Mass versus weight

The chains on the swing hold all the child’s weight. If one were to stand behind her at the bottom of the arc and try to stop her, one would be acting against her inertia, which arises purely from mass, not weight.

In everyday usage, the mass of an object is often referred to as its weight though these are in fact different concepts and quantities. In scientific contexts, mass refers loosely to the amount of "matter" in an object (though "matter" may be difficult to define), whereas weight refers to the force experienced by an object due to gravity.^[1] In other words, an object with a mass of 1.0 kilograms will weigh 9.8 newtons (newton is the unit of force, while kilogram is the unit of mass) on Earth (its mass multiplied by the gravitational field strength). Its weight will be less on Mars (where gravity is weaker), more on Saturn, and negligible in space when far from any significant source of gravity.

Objects on the surface of the Earth have weight, although sometimes this weight is difficult to measure. An example is a small object floating in a pool of water, or even a dish of water, which does not appear to have weight since buoyed by the water, but is found to have its usual weight when it is added to water in a container which is entirely supported and weighed on a scale. Thus, the "weightless object" floating in water actually transfers its weight to the bottom of the container (where the pressure increases). Similarly, a balloon has mass but may appear to have no weight or even negative weight, due to buoyancy in air. However, in the case of buoyancy, the weight of the balloon and the gas inside it has merely been transferred to a large area of the Earth's surface (in fact the entire surface, eventually), making the weight difficult to measure. The weight of a flying airplane is similarly distributed to the ground, but does not disappear. If the airplane is in level flight, the same weight-force is distributed to the surface of the Earth as when the plane was on the runway, but spread over a larger area.

A better scientific definition of mass describes it as having inertia, the resistance of an object to being accelerated when acted on by an external force. Gravitational "weight" is the force created when a mass is acted upon by a gravitational field and the object is not allowed to free-fall, but is supported or retarded by a mechanical force, such as the surface of a planet. Such a force confers weight. ^[2] Such weights can be added to by the weight created from any kind of mechanical force.

For example, in the illustrative photograph, the girl’s weight (force due to gravity and the inertial force resisting circular acceleration) is being supported by the swing seat and swing set. If one stands behind her at the bottom of her arc and abruptly stops her, the "bump" or stopping-force one experiences is due to acting against her inertia, and would be the same force even if gravity were not present. Such forces also confer a type of weight, which could be measured.

While the weight of a mass is a function of the strength of gravity, the mass of an object is constant for any given observer, so long as no energy or matter is added to the object.^[3] Accordingly, for an astronaut on a spacewalk in orbit (a free-fall), no effort is required to hold a communications satellite in front of him; it is "weightless". However, since objects in orbit retain their mass and inertia, an astronaut must exert ten times as much force to accelerate a 10‑ton satellite at the same rate as one with a mass of only 1 ton.

On Earth, a swing set can demonstrate this relationship of force, mass, and acceleration without being appreciably influenced by weight (downward force). If one were to stand behind a large adult sitting stationary in a swing and give him a strong push, the adult would accelerate relatively slowly and swing only a limited distance forwards before beginning to swing backwards. Exerting that same effort while pushing on the small girl would produce much greater acceleration.

Overview

Matter’s mass strongly influences many familiar kinetic properties.

Mass corresponds to the general, everyday notion of how "heavy" something is. However, mass is actually an inertial property; that is, the tendency of an object to remain at constant velocity unless acted upon by an outside force. Per Sir Isaac Newton’s 324-year-old laws of motion and an important formula that sprang from his work, F = ma, an object with a mass, m, of one kilogram will accelerate, a, at one meter per second per second (about one-tenth the acceleration due to earth’s gravity)^[4] when acted upon by a force, F, of one newton.

Inertia is sensed when a bowling ball is pushed horizontally on a level, smooth surface. This is quite distinct from its weight, which is the downwards gravitational force of the bowling ball one must counter when holding it off the floor. Importantly, the weight on the bowling ball on the Moon would be one-sixth of that on the Earth although its mass remained unchanged. Consequently, wherever the physics of recoil kinetics (mass, velocity, inertia, inelastic and elastic collisions) dominate and the influence of gravity is a negligible factor, the behavior of objects remains consistent even where gravity is relatively weak. For instance, billiard balls on a billiards table would scatter and recoil with the same speeds and energies after a break shot on the Moon as on Earth; they would, however, drop into the pockets much more slowly.

In the physical sciences, the terms "mass" and "weight" are rigidly defined as separate measures in order to enforce clarity and precision. In everyday use, given that most masses on Earth have weight and this relationship is usually highly proportional,^[5] "weight" often serves to describe both properties, its meaning being dependent upon context. For example, in commerce, the net weight of retail products actually refers to mass and is properly expressed in pounds (U.S.) or kilograms (see also Pound: Use in commerce). Conversely, the load index rating on automobile tires, which specifies the maximum structural load for a tire in kilograms, refers to weight; that is, the force due to gravity. Before the late twentieth century, this distinction was not as strictly applied, even in technical writing, so that expressions such as "molecular weight" (for molecular mass) are still seen.

