In scientific and engineering contexts, weight is the name given to the force on an object due to gravity. In everyday and legal contexts it is often used to mean mass.

When used to mean force, its magnitude (a scalar quantity), often denoted by an italic letter W, is the product of the mass m of the object and the magnitude of the local gravitational acceleration g; thus: . When considered a vector, weight is often denoted by a bold letter W. The unit of measurement for weight is that of force, which in the International System of Units (SI) is the newton. For example, an object with a mass of one kilogram has a weight of about 9.8 newtons on the surface of the Earth, about one-sixth as much on the Moon, and very nearly zero when in deep space far away from all bodies imparting gravitational influence.

In contrast to this "purely gravitational" definition, some books use an "operational" definition, defining the weight of an object as the force measured by the operation of weighing it (using a force-sensitive scale, such as a spring scale), in vacuum. This is the force an object exerts on a scale, and is equal to the force required to support it (although in the opposite direction to the "weight" force). This force measured by force-scales is the same as what some other sources term the object's "apparent weight".

The two definitions of weight differ, sometimes dramatically, when other factors intervene so that the force required to support a body is not exactly equal and opposite to the gravitational force acting on it. For example, in the operational definition, a mechanically accelerated object or person (such as from the acceleration due to change in velocity in a dragster or rocket) has more weight due to total g-force acceleration, but this is not so in the purely gravitational definition, in which weight only changes due to changes in the acceleration of gravity. In the case of an object in free fall, such as a falling apple, or an astronaut in an orbiting spacecraft, the operational definition implies that the weight is zero, in keeping with the familiar concept that such objects are weightless. In such situations a scale indicates a zero weight, as no force is exerted by the body on a support or scale, whereas by the purely gravitational definition, the body's weight in free fall would be the same as if the object were at rest (since, in the Newtonian theory of gravitation, gravity still exerts the same force on the body).

In everyday practical usage, including commercial usage, the term "weight" is commonly used to mean mass, which scientifically is an entirely different concept. On the surface of the Earth, the acceleration due to gravity (the "strength of gravity") is approximately constant; this means that the ratio of the weight force of a motionless object on the surface of the Earth to its mass is almost independent of its location, so that an object's weight force can stand as a proxy for its mass, and vice versa.

History

Discussion of the concepts of heaviness (weight) and lightness (levity) date back to the ancient Greek philosophers. These were typically viewed as inherent properties of objects. Plato described weight as the natural tendency of objects to seek their kin. To Aristotle weight and levity represented the tendency to restore the natural order of the basic elements: air, earth, fire and water. He ascribed absolute weight to earth and absolute levity to fire. Archimedes saw weight as a quality opposed to buoyancy, with the conflict between the two determining if an object sinks or floats. The first operational definition of weight was given by Euclid, who defined weight as: "weight is the heaviness or lightness of one thing, compared to another, as measured by a balance."

According to Aristotle, weight was the direct cause of the falling motion of an object, the speed of the falling object was supposed to be directly proportionate to the weight of the object. As medieval scholars discovered that in practice the speed of a falling object increased with time, this prompted as change to the concept of weight to maintain this cause effect relationship. Weight was split into a "still weight" or pondus, which remained constant, and the actual gravity or gravitas, which changed as the object fell. The concept of gravitas was eventually replaced by Jean Buridan's impetus, a precursor to momentum.

The rise of the Copernican view of the world led to the resurgence of the Platonic idea that like objects attract but in the context of heavenly bodies. In the 17th century, Galileo made significant advances in the concept of weight. He proposed a way to measure the difference between the weight of a moving object and an object at rest. Ultimately, he concluded weight was proportionate to the amount of matter of an object, and not the speed of motion as supposed by the Aristotelean view of physics.

Newton

The introduction of Newton's laws of motion and the development of Newton's law of universal gravitation led to considerable further development of the concept of weight. Weight became fundamentally separate from mass. Mass was identified as a fundamental property of objects connected to their inertia, while weight became identified with the force of gravity on an object and therefore dependent on the context of the object. In particular, Newton considered weight to be relative to another object causing the gravitational pull, e.g. the weight of the Earth towards the Sun.

Newton considered time and space to be absolute. This allowed him to consider concepts as true position and true velocity. Newton also recognized that weight as measured by the action of weighing was affected by environmental factors such as buoyancy. He considered this a false weight induced by imperfect measurement conditions, for which he introduced the term apparent weight as compared to the true weight defined by gravity.

