1  The Definition of Weight

For an object of mass m, its weight W is a vector and is given by

where g is the vector representing the local gravitational acceleration. This g can be determined by observing how a nearby freely-falling object accelerates relative to your chosen reference frame. See section 2 for more about the definition of “gravity”. See reference 1 for a discussion of how to define “mass”.

Equation 1 says nothing about what may be causing the local gravitational field; the only thing that matters is the absolute acceleration of your chosen reference frame. Of course it is often convenient to attach your reference frame to some nearby planet, in which case the planet makes a large contribution to g via the law of universal gravitation (equation 4) ... but that is rarely (if ever) the only contribution to g, for reasons discussed in section 2.

Remarks:

1.    A pencil in an orbiting space ship is weightless according to an observer in the ship’s reference frame.

2.    A pencil in orbit is not weightless according to an observer in a ground-based reference frame. Let’s be clear: it is the frame that matters. The pencil does not become weightless because of the motion of the pencil, but because of the motion of the reference frame.

Mass is 100% independent of the choice of frame. Weight is 100% dependent on the choice of frame.

3.    Notation: Many books adopt the convention that the vector W (boldface) has magnitude W (non-boldface) where W := |W|. Similarly the vector g has magnitude g := |g|. This works OK for books and for web pages, where boldface is easy to achieve, but when you’re writing things by hand it is not very convenient. For this reason (among others), I recommend the following convention: Vectors should be written without any decoration, i.e. without arrows over them, without lines under them, or anything like that. Boldface is optional, never required. So we are free to use non-bold W to represent a vector. If/when it is convenient to use boldface, bold W means the same thing as non-bold W. If we ever need the scalar magnitude of W, we write |W|. (It is not worth the trouble to come up with a more compact notation for |W|.) This convention works fine for all media, including handwritten notes, books, email, and so forth. This convention is consistent with (and practically required by) more advanced work involving multivectors. See reference 2 for more on this.

4.    The W and g vectors are directed downward. I don’t think this is part of the definition of W or g; instead, I think it is the definition of “downward”.

5.    Since g is a vector, it is improper to ask whether it is positive or negative. There is no “>” relationship defined for vectors (except possibly in one-dimensional vector spaces, which we exclude from consideration, since we are interested in multi-dimensional motion). Therefore you can’t ask whether a vector is “>0” or “<0”. Of course |g|, the magnitude of g, is non-negative, but that doesn’t tell us anything we didn’t already know (since the magnitude of any vector is automatically non-negative).

Therefore, when folks ask about the sign of g, you know they are asking an unanswerable question. It is often difficult to figure out what prompted the question.

• Sometimes they intended to ask about some particular component of g in some chosen reference frame. The answer could be positive or negative, depending on what reference frame they choose. If the +Z direction points downward, then g_z is positive, whereas if the +Z direction points upward, then g_z is negative. This is a common source of confusion, because different folks are free to choose different reference frames.
• Sometimes they intended to ask about projection of g along the direction of some other physically-significant vector in the problem. Again this depends on which other vector they’re talking about, but at least the answer to this question is physically-meaningful and observer-independent, once the question has been spelled out in sufficient detail.
• Also: when people seem to be confused about the “sign” of g, often the main problem lies elsewhere. It may have to do with velocity versus speed, as discussed in reference 3.

6.    Some people prefer to speak of the force due to gravity (or the force “of” gravity) instead of weight. Similarly they prefer to denote it F_g instead of W. (Usually F_g means exactly the same thing as W.) This allows us to write nice expressions such as:

Fg   :=   gravitational force ag   :=   local gravitational acceleration Fg   =   m ag

(2)

without even mentioning weight, as opposed to

W   :=   weight, i.e. gravitational force g   :=   local gravitational acceleration W   =   m g

(3)

Equation 2 has the advantage of logic and elegance, but it hasn’t caught on very widely.

7.    Most people live placidly near the surface of the earth, where to a fair approximation |g| is constant, which means there is a one-to-one correspondence between weight and mass. This explains why non-experts commonly blur the distinction between weight and mass.