Because mass and weight are separate quantities, they have different units of measure. In the International System of Units (SI), the kilogram is the unit of mass, and the newton is the unit of force. The non-SI kilogram-force is also a unit of force typically used in the measure of weight. Similarly, the avoirdupois pound, used in both the Imperial system and U.S. customary units, is a unit of mass and its related unit of force is the pound-force.

Converting units of mass to equivalent forces on Earth

Gravity anomalies covering the Southern Ocean are shown here in false-color relief. This image has been normalized to remove variation due to differences in latitude.

When an object’s weight (its gravitational force) is expressed in kilograms, the unit of measure is not a true kilogram; it is the kilogram-force (kgf or kg-f), also known as the kilopond (kp), which is a non-SI unit of force. All objects on Earth are subject to a gravitational acceleration of approximately 9.8 m/s². The CGPM (also known as the "General Conference on Weights and Measures") fixed the value of standard gravity at precisely 9.80665 m/s² so that disciplines such as metrology would have a standard value for converting units of defined mass into defined forces and pressures. In fact, the kilogram-force is defined as precisely 9.80665 newtons. As a practical matter, gravitational acceleration (symbol: g) varies slightly with latitude, elevation and subsurface density; these variations are typically only a few tenths of a percent. See also Gravimetry.

Professionals in engineering and scientific disciplines involving accelerations and kinetic energies rigorously maintain the distinctions between mass, force, and weight, as well as their respective units of measure. Engineers in disciplines involving weight loading (force on a structure due to gravity), such as structural engineering, first convert loads due to objects like concrete and automobiles—which are always tallied in kilograms—to newtons before continuing with their calculations. Primarily, this is because material properties like elastic modulus are measured and published in terms of the newton and pascal (a unit of pressure derived from the newton). For all practical engineering purposes on Earth, mass in kilograms is converted to weight in newtons by multiplying by 9.80665 m/s² (standard gravity).

Buoyancy and weight

Regardless of the fluid in which an object is immersed (gas or liquid), the buoyant force on an object is equal to the weight of the fluid it displaces.

A hot air balloon when it has neutral buoyancy has no weight for the men to support but still retains its great mass and inertia.

Usually, the relationship between mass and weight on Earth is highly proportional; objects that are a hundred times more massive than a one-liter bottle of soda almost always weigh a hundred times more—approximately 1,000 newtons, which is the weight one would expect on Earth from an object with a mass slightly greater than 100 kilograms. Yet, this is not always the case and there are familiar objects that violate this mass / weight proportionality.

A common helium-filled toy balloon is something familiar to many children. When such a balloon is fully filled with helium, it has buoyancy—a force that opposes gravity. When a toy balloon becomes partially deflated, it will often become neutrally buoyant and can float about the house a meter or two off the floor. In such a state, there are moments when the balloon is neither rising nor falling and—in the sense that a scale placed under it will have no force applied to it—is, in a sense perfectly weightless (actually as noted below, weight has merely been redistribubed along the Earth's surface so it cannot be measured). Though the rubber comprising the balloon has a mass of only a few grams, which might be almost unnoticeable, the rubber still retains all its mass when inflated.

Again, unlike the effect that low-gravity environments have on weight, buoyancy does not make a portion of an object’s weight vanish; the missing weight is instead being borne by the ground, which leaves less force (weight) being applied to any scale theoretically placed underneath the object in question (though one may perhaps have some trouble with the practical aspects of accurately weighing something individually in that condition). If one were however to weigh a small wading pool that someone then entered and began floating in, they would find that the full weight of the person was being borne by the pool and, ultimately, the scale underneath the pool. Whereas a buoyant object (on a properly working scale for weighing buoyant objects) would weigh less, the object/fluid system becomes heavier by the value of object's full mass once the object is added. Since air is a fluid, this principle applies to object/air systems as well; large volumes of air—and ultimately the ground—supports the weight a body loses through mid-air buoyancy.

The effects of buoyancy don’t just affect balloons; both liquids and gases are fluids in the physical sciences, and when all macro‑size objects larger than dust particles are immersed in fluids on Earth, they have some degree of buoyancy.^[6] In the case of either a swimmer floating in a pool or a balloon floating in air, buoyancy can fully counter the gravitational weight of the of the object being weighed, for a weighing device in the pool. However, as noted, an object supported by a fluid is fundamentally no different than an object supported by a sling or cable-- the weight has merely been transferred to another location, not made to disappear.

