Weight

A spring scale measures the weight of an object

In physics, there are various, nonequivalent definitions of the concept of weight.

In one of the more common definitions, the weight of an object, often denoted by W, is defined as being equal to the force exerted on it by gravity. This force is the product of the mass m of the object and the local gravitational acceleration g.[1] Expressed in a formula: W = mg. In the International System of Units, the unit of measurement for weight is the same as that for force: the newton.

On the surface of the Earth, the acceleration due to gravity is approximately constant; this means that the magnitude of an object's weight on the surface of the Earth is roughly proportional to its mass. In situations other than that of a constant position on the Earth, so long as the acceleration does not change, the weight of an object (the force it exerts against support in any accelerated frame) is proportional to its mass, also. In everyday practical use, therefore, including commercial use, the term "weight" is commonly used to mean "mass."[2]

Other definitions of the physical concept are detailed below.

Contents

Definitions

There are several definitions of the physical notion of weight, not all of which are equivalent.[1][3][4][5] In a purely scientific context, the concept of weight is rarely used (except for physics textbooks); however, the use of the physical concept of weight is common enough in technology.

Main differences are:

Gravitational

One of the most common definitions of weight found in introductory physics textbooks defines weight as the force exerted on a body by gravity.[5] This is often expressed in the formula W = mg, where W is the weight, m the mass of the object, and g gravitational acceleration.

This definition was established in Resolution 2 of the 3rd General Conference on Weights and Measures (CGPM) of 1901:

"The word weight denotes a quantity of the same nature[Note 1] as a force: the weight of a body is the product of its mass and the acceleration due to gravity."[7]

This resolution defines weight as a vector, since force is a vector quantity. However, some textbooks also take weight to be a scalar by defining:

"The weight W of a body is equal to the magnitude Fg of the gravitational force on the body."[8]

The gravitational acceleration varies from place to place. Sometimes, it is simply taken to a have a standard value of 9.80665 m/s2, which gives the standard weight.[6]

Operational

In the operational definition, the weight of an object is the force measured by the operation of weighing it, which is the force it exerts on its support.[3] This can make a considerable difference, depending on the details; for example, an object in free fall exerts little force on its support, if any, a situation that is commonly referred to as weightlessness. However, being in free fall does not affect the weight according to the gravitational definition. Therefore, the operational definition is sometimes refined by requiring that the object be at rest. However, this raises the issue of defining "at rest" (usually being at rest with respect to the Earth is implied by using standard gravity.

A minor issue with the formulation is that the operational definition, as usually given, does not take buoyancy into account. However, this is an instrument-dependent problem (since in theory, an object will always be weighed in a vacuum, with the correct instrument).[5]

The ISO definition

In the ISO International standard ISO 80000-4(2006), which is a part of the International standard ISO/IEC 80000, the definition of "weight" and remarks concerning that definition are given as

" Fg = m g

where m is mass and g is local acceleration of free fall.

It should be noted that, when the reference frame is Earth, this quantity comprises not only the local gravitational force, but also the local centrifugal force due to the rotation of the Earth.

The effect of atmospheric buoyancy is excluded in the weight.

In common parlance, the name "weight" continues to be used where "mass" is meant, but this practice is deprecated. "

The following points are emphasized in this definition of "weight":

The international standard ISO 80000-4(2006), describing the basic physical quantities and units in mechanics, cancels and replaces the second edition of ISO 31-3:1992. The major technical changes introduced in comparison with the previous standards were the following:

The new standard ISO 80000-4(2006) stresses some details concerning the definition of weight which were not clearly stated in the previous standard ISO 31-3:1992. The definition of the weight in the cancelled standard ISO 31-3:1992, with commentaries, is given in the following text.

The ISO standard ISO 31-3 (1992) defines weight as follows:

The weight of a body in a specified reference system is that force which, when applied to the body, would give it an acceleration equal to the local acceleration of free fall in that reference system.[9]

This definition allows use of the formula "W = m g", with g interpreted as the local acceleration of free fall in the specified frame.[10] The definition is dependent on the chosen frame of reference. When the chosen frame is co-moving with the object in question then this definition precisely agrees with the operational definition.[4] If however the specified frame is the one of the surface of the Earth, then the definition agrees with the gravitational definition.