However, the distinction remains important, and will never die out. For example, as previously mentioned, it is very convenient to say that astronauts see themselves as weightless but not massless.

Also, for careful work, you need to take into account the fact that |g| depends on where you are on the earth’s surface. The variation is on the order of one percent. See reference 4.

8.    The weight of an object in the lab frame will be very slightly time-dependent for various reasons including weather-related changes in the mass of air overhead, solar gravity, lunar gravity, et cetera.

9.    I find it unhelpful to define or recognize any distinction between “true weight” and “apparent weight”. Weight is weight. Let’s just call it the weight. If the weight relative to one accelerated reference frame is different from the weight relative to another, so be it. The situation is symmetrical; there is no basis for deciding which weight is “true” and which is “apparent”. Even if you could decide that one frame is “true” and the other only “apparent”, what are you going to do when there are three mutually-accelerated frames?

10.    Depending on context, there are various things that the word “gravity” could mean, and what the symbol “g” could mean. Some more-explicit, less-ambiguous terminology is discussed in section 2.

11.    The weight will not in general equal the total downward force. Weight includes only the force due to the local gravitational field, not the forces due to buoyancy, air currents, magnetic fields, bungee cords, or anything else. However, in a fair number of practical situations we can arrange that these non-gravitational contributions are negligible, in which case the weight can be well approximated by measuring the total downward force, perhaps by letting the object rest on a spring-scale. (Of course the scale should be at rest in the chosen reference frame.)

It is important to use a spring scale, not just any old scale, because there are plenty of scales on the market that measure mass instead of weight, or measure some weird combination of mass and weight. (Scale manufacturers can get away with considerable vagueness, since they assume their scales will be operated on the Earth’s surface, where |g| is reasonably uniform and well-known. See section 3.3.)

12.    Similarly, there some grade-school textbooks that try to define weight (not just approximate it) by saying weight is “whatever the scale reads”. Such a definition cannot be taken seriously, for a number of reasons. The problems include:

• This is an “instrumental” definition. Like all instrumental definitions, it is open to all sorts of questions about what happens if the instrument is miscalibrated, or outright broken. There are serious philosophical questions here, and also nontrivial practical questions when we start asking about your weight on the moon, or your weight aboard a space station, since scales that behaved fine on the earth might well misbehave in other situations.

  You might try to solve this class of problems by talking about an “ideal” scale, but defining what you mean by that is no easier than defining “weight” from scratch.

• If you were to take the scale reading literally, you would be in error, because you would have neglected the buoyancy correction.
• Almost all scales are intended to measure mass, not weight. For example, consider kitchen scales. If the cookie recipe calls for 1/4 pound of butter, the intention is presumably to use the same mass of butter even if I am baking cookies on the moon.
• In practice it is not particularly uncommon to find two scales in the same room, one of which (to a good approximation) responds to mass while the other (to a good approximation) responds to weight.
• Perhaps the most serious objection is that this notion of weight applies only to objects that are stationary or at least unaccelerated relative to the scale. For objects that are free to move, indeed even partially free to move (such as an Atwood machine, as shown in figure 1, or even a simple pendulum), the true weight is still there, as evidenced by the acceleration of the object, but the actual force exerted on the scale will not be equal to the weight mg.
  
  Figure 1: Atwood Machine on a Scale

  A student on a scale, hopping up and down, will illustrate the same point, namely a scale-reading that is wildly time-dependent and therefore different from what we would like to call “the” mass or “the” weight.

  Another notorious example is an hourglass on a scale. The flow of the sand causes the center of mass of the hourglass to move downward. In some rare cases, the motion of the center of mass will be uniform (to a sufficient approximation) in which case it doesn’t affect the reading. In other cases, depending on the shape of the sand-chambers and other factors, the motion of the center of mass will be nonuniform. This counts as an acceleration of the hourglass as a whole, in which case the situation is analogous to figure 1, and the force on the scale will not be equal to the weight mg.

