Weight, Gravitational Force, Gravity, g, Latitude, et cetera

*   Contents

• 1  Weight
  • 1.1  Definitions
  • 1.2  Basic Properties
• 2  Various Different Notions of “Gravity”
  • 2.1  Extrinsic, Framative Gravity
  • 2.2  Intrinsic, Massogenic Gravity
  • 2.3  Digression: Standard Gravity
  • 2.4  Discussion
• 3  Some Common Reference Frames
  • 3.1  The Terrestrial Laboratory Frame
  • 3.2  Earth-Centered Nonrotating Frame
  • 3.3  Proper Weight
  • 3.4  Car, Aircraft, Spacecraft, etc.
  • 3.5  Other Frames
• 4  The Direction of g; Vertical and Horizontal
• 5  The Magnitude of g; Weightlessness
• 6  Units and Measurements
  • 6.1  Definitions of Units
  • 6.2  The Effect of Units on the Form of Laws
  • 6.3  Instruments
• 7  Latitude
• 8  That Falling Sensation
• 9  The Figure of the Earth
• 10  Gravitational Waves
• 11  References

1  Weight

1.1  Definitions

The weight of any simple object is a vector W and is defined as:

where m is the mass of the object; see reference 1 for a discussion of how to define “mass”. Meanwhile, g is the vector representing the local gravitational acceleration. We define g as:

• Note: By arbitrary convention, the term “gravitational field” refers to the gravitational acceleration, even though the gravitational potential is also a field and tidal stress is also a field (in accordance with the definition of “field”).

  This is consistent with a similar arbitrary convention that applies to the electrical field and magnetic field.

• Operationally, we can determine g using equation 2, by dropping an object and observing its acceleration. This technique is used in sophisticated, top-of-the-line gravimeters; see reference 2.
• We can equally well determing g using equation 1, by measuring the weight of a known mass. This technique is used in the accelerometer instruments you find in aircraft. A simple homemade accelerometer is described in reference 3.

Equation 1 says nothing about what may be causing the local gravitational field. The definition of g does not care “why” the free objects are accelerating. The only thing that matters is the phenomenological fact that they are accelerating. See section 2.2 for a discussion of physical processes that can produce a gravitational field, including place-to-place variations in the gravitational field.

Both W and g are frame-relative, as can be seen by comparing item 1, item 2, and item 3 in section 1.2.

1.2  Basic Properties

1.    In the terrestrial “laboratory” reference frame, the largest contribution to g can be calculated from the size and mass of Planet Earth, by means of Newton’s law of universal gravitation (equation 9). However, that is definitely not the only contribution to g in the lab frame, for reasons discussed in section 3.1.

2.    A pencil in an orbiting spaceship is weightless according to an observer in the ship’s reference frame. For more on this, see section 5.

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: Note the following contrast:

In this document, the weight vector is denoted by W, with no boldface or other decorations. The scalar magnitude of W is denoted by |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 4 for more on this.

Many books represent vectors using boldface symbols such as W, and use the corresponding non-boldface symbol to represent 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. Some other books write little arrows over vectors. Both of these conventions become inconvenient or outright impossible when multivectors are involved.

For these reasons, among others, the smart policy is to write vectors without any decoration, i.e. without boldface, without arrows over them, without lines under them, or anything like that.

Keeping track of what’s a vector and what’s not is no different from keeping track of the dimensions. Vectors do not require special notation. In complex documents, it may help to include a legend or a glossary, to systematically document what the various symbols mean.

5.    The W and g vectors are directed downward. I don’t think of this as part of the definition of W or g; instead, I think of it as 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 5.

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:

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

(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 are weightless (relative to the spaceship) 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 6.

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.    The word “gravity” is ambiguous, because it could refer to the gravitational potential, the gravitational acceleration, the tidal stress, et cetera. Except in cases where the details don’t matter, or are obvious from context, it is better to avoid the word “gravity” and use some more specific term instead.

