very complex conversions that are difficult to just figure out “intuitively.”

On the other hand, if you’re just feeling lazy or in a hurry, you can always use Google’s calculator.

http://www.google.com/help/features.html#calculator

Mike

-- Michael Gatton (mgatton@adiglobal.com), April 06, 2004.

There is a much simpler way(at least to me) of doing convertions. For example: If 1kilo = 2.2pounds. Then 100Kilos would be how many pounds? You ask the question: would the answer be more or less? If more, less divides or if less more divides. Thats the rule. In this case the answer will be more so less would divide. So you would work it out like this 100kilos/1Kilo x 2.2pounds = 220pounds. On the other hand 2.2pounds = 1kilo therefore 1pound would be? Question: More or less? Its less, so more would divide. Hence it would be 1pound/2.2pounds x 1kilo = 0.454Kilos

I hope this makes sense!

-- Aba Jackson (aba_jackson@hotmail.com), August 19, 2004.

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