Weight here is the force necessary to put an object in an "particular reference frame," (which must be an accelerated frame if the body is to have any weight at all) into a free-fall frame, instead. If such a body is not already in free fall, and yet is stationary (as it must be in its particular reference frame, where it has weight) this requires that the body already is being acted upon by a force, which acts against its weight. This force, a supporting force, is responsible for its acceleration (which is also the acceleration of its frame). This force causes a measurable proper acceleration which is measurable by an accelerometer. This acceleration is, by definition, the acceleration of an object away from the acceleration of free fall. The object's weight must be exactly equal to this supporting force, but in the opposite direction, in order to keep the object motionless in its "particular reference frame."

For example, an object sitting on a spring-scale on a table on the surface of the Earth (an accelerated frame) is subject to a supporting-force from the scale and table, which is exactly enough to keep it from going into free fall, in the scale and table's gravitationally accelerated reference frame. This force causes the object's 1-g proper acceleration, which is in a direction upward. This acceleration can be directly measured as a 1-g acceleration upward, by an accelerometer affixed to the object, or to its reference frame (see the article on g-force). In the object and table's frame, this force is balanced, by Newton's third law, by the counter-force of the object's weight, which is measured as a downward force, by the scale. If the table and scale are removed, however, the force of the object's weight is exactly enough to put it into free-fall, by the ISO definition, and it will therefore go into free-fall. At that time, an accelerometer placed on it will read zero, and it will have no weight. (Objects in a free-fall, obeying Newton's first law in an inertial frame, are weightless).

In a similar situation where an object is on a scale on-board a rocket accelerating at 1-g in deep-space, the weight of the object will be measured as the same as on the Earth by the scale, and an accelerometer will show the same 1-g upward proper acceleration. However, in this case the acceleration is produced by the rocket engine, and the weight of the object is provided by the fictitious force (inertial force) associated with it being in the accelerated rocket-frame, rather than the similar gravitational force which causes positionally stationary frames near a mass to appear accelerated (these frames have a proper acceleration, even if they have no coordinate acceleration). In a centrifuge, or other similar accelerated frame system, weight is due to a similar fictitious inertial force (centrifugal force), and is a function of the system's proper acceleration, which is its difference in acceleration from a free-fall reference frame.

The identical operational and the ISO definitions for weight do in themselves take into consideration the practical fact that a scale under an object cannot be expected to measure its full weight, if the object is supported, in part or in whole, by some other means which does not transfer downward weight-force to the scale. Such unmeasured support detracts from an object's weight and may give it a false apparent weight. For example, an object might be suspended over a scale by a rope from a stand, and the scale would read an apparent weight of zero. This does not mean the object's weight is zero, but merely that the scale mechanism has been circumvented, by being placed somewhere other than the structures that supply the supporting force for the object; in this case, the object's weight would be correctly reported if the scales were placed under the stand. Similarly, an object undergoing levitation in a magnetic field does not actually lose its weight; rather the full weight would be shown if the scale were placed under the structures that supply the levitating field.

In a similar fashion, the apparent weight of objects immersed in a fluid may be reported incorrectly by a scale placed immediately under the object, but this is only because the fluid, like the rope in the example above, has transferred some of the support for the object, to a surface supporting the fluid, where the scale does not measure the increase in weight. This does not happen if the entire fluid mass is supported by the scale: for example, if a beaker of water is placed upon a scale and an object dropped into the beaker, the entire weight of the object will be is shown by the scale, no matter to what extent it is supported locally by buoyancy. In a similar fashion, objects immersed in air show a slightly smaller apparent weight, but this is only because scales do not measure the increased pressure and thus weight of the entire atmosphere (which would show the weight difference from true weight, directly). Such measurements are impractical, and therefore to correct for the buoyancy of air, the apparent weight of objects weighed by a spring-scale in air must have an additional calculated measure added, using the product of the density of air and the object's volume, as described in Archimedes' principle. However, the true weight of the object in such circumstances is unchanged, just as in the other "unmeasured support" examples.