Note that the term “gravitational field” by convention refers to the gravitational acceleration, even though the gravitational potential is also a field (in accordance with the usual definition of “field”). Similarly the tidal stress is a field.

12.    Even if we confine attention to things with dimensions of acceleration, there remain some very important ambiguities. Some more-explicit, less-ambiguous terminology is discussed in section 2.

13.    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 6.3.)

14.    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 (m) while the other, to a good approximation, responds to weight (mg).
• 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 Different Notions of “Gravity”

2.1  Extrinsic, Framative Gravity

The word “framative” is a contraction for “frame-relative”. We now restate equation 1, representing the framative weight and the framative gravitational acceleration using more-explict notation:

where the subscript “@f” means “relative to f”, where f is some chosen reference frame.

In situations where it is obvious from context which reference frame is intended, it is conventional to write simply g as shorthand for g_@f, and to call it “the” gravitational field. However, as always, shorthand entails some risk. When in doubt, use the more explict notation, g_@f.

It must be emphasized that different frames will have different g-values. This can happen even if both of the frames cover a single region. The g-value depends on what the frame is doing, not on what any particular particle is doing. To illustrate this point, consider two objects and two frames, making four cases altogether:

where @L refers to the lab frame, and @E refers to frame comoving with the elevator. In this example, the desk is stationary in the lab frame, while the elevator is in free fall.

The framative gravitational field g does not have to be the same everywhere. Suppose we have a bear located at the north pole, a penguin located at the south pole, and a troll located in a small hollow space at the center of the earth. Then the gravitational field g at these locations, measured relative to the earth-centered earth-fixed reference frame, is

where @ECNR refers to an earth-centered non-rotating reference frame, and x^ is a unit vector pointing toward the north celestial pole, roughly in the direction of the star Polaris. In the ECNR frame, the total acceleration a of the bear is zero, because the downward force of gravity is compensated by an upward force from the ice, pushing upward against his feet. Similar words apply to the penguin. The troll is not subject to any gravitational force or any contact force, so he just rattles around loose.

When you choose a frame, it doesn’t have to be local. For example, consider a freely falling elevator located at the north pole. Choose a special frame that is comoving with this elevator. The frame can be as small or as large as you like. We can even extend it to cover the entire earth. The bear is weightless, relative to this special frame. Meanwhile, the penguin is subject to an acceleration of twice the conventional terrestrial gravitational acceleration, relative to this special frame. In more detail:

Here “@ffNP” refers to “freely falling at the North Pole”. The bear is weightless in this frame, but is not in free fall. He is subject to an upward force from the ice. This upward force is not opposed by any gravitational force in this particular frame, so he accelerates upwards, towards Polaris.

The penguin is subject to a strong gravitational force towards Polaris, which is half-canceled by the force of the ice pushing on his feet.

More generally, each of the values in equation 7 is related to the corresponding value in equation 6 by adding 9.8 m/s/s in the +x^ direction.

2.2  Intrinsic, Massogenic Gravity

We can use equation 6 to calculate the difference between the gravitational field acting on the bear and the gravitational field acting on the troll. We can also use equation 7 to calculate the same thing, and we get the same value. In fact, we get the same value in any nonrotating frame.

The δg on the LHS of equation 8 is the difference between two gravitational fields. Newton’s law of universal gravitation gives us a more general expression for δg, namely:

where G is the universal gravitational constant, and (in this example) M is some mass (such as the mass of the Earth), and r^ ≡ r/|r| is a unit vector in the r-direction.

Specifically, δg(R, r) is the difference between the gravitational field at location R+r and location R, and δg_M(R, r) is one contribution to this difference, namely the contribution from a mass concentrated at location R. The result is the same for any R, so the R drops out, and we write the shorthand expression δg_M(r).

We call δg_M the massogenic contribution because it is proportional to the mass M. In a non-rotating frame, contributions of this form will be the only contributions to δg. However, in a rotating frame, there will also be a centrifugal contribution. (The procedures for dealing with rotating reference frames in general, and centrifugal fields in particular, are discussed in reference 7.)