Bottom line: Defining weight in terms of whatever “the” scale reads runs a large (and quite unnecessary) risk of spreading misconceptions.

2  Various Definitions of “Gravity” and “g”

Depending on context, there are various things that the word “gravity” could mean, and what the symbol “g” could mean.

To avoid confusion, it is helpful to define some more-specific terms. In increasing order of sophistication, we have:

• We define A-gravity (denoted g_A) by means of Newton’s law of universal gravitation, as follows: It has magnitude

  |gA| =

  G M r2

  (4)

  and is directed toward the center of some specified star, planet, or other gravitational source. Here M is the mass of the source, G is Newton’s constant of universal gravitation, and r is the distance from our location to the center of the source.

  In astronomical situations, such as the effect of earth’s gravity pulling on the moon, “the” gravity may sometimes be well approximated by g_A. That’s partly because correction terms (due to inhomogeneity etc.) become disproportionately small, since the radius r is large compared to the size of the source. It is also partly because centrifugal terms (associated with the rotation of the source) do not contribute, since we are using a reference frame that does not partake of the surface rotation of the source.

  In contrast, it must be emphasized that in practical terrestrial situations, g_A is definitely not “the” gravity; it is only an approximation. It an adequate approximation for some terrestrial applications, but not others.

  The A in A-gravity could stand for Astronomical. In a terrestrial situation, the A could stand for Approximate, since it is only an approximation to the usual terrestrial lab-frame gravity. Also the A could be recognized as the first letter of the alphabet, reminding us that g_A is a useful first approximation to the observed lab-frame gravity.

• L-gravity is denoted g_L and is associated with the acceleration of the laboratory reference frame. It can be measured by dropping an object and measuring its freely-falling motion relative to the laboratory.

  In typical terrestrial laboratory situations, g_L consists of g_A (which is the dominant term), plus the following corrections:

  • the centrifugal field of the earth’s rotation;
  • the fact that the earth is generally oblate rather than spherical;
  • local variations in the density of the earth;
  • the direct and indirect effects of lunar gravity;
  • the direct and indirect effects of solar gravity;
  • the weather-related variation in the mass of air overhead; and
  • various smaller terms.

  The L in L-gravity stands for Laboratory.

• As discussed in section 3.1, the “standard” acceleration of gravity is defined to be 9.80665 m/s² and is denoted g_N. The direction is unspecified.

  You cannot assume your |g_L| is equal to g_N. The two can differ by a goodly fraction of a percent, depending on lattitude, elevation, et cetera.

• E-gravity is denoted g_E and is associated with the total acceleration of the reference frame, relative to a nearby freely-falling object. In accordance with the equivalence principle, it does not distinguish among the possible contributions to this acceleration.

  The E in E-gravity stands for Einstein, since Einstein’s general theory of relativity strongly emphasizes the equivalence principle, emphasizes freely-falling reference frames, and treats on an equal footing any and all contributions to the acceleration of an accelerated reference frame.

• The proper weight of an object is the mass times the proper acceleration g_p. This refers to g as measured in a frame instantaneously comoving with the center of mass of the object. This is an elegant and sophisticated idea. It is not appropriate for an introductory discussion, but is sometimes useful, as discussed below.

Each such acceleration can be associated with a force (or pseudo-force). For the four acceleration vectors just mentioned, the associated force is:

The examples mentioned here are definitely not the only g-like quantities that can be defined. For starters, you can define 2^N different g-like quantities just by selectively including or excluding the “corrections” mentioned in the definition of g_L above.

For an observer at rest relative to the laboratory, g_E is identical to g_L. For an observer in a car, moving relative to the laboratory, using a reference frame attached to the car, g_E may differ significantly from g_L. It will differ in direction if the car is turning, and differ in magnitude if the car is driving over the crest of a hill.

In almost all practical situations, when people talk about “the” gravity, they are referring to E-gravity. That is, in practical situations, you can usually assume g = g_E, and in the laboratory that includes g = g_E = g_L. We took g to be equal to g_E when we defined g at the beginning of section 1.