Although Newtonian physics made a clear distinction between weight and mass, the term weight continued to be commonly used when people meant mass. This led the 3rd General Conference on Weights and Measures (CGPM) of 1901 to officially declare "The word weight denotes a quantity of the same nature as a force: the weight of a body is the product of its mass and the acceleration due to gravity", thus distinguishing it from mass for official usage.

Relativity

In the 20th century, the Newtonian concepts of absolute time and space were challenged by relativity. Einstein's principle of equivalence put all observers, moving or accelerating, on the same footing. This led to an ambiguity as to what exactly is meant by the force of gravity and weight. A scale in accelerating elevator cannot be distinguished from a scale in a gravitational field. Gravitational force and weight thereby became essentially frame-dependent quantities. This prompted the abandonment of the concept as superfluous in the fundamental sciences such as physics and chemistry. Nonetheless, the concept remained important in the teaching of physics. The ambiguities introduced by relativity led, starting in the 1960s, to considerable debate in the teaching community as how to define weight for their students, choosing between a nominal definition of weight as the force due to gravity or an operational definition defined by the act of weighing.

Definitions

Several definitions exist for weight, not all of which are equivalent.

The main differences in these definitions are:

whether the definition is based on the standard gravity of the Earth, or is based on any other proper acceleration, for example, Moon's gravity for an object on the moon;

whether the quantity is determined directly and is thus weighing-instrument-dependent (see Apparent weight), or is determined indirectly, from measurements of mass and acceleration;

whether the quantity is a vector or a scalar.

Gravitational definition

The most common definition of weight found in introductory physics textbooks defines weight as the force exerted on a body by gravity. This is often expressed in the formula , 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: as a force: the weight of a body is the product of its mass and the acceleration due to gravity." |Resolution 2 of the 3rd General Conference on Weights and Measures}} 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 gravitational acceleration varies from place to place. Sometimes, it is simply taken to a have a standard value of , which gives the standard weight.

Operational definition

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. This can make a considerable difference, depending on the details; for example, an object in free fall exerts little if any force on its support, 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). In the operational definition, the weight of an object at rest on the surface of the Earth is lessened by the effect of the centrifugal force from the Earth's rotation.

A minor issue with the formulation is that the operational definition, as usually given, does not explicitly exclude the effects of buoyancy, which reduces the measured weight of an object when it is immersed in a fluid such as air. As a result, a floating balloon or an object floating in water might be said to have no weight. However, this is commonly regarded as an instrument-dependent problem, since in theory, an object will always be weighed in a vacuum with the correct instrument.

ISO definition

In the ISO International standard ISO 80000-4(2006), describing the basic physical quantities and units in mechanics as a part of the International standard ISO/IEC 80000, the definition of weight is given as:

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. If the specified frame is the surface of the Earth, the weight according to the ISO and gravitational definitions differ only by the centrifugal effects due to the rotation of the Earth.

Correcting for levitation, mechanical suspension, and buoyancy

The identical operational and the ISO definitions for weight do not, in themselves, take into consideration the practical fact that a scale under an object cannot always be expected to measure its full weight. This situation is present if the object is supported, in part or in whole, by some other means which does not transfer downward weight-force to the weighing scale. Such unmeasured support detracts from an object's measured 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.

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. 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 the same everywhere on the surface of the Earth. In a uniform 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. Similarly, a balance measures mass indirectly by comparing the weight of the measured item to that of an object(s) of known mass. Since the measured item and the comparison mass are in virtually the same location, so experiencing the same gravitational field, the effect of varying gravity does not affect the comparison or the resulting measurement.

The Earth's gravitational field is not uniform but can vary by as much as 0.5% 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.

! 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 man of mass 180 pounds weighs only about 30 pounds-force when visiting the Moon.

Units

SI units

In most modern scientific work, physical quantities are measured in SI units. The SI unit of weight 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).

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).

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

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 ofEarth gravity	! Surface gravitym/s2
Sun	27.90	274.1
	0.3770	3.703
Venus	0.9032	8.872
	1 (by definition)	9.8226
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

Body weight

Notes

References

Category:Commerce Category:Mass Category:Force Category:Physiology

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