Vector or scalar

The definitions of the physical concept of weight given above define it as a vector quantity, having both magnitude and direction. For an object at rest on the surface of the Earth, its weight is a force that points down, approximately towards the centre of the Earth. In spite of this, the vector aspect is usually ignored in common scientific discourse, and "the weight" is used to denote a scalar quantity, where, according to the definition employed, "the magnitude of the weight" would be more appropriate. Some physics textbooks define weight outright as a scalar quantity, as in the following definition:

"The weight W of a body is equal to the magnitude Fg of the gravitational force on the body."[8]

Weight and mass

In modern scientific usage, weight and mass are fundamentally different quantities: mass is an intrinsic property of matter, whereas weight is a force that results from the action of gravity on matter: it measures how strongly the force of gravity pulls on that matter. However, in most practical everyday situations the word "weight" is used when, strictly, "mass" is meant.[2][11] For example, most people would say that an object "weighs one kilogram", even though the kilogram is a unit of mass.

The scientific distinction between mass and weight is unimportant for many practical purposes because the strength of gravity is almost constant everywhere on the surface of the Earth. In a constant gravitational field, the gravitational force exerted on an object (its weight) is directly proportional to its mass. For example, object A weighs 10 times as much as object B, so therefore the mass of object A is 10 times greater than that of object B. This means that an object's mass can be measured indirectly by its weight, and so, for everyday purposes, weighing (using a weighing scale) is an entirely acceptable way of measuring mass. Conversely, a balance actually measures mass, not weight (in the scientific sense), but the quantity thus determined is still called "weight" in everyday use.

The Earth's gravitational field is not actually constant but can vary by as much as 0.5%[12] at different locations on Earth (see Earth's gravity). These variations alter the relationship between weight and mass, and must be taken into account in high precision weight measurements that are intended to indirectly measure mass. Spring scales, which measure local weight, must be calibrated at the location at which the objects will be used to show this standard weight, to be legal for commerce.

This table shows the variation of acceleration due to gravity (and hence the variation of weight) at various locations on the Earth's surface.[13]

Location Latitude m/s2
Equator 0° 9.7803
Sydney 33° 52´S 9.7968
Aberdeen 57° 9´N 9.8168
North Pole 90° N 9.8322

The historic use of "weight" for "mass" also persists in some scientific terminology – for example, the chemical terms "atomic weight", "molecular weight", and "formula weight", can still be found rather than the preferred "atomic mass" etc.

In a different gravitational field, for example, on the surface of the Moon, an object can have a significantly different weight than on Earth. The gravity on the surface of the Moon is only about one-sixth as strong as on the surface of the Earth. A one-kilogram mass is still a one-kilogram mass (as mass is an intrinsic property of the object) but the downward force due to gravity, and therefore its weight, is only one-sixth of what the object would have on Earth. So a 180-pound man on Earth weighs only about 30 pounds when visiting the Moon.

Units

Four examples of mass and force units
System FPS Engineering FPS Gravitational FPS Absolute Metric
Force (F) F = m·a F = m·a/gc = w·a/g F = m·a F = m·a
Weight (w) w = m·g w = m·g/gc ≈ m w = m·g
Units
Acceleration (a) ft/s2 ft/s2 ft/s2 m/s2
Mass (m) slug pound-mass pound kilogram
Force (F) pound pound-force poundal newton

SI units

In most modern scientific work, physical quantities are measured in SI units. The SI unit of force (and hence weight in the mechanics sense) is the same as that of force: the newton (N) – a derived unit which can also be expressed in SI base units as kg·m/s2 (kilograms times meters per second squared).[11]

In commercial and everyday use, the term "weight" is usually used to mean mass, and the verb "to weigh" means "to determine the mass of" or "to have a mass of". Used in this sense, the proper SI unit is the kilogram (kg).[11]

The pound and other non-SI units

In United States customary units, the pound can be either a unit of force or a unit of mass. Related units used in some distinct, separate subsystems of units include the poundal and the slug. The poundal is defined as the force necessary to accelerate an object of one-pound mass at 1 ft/s2, and is equivalent to about 1/32.2 of a pound-force. The slug is defined as the amount of mass that accelerates at 1 ft/s2 when one pound-force is exerted on it, and is equivalent to about 32.2 pounds (mass).