Many textbooks use inconsistent definitions, switching back and forth from one definition to another. In the chapter on universal gravitation, “g” and “gravity” may refer to A-gravity. Meanwhile, in other chapters, including anything related to practical weighing, “g” and “gravity” may refer to E-gravity.

To repeat: In this document, we use g as shorthand for g_E ... but you should beware that other authors are wildly inconsistent about the meaning of g.

The lab-frame weight F_L is approximately but not exactly equal to F_A. In theory, F_L would exactly equal F_A for the lab frame on an isolated airless nonrotating spherical homogeneous planet, but there aren’t any of those around here. In the usual lab frame, the largest single contribution to F_L − F_A comes from the centrifugal field due to the earth’s rotation.

L-gravity is consistent with the conventional practice of measuring altitude relative to sea level, and with conventional notions of vertical and horizontal. The surface of a liquid at rest is considered locally horizontal, and is globally an isopotential, if and only if centrifugal terms are included. To repeat: The notion of “sea level” does not makes sense unless we include centrifugal terms in our notion of gravity. Our definition of L-gravity is also consistent with conventional definitions of vertical and horizontal: a table in the laboratory is considered horizontal if things don’t spontaneously roll off it, which is dependent on L-gravity (not A-gravity).

L-down diverges from A-down by about 0.1 degrees at temperate latitudes. The magnitude of L-gravity differs from the magnitude of A-gravity by about one third of a percent at the equator, and about one quarter of a percent at temperate latitudes. In the terrestrial lab frame, L-down defines the practical, operational definition of vertical (and by extension, horizontal). Operationally, this means that an undisturbed pool of water is horizontal, and an undisturbed plumb line is vertical.

Here’s a parable: Once upon a time, some folks who misunderstood the distinction between L-gravity and A-gravity decided to build a swimming pool. It was 50 feet across. They wanted the rim of the pool to be horizontal, but they constructed it to be perpendicular to g_A rather than to g_L. As a result, one side was just over one inch too high, measured relative to the water. This looked really terrible. The point is that in accordance with the laws of physics, the water distributed itself according to g=g_E=g_L, not g_A.

To summarize: in practical terrestrial applications, gravity is not equal to GM/r². In addition to the GM/r² term it includes the centrifugal term and various smaller corrections. This result does not depend on any sophisticated notions of modern physics. In particular, we do not need to invoke Einstein’s principle of equivalence (although we could). Instead we are depending on basic, practical, operational notions of horizontal, vertical, up/down, and gravity – notions that predate Einstein and predate Newton by thousands of years.

On the other side of the same coin, our results are entirely consistent with modern physics notions. In practice, local gravity almost always means E-gravity, and includes all contributions to the acceleration of the reference frame, relative to free fall. This is as it should be in accordance with Einstein’s principle of equivalence, which asserts that a gravitational field is locally indistinguishable from acceleration of the reference frame.

This brings us back to a point made earlier: even though mass is 100% independent of the choice of reference frame, weight is 100% dependent on the choice of reference frame. Important examples include:

• In the usual terrestrial lab frame, the dominant contribution to L-gravity is A-gravity, with significant corrections due to the centrifugal field associated with the earth’s rotation.
• Astronauts living in a space station correctly consider themselves weightless, because in the frame of an orbiting spacecraft, E-gravity is essentially zero. That is related to the fact that A-gravity is equal and opposite to the acceleration associated with the orbital trajectory.

Purists might be tempted to define a “purely” ideal (non-Einsteinian) weight in terms of F_A instead of F_E … but this would be unconventional and very impractical. The fact remains that in universal laboratory practice, weight is defined in terms of F_L, and is likely to remain so for the foreseeable future. The idealized term F_A is the dominant contribution to the local gravity F_L, but there are in fact other contributions. Get used to it.