The RHS of equation 9 is manifestly frame-independent. It should be clear from section 2.1 that there cannot be any frame-independent expression for g(bear) or g(troll) separately, but we can have a frame-independent expression for the difference between the two. The difference is proportional to the mass of the planet, and inversely proportional to the square of the distance.

Strictly speaking, equation 9 applies only to a small, localized mass. It applies in the “classical” limit, which requires that the fields must be not too strong and not too rapidly changing. It also requires that the mass M must be not too rapidly spinning. These requirements are satisfied in the solar system, with a wide margin of safety for all practical purposes. In a non-classical situation, equation 9 must be replaced by the equations of general relativity.

There is a theorem due to Newton that says that if you are outside a uniform spherical shell of mass, its contribution to the gravitational field is the same as if all the mass were concentrated at the center. If you are inside such a shell, its contribution to the gravitational field is zero. Therefore, to the extent that a planet can be considered to be made up of a series of concentric shells, where each shell has a uniform density, then equation 9 applies directly and simply. The mass on the RHS is the total mass.

On the other hand, if the planet is not isotropic, equation 9 can still be applied, but brute force is required. The procedure is to break the planet into a great number of small regions, apply the equation to each region separately, and then add up all the contributions to the field.

2.3  Digression: Standard Gravity

There is one more notion of “the” gravitational acceleration that we should mention. As will be discussed in section 6.1, the “standard” acceleration of gravity is defined by fiat 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_@L| at sea level in temperate latitudes is close to g_N, with a small fraction of a percent.

2.4  Discussion

1. The local framative gravitational acceleration g_@L is what matters to an apartment building or a grandfather clock or anything else that sits in one place near the earth’s surface. Therefore g_@L is “the” gravitational acceleration relevant to architects and clockmakers.
2. In contrast, the massogenic contribution δg_M(r) is “the” gravitational acceleration relevant to astrophysicists. You can get away with setting g = δg_M if there is only one large mass in the problem, and you always choose an unaccelerated non-rotating reference frame centered on the mass M.
3. In many situations, there is a choice, and different people could reasonably choose differently. For example, when analyzing an orbital rendezvous maneuver, you could choose an earth-centered non-rotating frame, or an earth-centered earth-fixed co-rotating frame, or a frame comoving with one of the spacecraft. The g-value will be different in each case. The only sane path forward is to be explicit about what frame is being used.
4. Beware: Many introductory-level physics textbooks use inconsistent terminology, switching back and forth from one definition to another. In the chapter on universal gravitation, “g” and “gravitational acceleration” probably refer to the intrinsic, universal, incremental field, δg. Meanwhile, in other chapters, including anything related to practical weighing, “g” and “gravitational acceleration” almost certainly refer to the extrinsic framative field, g_@L.

5. Note the sun’s gravitational field differs in direction and magnitude as you go from place to place on earth. This difference in gravitational field gives rise to something we call tidal stress, which indirectly drives the observed tides.

   The same idea applies to the intrinsic, universal, incremental δg. It is a difference in gravitational field between one place and another, so in some fundamental conceptual sense, it is tidal.

6. As you may have heard, general relativity explains “gravitation” in terms of the curvature of spacetime. It must be emphasized that curvature has to do with the intrinsic δg. The curvature of spacetime produces tidal stress. Curvature has got absolutely nothing to do with the local, extrinsic, framative g. See reference 8 for a simple description of how this works.

3  Some Common Reference Frames

We now consider various reference frames that are commonly used, and discuss some of the physical processes that contribute to the value of the framative g in each frame.

3.1  The Terrestrial Laboratory Frame

The gravitational acceleration 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 ... or by weighing a known mass. Considerable accuracy can be achieved, relatively easily, using a reversible pendulum (Kater pendulum, or something along those lines). See reference 9.