The mass of "weightless" (neutrally buoyant) balloons can be better appreciated with much larger hot air balloons. Although no effort is required to counter their weight when they are hovering over the ground (when they can often be within one hundred newtons of zero weight), the inertia associated with their appreciable mass of several hundred kilograms or more can knock fully grown men off their feet when the balloon’s basket is moving horizontally over the ground.

Buoyancy and the resultant reduction in the downward force of objects being weighed underlies Archimedes' principle, which states that the buoyancy force is equal to the weight of the fluid that the object displaces. If this fluid is air, the force may be small.

Buoyancy effects of air on measurement

Normally, the effect of air buoyancy on objects of normal density is too small to be of any consequence in day-to-day activities. For instance, buoyancy’s diminishing effect upon one’s body weight (a relatively low-density object) is ¹⁄₈₆₀ that of gravity (for pure water it is about ¹⁄₇₇₀ that of gravity). Furthermore, variations in barometric pressure will rarely affect one’s weight more than ±1 part in 30,000.^[7] However, in metrology (the science of measurement), the precision mass standards for calibrating laboratory scales and balances are manufactured with such accuracy that air density is accounted for to compensate for buoyancy effects. Given the extremely high cost of platinum-iridium mass standards like the International Prototype Kilogram (the mass standard in France that defines the magnitude of the kilogram), high-quality "working" standards are made of special stainless steel alloys^[8] with densities of about 8,000 kg/m³, which occupy greater volume than those made of platinum-iridium, which have a density of about 21,550 kg/m³. For convenience, a standard value of buoyancy relative to stainless steel was developed for metrology work and this results in the term "conventional mass".^[9] Conventional mass is defined as follows: "For a mass at 20 °C, ‘conventional mass’ is the mass of a reference standard of density 8,000 kg/m³ which it balances in air with a density of 1.2 kg/m³." The effect is a small one, 150 ppm for stainless steel mass standards, but the appropriate corrections are made during the manufacture of all precision mass standards so they have the true labeled mass.

Whenever a high-precision scale (or balance) in routine laboratory use is calibrated using stainless steel standards, the scale is actually being calibrated to conventional mass; that is, true mass minus 150 ppm of buoyancy. Since objects with precisely the same mass but with different densities displace different volumes and therefore have different buoyancies and weights, any object measured on this scale (compared to a stainless steel mass standard) has its conventional mass measured; that is, its true mass minus an unknown degree of buoyancy. In high-accuracy work, the volume of the article can be measured to mathematically null the effect of buoyancy.

Types of scales and what they measure

A balance-type weighing scale: Unaffected by the strength of gravity.

Load-cell based bathroom scale: Affected by the strength of gravity.

Technically, whenever someone stands on a balance-beam-type scale at a doctor’s office, they are truly having their mass measured. This is because balances ("dual-pan" mass comparators) compare the weight of the mass on the platform with that of the sliding counterweights on the beams; gravity serves only as the force-generating mechanism that allows the needle to diverge from the "balanced" (null) point. Balances can be moved from Earth’s equator to the poles without spuriously indicating that objects became over 0.3% more massive; they are immune to the gravity-countering centrifugal force due to Earth’s rotation about its axis. Conversely, whenever someone steps onto spring-based or digital load cell-based scales (single-pan devices), they are technically having their weight (force due to strength of gravity) measured. On force-measuring instruments such as these, variations in the strength of gravity affect the reading. As a practical matter, when force-measuring scales are used in commerce or hospitals, they are calibrated on-site and certified on that basis so the measure is mass, expressed in pounds or kilograms, to the desired level of accuracy.^[10]

Use in commerce

In the United States of America the United States Department of Commerce, the Technology Administration, and the National Institute of Standards and Technology (NIST) have defined the use of mass and weight in the exchange of goods under the Uniform Laws and Regulations in the areas of legal metrology and engine fuel quality in NIST Handbook 130.

NIST Handbook 130 states:

V. "Mass" and "Weight." [NOTE 1, See page 6]

The mass of an object is a measure of the object’s inertial property, or the amount of matter it contains. The weight of an object is a measure of the force exerted on the object by gravity, or the force needed to support it. The pull of gravity on the earth gives an object a downward acceleration of about 9.8 m/s². In trade and commerce and everyday use, the term "weight" is often used as a synonym for "mass." The "net mass" or "net weight" declared on a label indicates that the package contains a specific amount of commodity exclusive of wrapping materials. The use of the term "mass" is predominant throughout the world, and is becoming increasingly common in the United States. (Added 1993)

W. Use of the Terms "Mass" and "Weight." [NOTE 1, See page 6]

When used in this handbook, the term "weight" means "mass". The term "weight" appears when inch-pound units are cited, or when both inch-pound and SI units are included in a requirement. The terms "mass" or "masses" are used when only SI units are cited in a requirement. The following note appears where the term "weight" is first used in a law or regulation.