The kilogram-force is a non-SI unit of force, defined as the force exerted by a one kilogram mass in standard Earth gravity (equal to 9.80665 newtons exactly). The dyne is the cgs unit of force and is not a part of SI, while weights measured in the cgs unit of mass, the gram, remain a part of SI.

Sensation of weight

The sensation of weight is caused by the force exerted by fluids in the vestibular system –a three-dimensional set of tubes in the inner ear. It is actually the sensation of g-force, regardless of whether this is due to being stationary in the presence of gravity, or, if the person is in motion, the result of any other forces acting on the body such as in the case of acceleration or deceleration of a lift, or centrifugal forces when turning sharply.

Measuring weight

A weighbridge, used for weighing trucks

Weight is commonly measured using one of two methods. A spring scale or hydraulic or pneumatic scale measures local weight, the local force of gravity on the object (strictly apparent weight force). Since the local force of gravity can vary by up to 0.5% at different locations, spring scales will measure slightly different weights for the same object (the same mass) at different locations. To standardize weights, scales are always calibrated to read the weight an object would have at a nominal standard gravity of 9.80665 m/s2 (approx. 32.174 ft/s2). However, this calibration is done at the factory. When the scale is moved to another location on Earth, the force of gravity will be different, causing a slight error. So to be highly accurate, and legal for commerce, spring scales must be re-calibrated at the location at which they will be used.

A balance on the other hand, compares the weight of an unknown object in one scale pan to the weight of standard masses in the other, using a lever mechanism – a lever-balance. The standard masses are often referred to, non-technically, as "weights". Since any variations in gravity will act equally on the unknown and the known weights, a lever-balance will indicate the same value at any location on Earth. Therefore, balance "weights" are usually calibrated and marked in mass units, so the lever-balance measures mass by comparing the Earth's attraction on the unknown object and standard masses in the scale pans. In the absence of a gravitational field, away from planetary bodies (e.g. space), a lever-balance would not work, but on the Moon, for example, it would give the same reading as on Earth. Some balances can be marked in weight units, but since the weights are calibrated at the factory for standard gravity, the balance will measure standard weight, i.e. what the object would weigh at standard gravity, not the actual local force of gravity on the object.

If the actual force of gravity on the object is needed, this can be calculated by multiplying the mass measured by the balance by the acceleration due to gravity – either standard gravity (for everyday work) or the precise local gravity (for precision work). Tables of the gravitational acceleration at different locations can be found on the web.

Gross weight is a term that generally is found in commerce or trade applications, and refers to the total weight of a product and its packaging. Conversely, net weight refers to the weight of the product alone, discounting the weight of its container or packaging; and tare weight is the weight of the packaging alone.

Relative weights on the Earth, other celestial bodies and the Moon

The table below shows comparative gravitational accelerations at the surface of the Sun, the Earth's moon, each of the planets in the solar system. The “surface” is taken to mean the cloud tops of the gas giants (Jupiter, Saturn, Uranus and Neptune). For the Sun, the surface is taken to mean the photosphere. The values in the table have not been de-rated for the centrifugal effect of planet rotation (and cloud-top wind speeds for the gas giants) and therefore, generally speaking, are similar to the actual gravity that would be experienced near the poles.