In the lab frame, g_@L includes the following contributions:

• The universal gravitation term, δg(r, M), where M is the mass of the earth. 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. This changes the radius-vector r that appears in the law of universal gravitation and in the centrifugal-force law.
• Topographic elevation, which also changes r.
• Local variations in the density of the earth.
• The direct and indirect effects of lunar gravity, including the monthly motion of the earth around the center-of-mass of the earth/moon system.
• The direct and indirect effects of solar gravity, including the annual motion of the earth around the center-of-mass of the earth/sun system.
• Weather-related variations in the mass of air overhead.
• Various smaller terms.

To repeat: The difference |g_@L − δg(r, M)| is not zero. It is usually less than 1% of |g_@L| but not much less, depending on latitude, elevation, et cetera.

3.2  Earth-Centered Nonrotating Frame

We can with a little effort construct a situation where the actual gravitational acceleration g is very nearly equal to δg(r, M). 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 sometimes convenient 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 9. That is, m g is very nearly m δg(r, M). All of the correction terms mentioned in section 3.1 must still be considered, which means that the ECNR frame is not exactly an inertial frame.

• 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 ECNR frame is “almost” inertial. That is, the correction terms are fairly small. For example, the acceleration of the earth in its annual orbit is only 0.06% as large as the earth’s surface gravity. That’s 600 parts per million.

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

3.3  Proper Weight

For a rigid object, its proper weight 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 10.

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. Thirdly, 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.

3.4  Car, Aircraft, Spacecraft, etc.

For an observer traveling in a vehicle – perhaps a car, aircraft, or spacecraft – it is very common to use a frame of reference that is instantaneously comoving with the vehicle. There are many advantages to doing so. There is however some cost to doing so, because if the vehicle’s reference frame is accelerating or turning, this must be included in the definition of the vehicle’s framative gravitational acceleration, g_@V.

For a car moving in a straight line relative to the laboratory, g_@V and g_@L will be the same. In contrast, if the car is zooming over the crest of hill, or if it is making a sharp turn, g_@V and g_@L will be significantly different.

3.5  Other Frames

The examples mentioned so far 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 3.1.

4  The Direction of g; Vertical and Horizontal

As mentioned in section 3.1, the lab-frame weight m g_@L is approximately but not exactly equal to m δg(r, M). In theory, m g_@L would exactly equal m δg(r, M) 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 gravitational acceleration vector; if you are interested in the magnitude, see section 5.

The direction of our lab-frame gravitational acceleration 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 δg(r, M)).

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 δg(r, M) 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 δg(r, M) 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 δg(r, M) 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 δg(r, M).

To summarize: in practical terrestrial applications, the gravitational acceleration 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.

5  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 gravitational acceleration vector; if you are interested in the direction, see section 4.

The magnitude of g_@L differs from the magnitude of δg(r, M) 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 δg(r, M), 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.

6  Units and Measurements

6.1  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 the “standard” value, but of course varies with time, location, elevation, et cetera. Reference 6 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 11 and reference 12.

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. Some constructive alternatives are discussed in section 6.2.

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.

6.2  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 the 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.

6.3  Instruments

Note the contrast:

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.

7  Latitude

Because of the earth’s centrifugal field, and for other reasons discussed in section 3.1 and especially section 4, the conventional “down” direction does not point toward the center of the earth. So you might be wondering how to measure the difference between these two directions.

First let me outline a seemingly plausible measurement scheme and explain why it does not answer the question:

• Use a sextant to measure the height of the celestial pole above the horizon, and compare that to your latitude.
• Equivalently, perhaps more accurately, use a sextant to measure what stars are overhead, and compare their declination to your colatitude.

As it turns out, that scheme doesn’t work, because by tradition the “latitude” is defined to be the geodetic latitude, i.e. defined to match what you see in the sextant! It is defined in terms of the local notion of horizontal and vertical, which includes the centrifugal acceleration. All that makes sense if you think about how latitude has been used for navigation over the centuries.

