Copyright © 2002 jsd
Weight, Gravitational Force, Gravity, g, Latitude, et cetera
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:
This is consistent with a similar arbitrary convention that applies to the electrical field and magnetic field.
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.
|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.
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:
without even mentioning weight, as opposed to
Equation 3 has the advantage of logic and elegance, but it hasn’t caught on very widely.
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.
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.
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.)
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.
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.
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:
g@E = 0
g@L = 9.8 m/s/s
object on desk,
g@E = 0
object on desk,
g@L = 9.8 m/s/s
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. 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.
Note that the frame does not have to be local, in the sense that even if you have a small elevator, the frame comoving with the elevator can be as small or as large as you like. For example, consider a special frame that is comoving with a freely falling elevator. The elevator is at the north pole, but we extend the frame 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:
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.
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 δgM(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 δgM(r).
We call δgM the barogenic 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.
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 gN. The direction is unspecified.
This gN 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 gN, with a small fraction of a percent.
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.
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.
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.
In the lab frame, g@L includes the following contributions:
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.
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.
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.
The remaining correction terms are on the order of a few parts per million, or smaller.
For a rigid object, its proper weight is the mass times the proper acceleration gp. 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 9.
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.
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.
The examples mentioned so far are definitely not the only g-like quantities that can be defined. For starters, you can define 2N different g-like quantities just by selectively including or excluding the “corrections” mentioned in section 3.1.
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 gN 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=gE=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 gE, 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.
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:
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 gN but everybody I know calls it Gee or G (not to be confused with Newton’s constant of universal gravitation, also denoted G).
This gN 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” (kgf), 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 10 and reference 11.
There also exists, informally, the “pound of force” (lbf), 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 lbf. In general, though, lbf 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.
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,
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 kgf).|
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:
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:
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
Reference 12 addresses the bodily feelings associated with free fall, weight, and/or weightlessness.
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 13. 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 13. 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 barogenic 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 14; 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.
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 barogenic 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 15 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, you can find g values for selected locations on the appropriate page in reference 15, or in a simpler format in reference 16.
Copyright © 2002 jsd