NOTE 1: When used in this law (or regulation), the term "weight" means "mass." (See paragraph V. and W. in Section I., Introduction, of NIST Handbook 130 for an explanation of these terms.) (Added 1993) 6"

U.S. federal law, which supersedes this handbook, also defines weight, particularly Net Weight, in terms of the avoirdupois pound or mass pound. From 21CFR101 Part 101.105 – Declaration of net quantity of contents when exempt:

(a) The principal display panel of a food in package form shall bear a declaration of the net quantity of contents. This shall be expressed in the terms of weight, measure, numerical count, or a combination of numerical count and weight or measure. The statement shall be in terms of fluid measure if the food is liquid, or in terms of weight if the food is solid, semisolid, or viscous, or a mixture of solid and liquid; except that such statement may be in terms of dry measure if the food is a fresh fruit, fresh vegetable, or other dry commodity that is customarily sold by dry measure. If there is a firmly established general consumer usage and trade custom of declaring the contents of a liquid by weight, or a solid, semisolid, or viscous product by fluid measure, it may be used. Whenever the Commissioner determines that an existing practice of declaring net quantity of contents by weight, measure, numerical count, or a combination in the case of a specific packaged food does not facilitate value comparisons by consumers and offers opportunity for consumer confusion, he will by regulation designate the appropriate term or terms to be used for such commodity.

(b)(1) Statements of weight shall be in terms of avoirdupois pound and ounce.

See also 21CFR201 Part 201.51 – "Declaration of net quantity of contents" for general labeling and prescription labeling requirements.

See also

Apparent weight Gravimetry Inertia International System of Units (SI) Kilogram Kilogram-force

Mass Newton (unit) Pound (mass) Pound-force Standard gravity Weight

Notes

1. ^ [1]
2. ^ National Physical Laboratory: What are the differences between mass, weight, force and load? (FAQ - Mass & Density)
3. ^ See Mass in special relativity for a discussion of mass in this context. An object or particle does not have to be traveling very close to the speed of light, c, for its relativistic mass, M (or γm) to measurably vary from its rest mass m₀. Per the Lorentz transformations and Einstein’s 1905 paper, The Special Theory of Relativity, relativistic mass is 0.5% greater than m₀ at only 9.96% c, thus affecting measurements performed at a precision of one percent. Whereas 10% the speed of light is exceedingly fast in most contexts, it is not "close to the speed of light" as physics understands the term.
4. ^ In professional metrology (the science of measurement), the acceleration of Earth’s gravity is taken as standard gravity (symbol: g_n), which is defined as precisely 9.80665 meters per square second (m/s²). The expression "1 m/s² " means that for every second that elapses, velocity changes an additional 1 meter per second. In more familiar terms: an acceleration of 1 m/s² can also be expressed as a rate of change in velocity of precisely 3.6 km/h per second (≈2.2 mph per second).
5. ^ Note that not all masses have weight. A helium-filled toy balloon has mass but has negative weight (buoyancy). A mix of helium and air in a toy balloon can result in a ballon with mass and neutral buoyancy—no weight to speak of.
6. ^ Objects the size of small dust particles, or smaller, are so strongly influenced by Brownian motion that they are no longer influenced by buoyancy.
7. ^ Assumptions: An air density of 1160 g/m³, an average density of a human body (with collapsed lungs) equal to that of water, and variations in barometric pressure rarely exceeding ±22 torrs (2.9 kPa). Assumptions primary variables: An altitude of 194 meters above mean sea level (the worldwide median altitude of human habitation), an indoor temperature of 23 °C, a dewpoint of 9 °C, and 760 mmHg (101 kPa) sea level–corrected barometric pressure.
8. ^ For example, for the recalibration of the US National Prototype Kilogram in 1985, two austenitic stainless steel artifacts were used for comparison. One, named D2, is similar to 18-8 (type 304) stainless (ie 18% Chromium, 8% Nickel); the other, named CH-1, is a more complex alloy that could be roughly designated as CrNiMo30-25-2 (chemical composition is 29.9% Cr, 25.1% Ni, 2.2% Mo, 1.45% Mn, 0.53 Si, 0.2% Cu, 0.07% C, 0.0019% P). Davis, R. N. (1985). "Recalibration of the US National Prototype Kilogram". Journal of Research of the National Bureau of Standards (Washington: US Government Printing Office) 90 (4): p.267. http://nvl.nist.gov/pub/nistpubs/jres/090/4/V90-4.pdf. Retrieved 02 May 2011. 
9. ^ International Recommendation OIML R33, International Organization of Legal Metrology.
10. ^ National General Conference on Weights and Measures, Specifications, Tolerances, and Other Technical Requirements for Weighing and Measuring Devices, NIST Handbook 44

External links

• WonderQuest: Weightless in space, Thumping to talk