To say the same thing the other way, the unadorned term “latitude” does not correspond to the geocentric latitude, i.e. the angle as seen from the center of the earth.

As a consequence of the definition, parallels of latitude are not equally spaced. This is easy to visualize if you extrapolate to a planet with extreme flattening, so that the geoid looks like a pancake.

So, if you want to measure what’s going on, it requires nothing more than a highly accurate map, or (preferably) accurate survey data. Measure the north-south distance between two points, and compare it to the difference in latitude. Do this twice: Once for a pair of points near the equator, and again for pair of points far from the equator.

You can simulate the experiment using a good geodesy software package. Beware that there are some not-so-good packages out there. I haven’t done exhaustive checks, but so far I’ve been happy with GeographicLib by Charles Karney.

I calculate that one degree of latitude is

8  That Falling Sensation

Reference 13 addresses the bodily feelings associated with free fall, weight, and/or weightlessness.

9  The Figure of the Earth

The earth is not a sphere. To a good approximation, it is an ellipsoid. That is the natural shape for a spinning, gravitating object in mechanical equilbrium. A wonderful review can be found in reference 14. It outlines the contributions of Newton, Maclaurin, Jacobi, Meyer, Liouville, Dirichlet, Dedekind, Riemann, Poincaré, and Cartan.

It is well known that the gravitational field is less at the equator than at the poles. The variation is not huge, but it is nontrivial. It is easily measurable using simple instruments, especially if you measure it at widely-separated places and compare notes. If you wish to calculate this from first principles, you should follow the recipe given in reference 14. The general case requires evaluating lots of spherical harmonics, but when the ellipsoid is not very eccentric, things are a lot simpler.

In contrast, it would not correct to treat the earth as a point mass and calculate the field using only the universal massogenic formula, equation 9. It’s tempting to try that, because the radius is greater at the equator, so you might think the weaker field is explained by the 1/r² dependence in equation 9 ... but that’s not the correct physics. It’s not even close. It overestimates the effect. That’s because there’s more stuff under your feet at the equator, and less at the poles.

Newton showed that the field of a uniform spherical object is the same as you would get from a point mass at the center ... but he also showed that the same cannot be said of an ellipsoid.

Also it would not be correct to neglect the centrifugal field. The usual terrestrial lab frame is a rotating frame, and the observed g in that frame contains a smallish but nontrivial contribution from the centrifugal field.

Figure 3 shows some data, and compares it to various models. The black circles are calculated using the EGM2008 model, evaluated at various locations around the globe. The model is based on thousands of measurements, and uses thousands of adjustable parameters, so it should be pretty good. I checked the model against one actual direct measurement (Blackford Hill) and it agreed quite closely. The scatter in the data is presumably due to inhomogeneities in the earth, as a function of longitude. The red circles are from reference 15; I have no idea whether they are based on a model or direct measurements.

All the plotted points are corrected to sea level. If you want the actual measured acceleration at any location, starting from figure 3, you need to apply an elevation correction.

The EGM2008 model can be evaluated as a function of latitude, longitude, and elevation using the Gravity program which is part of the Geographiclib suite.

Figure 3: Gravity as a Function of Latitude

The dashed black line shows the plain old “standard” gravity. It’s a good approximation if you live in Ottawa, Seattle, or Paris ... but not so good if you live at a significantly higher or lower latitude.

The red curve is what you would get by treating the earth as a point mass and calculating the field on the surface of the ellipsoid, plugging the geocentric radius into equation 9.

The yellow curve is what you would get by combining the universal massogenic contribution plus the centrifugal contribution. It is in the right ballpark in the mid-latitudes, as you might expect ... but at higher or lower latitudes if you want a decent result, you have to move beyond the point-mass approximation. That is, you need to account for the fact that there is more stuff under your feet at the equator than at the poles.

