where m is the mass of the object and g is the vector representing the local gravitational acceleration. See reference 1 for a discussion of how to define “mass”. In any given reference frame, we define g as:

g := the acceleration of a freely-falling test particle (2)

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”.

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.

Let’s consider some examples:

1. One important special case is the terrestrial “laboratory” reference frame. In this frame the largest contribution to g can be calculated using Newton’s law of universal gravitation (equation 5) ... but that is definitely not the only contribution to g in the lab frame, for reasons discussed in section 2.3.

2. A pencil in an orbiting space ship is weightless according to an observer in the ship’s reference frame. For more on this, see section 4.

3. Meanwhile, the orbiting pencil 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.

4. Notation: Many books adopt the convention that a boldface symbol such as W represents a vector, while the corresponding non-boldface symbol represents the magnitude of the vector: W ≡ |W|. That convention 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 boldface, without arrows over them, without lines under them, or anything like that.

In particular, in this document, the weight W is a vector. 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.

5. 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”.

6. Since g is a vector, it is improper to ask whether it is positive or negative. There is no “greater-than” 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 for any vector g, the magnitude |g| is non-negative, but that doesn’t tell us anything we didn’t already know (since the magnitude of any vector is necessarily 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, but here are some hypotheses for you to consider:

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 basis vectors they have chosen. If the +Z basis vector points downward, then g_z is positive, whereas if the +Z basis vector points upward, then g_z is negative. This is a common source of confusion, because everyone is free to choose whatever basis they like.
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.

7. 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:

F_g	:=	gravitational force
a_g	:=	local gravitational acceleration
F_g	=	m a_g

(3)

without even mentioning weight, as opposed to

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

(4)

Equation 3 has the advantage of logic and elegance, but it hasn’t caught on very widely.

8. Most people live placidly near the surface of the earth, where |g| is constant within 1% or so, which means there is, rougly speaking, 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.

9. 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.

10. 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?

11. 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.

12. 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. (Assuming of course that the scale itself is at rest in the chosen reference frame.)

It is important to use the right type of scale, because some scales are primarily sensitive to weight (i.e. force), some scales are primarily sensitive to mass, and some are sensitive to some weird combination of the two. (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 5.3.)

13. 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 “Gravity” Scenarios

Gravity depends on where you are and what you have chosen as your reference frame. We now discuss some examples.

2.1 Standard Gravity

As discussed in section 5.1, the “standard” acceleration of gravity is defined to be 9.80665 m/s² exactly, and is denoted g_N. The direction is unspecified.

This g_N is useful as a benchmark, as a standard of comparison. The actual value of g at sea level in temperate latitudes is close to g_N, with a small fraction of a percent.

Also, g_I at the equator is close to g_N, as discussed in section 2.2

2.2 Primary Gravity

We start by defining the notion of idealized, introductory, primary gravity, denoted g_I. It is defined using Newton’s law of universal gravitation, as follows: It has magnitude

|g_I| =

G M

r²

(5)

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 the symbol for primary gravity, g_I, the subscript I can stand for idealized or for introductory. In can also be interpreted as the Roman ordinal number I meaning primus i.e. primary or first. All these refer to the fact that equation 5 is usually the first equation people learn when studying the physics of gravity.

It must be emphasized that g_I is not the definition of gravity. Equation 5 does not define gravity.

2.3 The Terrestrial Laboratory Frame

The gravity in the ordinary lab frame is denoted g_L, where the L stands for laboratory.

As always, g can be measured by dropping an object and measuring its freely-falling motion relative to the laboratory. This is what gravity means. This is what gravity has always meant.

In the lab frame, g_L includes the following contributions:

The primary gravity term, g_I. This is far and away the largest contribution to g_L.
The centrifugal field of the earth’s rotation.
The fact that the earth is generally oblate rather than spherical, as a consequence of the aformentioned centrifugal field.
Local variations in the density of the earth.
The direct and indirect effects of lunar gravity.
The direct and indirect effects of solar gravity.
Weather-related variations in the mass of air overhead.
Various smaller terms.

To repeat: The difference |g_L − g_I| is not zero. It is usually less than 1% of |g_L| but not much less, depending on latitude, elevation, et cetera.

2.4 Earth-Centered Nonrotating Frame

We can with a little effort construct a situation where the actual gravity g is very nearly equal to g_I. One possibility is suggested by figure 2.

Figure 2: Earth-Centered Nonrotating Reference Frame

As shown in the figure, we imagine a huge triangular truss has been constructed in space high above the equator. The center of the truss coincides with the center of the earth. It must be emphasized that the truss is not orbiting i.e. not rotating. Such a frame is called an earth-centered nonrotating (ECNR) reference frame.

Observers on the truss observe that the earth spins below them once every sideral day.

The ECNR frame is good for analyzing the motion of a satellite in earth orbit.