To repeat: L-down diverges from A-down by about 0.1 degrees at temperate latitudes. The magnitude of A-gravity differs from the magnitude of L-gravity by about one third of a percent at the equator, and about one quarter of a percent at temperate latitudes. It is the angle that makes g_A unsuitable for terrestrial applications such as swimming pool construction, skyscraper construction, and innumerable other practical applications.

In contrast, it is the magnitude that makes g_A unsuitable for describing weight (or lack thereof) in the frame of a freely orbiting spacecraft.

It is sometimes useful to consider the proper weight of an object, which is the mass times the proper acceleration g_p. This refers to g as measured in a frame instantaneously comoving with the center of mass of the object. This is an elegant and sophisticated idea, and is sometimes useful, for instance if you are designing an inertial guidance system. For details, see reference 5. On the other hand, proper wieght is not appropriate for an introductory discussion of gravity and weight. For one thing, it blurs the distinction between the motion of the frame and the motion of the object. Secondly, if the frame is undergoing nonuniform acceleration, this introduces complexities into the laws of motion, which introductory-level students are not prepared to handle. Also, there are subtleties in the definition of proper accleration. (Obviously you can’t think of an object as accelerating relative to itself.)

3  Units and Measurements

3.1  Definitions of Units

The conventional “standard” gravitational acceleration is by international agreement defined to be 9.80665 m/s², and is called one Gee, often shortened to just G. According to NIST, the recommended symbol is g_N but everybody I know calls it Gee or G (not to be confused with Newton’s constant of universal gravitation, also denoted G).

This g_N is useful for calibrating accelerometers in a standard way. It was never intended to tell you the actual g_L of your laboratory.

The actual observed gravitational acceleration in an earthbound laboratory is typically within 1% of “standard” value, but of course varies with time, location, elevation, et cetera. Reference 4 provides a tool for looking up measurements of the local gravitational acceleration almost anywhere in the United States.

The kilogram (kg) is defined to be a unit of mass. There also exists an informal unit, the “kilogram of force” (kg_f), which is defined to be one kg multiplied by one Gee.

The pound (lb) is a unit of mass. The pound has been in use since the late 13th century or early 14th century. That means the idea of “pound” is hundreds of years older than the idea of “F=ma”. (In contrast, the “kilogram” is more than a century younger than “F=ma”.)

Some people claim the pound (as used in the US) to be defined as a unit of force, but this is not true. As far as I can tell, it has never been true. Under US law, the pound has been recognized as proportional to the SI unit of mass – the kilogram – since 1866. Under US law from 1901 to 1959, the pound was explicitly defined in terms of the kilgram, namely 0.4535924277 kg. By international agreement adopted in 1959, the “international pound” is defined to be 0.45359237 kg, which is now the value used for all purposes in the US. See reference 6 and reference 7.

There also exists, informally, the “pound of force” (lb_f), which is defined to be one lb multiplied by one Gee. If you hear somebody talking about a “pound of weight”, you can assume they mean lb_f. In general, though, lb_f should be avoided. Come constructive alternatives are discussed in section 3.2.

Scientists who measure the variations in the earth’s gravitational field customarily use units of gals and milligals. The gal is defined to be 1 cm/s², so it is 100 times smaller than the SI unit of acceleration. The gal is named in honor of Galileo, but gal is the full name of the unit, not an abbreviation. Conversely, there is no abbreviation for gal. The abbreviaton for milligal is mgal.

If you are doing unit-conversion calculations using the “units” program, the gal unit must be capitalized, i.e. Gal. This non-standard capitalization is a kludge to avoid conflict with the abbreviation for gallon.

3.2  The Effect of Units on the Form of Laws

SI (Le Système International d’Unités) was designed so that most of the units are consistent with each other, and consistent with the laws of physics. For example, the SI unit of force is equal to the unit of mass times the unit of acceleration, so we can write F=ma without any conversion factors.