Last but not least, the magenta curve is equation 11, the Somigliana equation. This could be considered an empirical three-parameter fit, fitting the gravity data as a function of latitude, without regard to longitude. It fits the data remarkably well for such a simple model. However, it’s even better than that, because it is not entirely an empirical model. It’s well founded in the physics, or at least it would be for a star or planet where the surface was a gravitational equipotential, as the earth would be without the dry land – and especially without the mountain ranges. The equation is:

where φ is the geodetic latitude, g(0) is the gravitational acceleration at the equator, e is the eccentricity, and k is a parameter that has to do with the density distribution. The earth has a k value greater than 0, which indicates it is denser on the inside than it is near the surface.

For a fluid with uniform density, the problem would be overconstrained, since we get to measure both the radius and the gravitational field as functions of latitude. We can get a lot of mileage out of the fact that the fluid surface has to be an equipotential. For the real earth, the radius and field measurements suffice to firmly reject the uniform-density hypothesis.

There are other three-parameter fits on the market, such as the International Gravity Model, but they aren’t any simpler don’t work as well as the Somigliana equation, and are less well founded in the physics.

See reference 16 for the spreadsheet used to produce figure 3. It contains the data and some other gravity-related calculations.

If you want some data but don’t feel like installing the Geographiclib package, there are web apps such as reference 17 or reference 18 that will give you the g value at arbitrary locations. Also, you can find g values for selected locations on the appropriate page in reference 16, or in a simpler format in reference 19.

10  Gravitational Waves

This is a very complicated subject. Reference 20 discusses some of the key concepts and attempts to dispel some of the most pernicious misconceptions.

11  References

1.

John Denker,
“How to Define Mass”
www.av8n.com/physics/mass.htm

2.

T.M. Niebauer, G.S. Sasagawa, J.E. Faller, R. Hilt, and F. Klopping
“A new generation of absolute gravimeters”
http://www.microglacoste.com/pdf/newgeneration.pdf

3.

John Denker
“Simple Accelerometer”
www.av8n.com/physics/accelerometer.htm

4.

John Denker,
“Introduction to Vectors”
www.av8n.com/physics/vector-intro.htm

5.

John Denker,
“Velocity, Speed, Acceleration, and Deceleration”
www.av8n.com/physics/acceleration.htm

6.

GEON project,
“GeoNet Gravity and Magnetic Dataset Repository”
http://www.geongrid.org/index.php/gateways/gravitymagnetics

7.

John Denker,
“Motion in a Rotating Frame”
www.av8n.com/physics/rotating-frame.htm

8.

John Denker,
“Tabletop Geodesics, General Relativity, and Embedding Diagrams”
www.av8n.com/physics/geodesics.htm

9.

Wikipedia article: Kater’s pendulum
https://en.wikipedia.org/wiki/Kater’s_pendulum

10.

John Denker,
“Acceleration in Spacetime”
www.av8n.com/physics/spacetime-acceleration.htm

11.

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

12.

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

13.

John Denker,
“Can You Feel Gravity?”
www.av8n.com/physics/gravity-perception.htm

14.

S. Chandrasekhar,
“Ellipsoidal Figures of Equilibrium - An Historical Account”
Communications on Pure and Applied Mathematics, xx, 251–265 (1967)
http://people.ucsc.edu/~igarrick/EART290/chandrasekhar_1967.pdf

15.

Elizabeth Chesick
“Acceleration due to Gravity”
http://www.haverford.edu/educ/knight-booklet/accelarator.htm

16.

John Denker,
“Spreadsheet for Calculating Gravity as a Function of Latitude etc.”
./gravity-centrifugity.xls

17.

Andreas Lindau,
Physikalisch-Technische Bundesanstalt
“Gravity Information System”
https://www.ptb.de/cartoweb3/SISproject.php

18.

NIST
“Surface Gravity Prediction”
https://geodesy.noaa.gov/cgi-bin/grav_pdx.prl

19.

“Gravity Data from EGM2008”
./lat-lon-g.csv

20.

John Denker,
“Pre-Introduction to Gravitational Waves”
www.av8n.com/physics/gravitational-wave-intro.htm