Body Multiple of
Earth gravity		 m/s2
Sun 27.90 274.1
Mercury 0.3770 3.703
Venus 0.9032 8.872
Earth 1 (by definition) 9.8226[14]
Moon 0.1655 1.625
Mars 0.3895 3.728
Jupiter 2.640 25.93
Saturn 1.139 11.19
Uranus 0.917 9.01
Neptune 1.148 11.28

See also

Other meanings of weight are those related to health:

Notes

  1. ↑ The phrase "quantity of the same nature" is a literal translation of the French phrase grandeur de la même nature. Although this is an authorized translation, VIM 3 of the International Bureau of Weights and Measures recommends translating grandeurs de même nature as "quantities of the same kind".[6]

References

  1. ↑ 1.0 1.1 Gat, Uri (1988). "The weight of mass and the mess of weight". In Richard Alan Strehlow. Standardization of Technical Terminology: Principles and Practice – second volume. ASTM International. pp. 45–48. ISBN 978-0-8031-1183-7. http://books.google.com/books?hl=en&lr=&id=CoB5w9Km0mUC&oi=fnd&pg=PA45. 
  2. ↑ 2.0 2.1 The National Standard of Canada, CAN/CSA-Z234.1-89 Canadian Metric Practice Guide, January 1989:
    • 5.7.3 Considerable confusion exists in the use of the term "weight." In commercial and everyday use, the term "weight" nearly always means mass. In science and technology "weight" has primarily meant a force due to gravity. In scientific and technical work, the term "weight" should be replaced by the term "mass" or "force," depending on the application.
    • 5.7.4 The use of the verb "to weigh" meaning "to determine the mass o f," e.g., "I weighed this object and determined its mass to be 5 kg," is correct.
  3. ↑ 3.0 3.1 Allen L. King (1963). "Weight and weightlessness". American Journal of Physics 30: 387. doi:10.1119/1.1942032. 
  4. ↑ 4.0 4.1 A. P. French (1995). "On weightlessness". American Journal of Physics 63: 105–106. doi:10.1119/1.17990. 
  5. ↑ 5.0 5.1 5.2 Galili, I.; Lehavi, Y. (2003). "The importance of weightlessness and tides in teaching gravitation". American Journal of Physics 71 (11): 1127–1135. doi:10.1119/1.1607336. http://sites.huji.ac.il/science/stc/staff_h/Igal/Research%20Articles/Weight-AJP.pdf. 
  6. ↑ 6.0 6.1 Working Group 2 of the Joint Committee for Guides in Metrology (JCGM/WG 2) (2008) (in English and French). International vocabulary of metrology — Basic and general concepts and associated terms (VIM) — Vocabulaire international de métrologie — Concepts fondamentaux et généraux et termes associés (VIM) (JCGM 200:2008) (3rd ed.). BIPM. Note 3 to Section 1.2. http://www.bipm.org/utils/common/documents/jcgm/JCGM_200_2008.pdf. 
  7. ↑ Barry N. Taylor and Ambler Thompson, ed (2008). The International System of Units (SI). NIST Special Publication 330 (2008 ed.). NIST. p. 52. http://physics.nist.gov/Pubs/SP330/sp330.pdf. 
  8. ↑ 8.0 8.1 Halliday, David; Resnick, Robert; Walker, Jearl (2007). Fundamentals of Physics, Volume 1 (8th ed.). Wiley. p. 95. ISBN 978-0-470-04473-5. 
  9. ↑ Technical committee 12 (1992). Quantities and units -- Part 3: Mechanics. International Standards Organization. 
  10. ↑ "The Semantics Problems on the Definitions of Weight". ERAS Conference 2009. Educational Research Association of Singapore. November 2009. http://www.eras.org.sg/papers/2-4-34.doc. Retrieved 2010-06-09. 
  11. ↑ 11.0 11.1 11.2 A. Thompson and B. N. Taylor (July 2, 2009 (last updated: March 3, 2010)). "The NIST Guide for the use of the International System of Units, Section 8: Comments on Some Quantities and Their Units". Special Publication 811. NIST. http://physics.nist.gov/Pubs/SP811/sec08.html#8.3. Retrieved 2010-05-22. 
  12. ↑ Hodgeman, Charles, Ed. (1961). Handbook of Chemistry and Physics, 44th Ed.. Cleveland, USA: Chemical Rubber Publishing Co..  p.3480-3485
  13. ↑ Clark, John B (1964). Physical and Mathematical Tables. Oliver and Boyd. 
  14. ↑ This value excludes the adjustment for centrifugal force due to Earth’s rotation and is therefore greater than the 9.80665 m/s2 value of standard gravity.