In this reference frame, the weight of the anvils (as shown in the diagram) is very nearly given by equation 5. That is, m g is very nearly m g_I. All of the correction terms mentioned in section 2.3 must still be considered; however:

We have gotten rid of the centrifugal term because the frame is not rotating.
We don’t need to worry too much about the oblateness because we are measuring things relative to the center of the earth, not relative to the surface
The satellite is sufficiently high that it is not very sensitive to the details of the earth’s geology and geography.

The remaining correction terms are on the order of a few parts per million, or smaller.

2.5 General Relativity; g_E

Sometimes for emphasis we define g_E such that −g_E is the acceleration of the chosen reference frame, relative to a nearby freely-falling body.

This is the same as our definition of plain g, but sometimes calling it g_E helps clarify the discussion when talking to someone who is tempted to define g differently. We can say that we define g = g_E even if they define g differently.

As always, in accordance with the equivalence principle, we care about the total acceleration of the reference frame. We not distinguish among the possible contributions to this acceleration.

The E in g_E 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.

2.6 Proper Weight

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, and is sometimes very useful. For example, proper acceleration is essential if you are designing an inertial guidance system. For details, see reference 5.

On the other hand, proper weight 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 acceleration. Obviously you can’t think of an object as accelerating relative to itself, but nevertheless proper acceleration is a well-defined concept.

2.7 Summary

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 section 2.3.

For an observer at rest relative to the laboratory, the total gravity 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 the plain old total gravity. That is, in practical situations, you can usually assume g = g_E, and in the terrestrial laboratory frame 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 primary gravity. Meanwhile, in other chapters, including anything related to practical weighing, “g” and “gravity” may refer to total 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.

3 The Direction of g; Vertical and Horizontal

As mentioned in section 2.3, the lab-frame weight m g_L is approximately but not exactly equal to m g_I. In theory, m g_L would exactly equal m g_I for the lab frame on an isolated airless nonrotating spherical homogeneous planet, but there aren’t any of those around here.

In this section we consider the direction of the gravity vector; if you are interested in the magnitude, see section 4.

The direction of our lab-frame gravity vector is consistent with the conventional notions of vertical and horizontal. A table in the laboratory is considered horizontal if things don’t spontaneously roll off it, which is dependent on g_L (not g_N or g_I).

The surface of a liquid at rest is an isopotential surface, locally and globally. The local orientation of this surface defines what we mean by horizontal. That is to say, the ordinary notion of “sea level” does not makes sense unless we include centrifugal terms in our notion of gravity. Operationally, this means that an undisturbed pool of water is horizontal, and an undisturbed plumb line is vertical.

The angle between the vectors g_L and g_I is about 0.1 degrees at temperate latitudes.

Here’s a parable: Once upon a time, some folks who misunderstood the distinction between g_L and g_I 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_I rather than to g_L, based on careful astronomical observations. As a result, one side was just over one inch too high, measured relative to the water. This looked really terrible. The point of the story is that in accordance with the laws of physics, the water distributed itself according to g=g_E=g_L, not g_I.

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 approach is entirely consistent with modern physics notions. We say that g means g_E, and 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.

4 The Magnitude of g; Weightlessness

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.

In this section we consider the direction of the gravity vector; if you are interested in the direction, see section 3.

The magnitude of g_L differs from the magnitude of g_I by about one third of a percent at the equator, and about one quarter of a percent at temperate latitudes. This would be very significant in ordinary laboratory practice if you used a scale that measured weight ... which is why laboratory scales are generally designed to measure mass, not weight. For example, a two-pan balance compares one mass with another in a way that is insensitive to small changes in |g|. The sensitivity of the balance depends on |g|, but the balance-point does not.

It turns out that variations in the direction of g are often more interesting than the variations in the magnitude.

Important examples include:

In the usual terrestrial lab frame, the dominant contribution to g_L is g_I, with significant corrections due (directly and indirectly) to the centrifugal field associated with the earth’s rotation.
Astronauts living in a space station correctly consider themselves weightless, because they are using a frame comoving with the spacecraft, and restricting attention to regions near the spacecraft. In that frame (and in those regions), g is millions of times smaller than g_N.

5 Units and Measurements

5.1 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 kilogram, 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 5.2.

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 abbreviation 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 even-more-non-standard abbreviation for gallon.

5.2 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.

5.3 Instruments

Note the contrast:

Some instruments measure mass. These are typically labeled in kg and g, or lb and oz ... which makes sense.

There are other instruments that measure force. Unfortunately, these are often labeled in mass units (such as kg) when they should be labeled in force units (such as kg_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.

However, 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 surface of the moon, a balance-type instrument is good for measuring mass. It can measure mass without needing to be recalibrated. However, 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.

6 That Falling Sensation

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

7 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

[Contents]

Definition of Weight, Gravitational Force, Gravity, g, et cetera

* Contents

1 Weight