Other systems of units are not always so consistent, in which case you must re-think many of the laws of physics, with an eye toward putting in conversion factors where needed. For example,

• If you want to measure acceleration in feet per second per second, mass in lb, and force in lb_f, then you cannot write Newton’s second law in the form F = m a. Instead you must write F = m a/g. This is what I usually do, on the rare occasions when I am forced to calculate with pounds at all.
• If you want to measure mass in lb and force in lb_f, and still write F = m a without a conversion factor, then you must measure acceleration in Gees (not in feet per second squared). This is almost equivalent to the previous option; it just absorbs the conversion factor into the definition of acceleration.
• If you want to measure acceleration in feet per second per second and mass in lb, and still write F = m a, then you must measure force in poundals.
• Some older engineering books measure force in lb_f, mass in slugs, and acceleration in feet per second per second. This allows them to write F = m a without a conversion factor.
• My advice is to convert everything to SI at the first opportunity, so we can forget about ugly things like poundals and slugs.

3.3  Instruments

You can buy instruments that measure mass, typically labelled in kg and g, or lb and oz. You can also buy instruments that measure force ... and unfortunately they are often labelled in kg when they should be labelled in kg_f, or labelled in lb when they should be labelled in lb_f.

As if things weren’t complicated enough, sometimes you find instruments that measure some weird linear combination of mass and weight. They might use a balance (mass) to obtain the high-order digits of the reading, and then use a spring (force) to obtain the low-order digits. Fortunately though, under standard conditions on the earth’s surface, it usually doesn’t matter very much whether the instrument measures true mass, true force, or some combination ... since mass and force have a known proportional relationship under standard conditions.

But it is interesting to ask whether such-and-such instrument would work correctly on the surface of the moon, or in the weightless environment of a space station. Actually that’s not quite the right question, since whether the instrument “works” depends partly on the instrument but also depends on how you choose to use the instrument. For any instrument, you can pull on it with a rubber band (or magnet or whatever) in such a way that you are almost certainly measuring the force of the rubber band, not the mass of the rubber band. On the other hand, for any instrument, you can hook up a passive object in such a way that you are measuring the mass of the object.

In non-standard conditions, such as on the suface of the moon, a balance-type instrument is good for measuring mass. It can measure mass without needing to be recalibrated. But it is a disaster for measuring force. Conversely, a spring-type instrument is good for measuring force, but is a disaster for measuring mass.

As Michael Edmiston has pointed out, there is a large class of mass-measuring instruments (including some of the crudest, and also many of the finest) that contain a built-in standard of mass, and measure things relative to this standard. Instruments in this class can be expected to measure mass (not force, not weight), independent of the value of |g| over some reasonably-wide range.

The interesting thing about such instruments is they assume that g is not changing too much as a function of time, and/or assume g is not changing too much as a function of position. That is:

• A beam-balance compares the mass in one pan against a standard in the other pan. This can measure mass extremely accurately, but is subject to the assumption two pans are located sufficiently close together that there is no significant difference in g between the two locations.
• Likewise, some very fine laboratory-grade single-pan instruments use a force sensor, but calibrate it against a standard mass every so often. This also can measure mass extremely accurately, but is subject to the assumption that g doesn’t significantly change between the time of the calibration and the time of the measurement.

4  That Falling Sensation

Reference 8 addresses the bodily feelings associated with free fall, weight, and/or weightlessness.

5  References

1.

John Denker, “How to Define Mass” ./mass.htm

2.

John Denker, “Fundamental Notions of Vectors” ./intro-vector.htm

3.

John Denker, “Velocity, Speed, Acceleration, and Deceleration” ./acceleration.htm

4.

Pan American Center for Earth and Environmental Sciences (PACES), “GeoNet Gravity and Magnetic Dataset Repository” http://irpsrvgis00.utep.edu/repositorywebsite/

5.

John Denker, “Acceleration in Spacetime” ./spacetime-acceleration.htm

6.

“NIST Guide to SI Units” http://physics.nist.gov/Pubs/SP811/appenB8.html

7.

Sizes Inc., “Pound Avoirdupois” http://www.sizes.com/units/pound_avoirdupois.htm

8.

John Denker, “Can You Feel Gravity?” ./gravity-perception.htm