Weight

This is the article describing weight in the physical sciences. For other uses, see Weight (disambiguation)

Technical usage in physical sciences

Established 'official' technical definition : In the physical sciences, the weight of an object has a particular technical meaning. It is the net gravitational force exerted on that object by all other objects.
In the physical sciences, weight by Definition VIII, per Newton's Mathematical Principles of Natural Philosophy or Principia, is an upward force exerted on matter to deny the body from entering freefall as a result of gravity, a centripetal acceleration field. This force is known as the Normal Force which is always perpendicular to a surface. During freefall, in a vacuum, you are weightless.

An object's weight is equal to its mass multiplied by the net acceleration.

Widely-used practical technical definition : But even in technical contexts, actual usage usually defaults to something different, more practical, and equally precise. The observed local acceleration field (incorporating e.g. accelerative effects due to the rotation of an earth-fixed frame of reference as well as gravity) is used instead of just the true gravitational acceleration. The local acceleration field is actually what one wants if one is trying to predict e.g. the weight measured by a spring balance.

Discussion : The difference between the two technical definitions, official and practical, is awkward, but it is widely accepted and well-understood in technical practice.

The difference between the official technical definition and non-technical usage is more troublesome. It can be argued that it would be preferable to define weight technically as something like what is now called apparent weight. That would bring the practical technical usage into alignment with non-technical usage and tie weight to perceivable properties, while breaking the immediate connection to gravity. But however reasonable, that is not established technical usage, and it has very far to go to become such.

The word weight entered Old English sometime around the 9th century, and meant the quantity measured with a balance. The word "weight" is commonly used synonymously with "mass", though the two concepts are technically quite distinct.

Weight and mass

In scientific usage weight and mass are quite different quantities: mass is an intrinsic property of matter, whereas weight is a force that results from the action of gravity on matter. (Scientific terms such as "atomic weight", "molecular weight", and "formula weight" are still encountered, but some discouraged these terms and terms like atomic mass are preferred.)

In everyday usage, however, weight and mass are usually not different. For instance, when we buy or sell goods "by weight", we are primarily interested in the amount of goods exchanged (the mass), not how hard they press down on the table (the weight). Similarly, in measurements of body weight we are primarily interested in the amount of tissue (fat, muscle, etc.) present. We may also say, for example, that an object "weighs one kilogram".

The gravitational force exerted on an object is directly proportional to its mass, so a mass of x kilograms always weighs x times as much as a mass of one kilogram. Weight can therefore stand as a proxy for mass, and vice versa.

The distinction between mass and force due to gravity becomes obvious when we move away from the earth's surface. For example, on the surface of the Moon gravity is only about one sixth as strong as on the surface of the earth. A one kilogram mass is still a one kilogram mass – mass is an intrinsic property of the object. However, the weight of the object – the downwards force due to gravity – is only one sixth of what it is at the surface of the earth; that is, only one sixth of what we would expect one kilogram to weigh.

Although gravity at the earth's surface is nearly constant, it does vary slightly with location, which means that objects do in fact have slightly different weights in different places. For further information see acceleration due to gravity, Physical geodesy, Gravity anomaly and Gravity.

Units of weight and mass

Systems of units of weight and mass have a tangled history, partly because the distinction was not properly understood when many of the units first came into use.

Since weight is a force, the units of weight are, in modern scientific work, simply units of force. The SI unit of force is the newton (N), which can also be expressed in SI base units as kg m/s² (kilograms times metres per second squared). The SI unit of mass is the kilogram. In most countries scientists have now adopted SI units. The kilogram-force is a non-SI unit of weight, defined as the weight force exerted by one kilogram.

In non-scientific usage, weight is usually measured in units that are technically units of mass, such as the kilogram, as discussed above. The pound is also officially defined as a unit of mass (not weight) in the United States, United Kingdom and elsewhere – the corresponding unit of force (weight) being the pound-force. Even so, the use of pounds to measure force (rather than mass) is still common in engineering, and it occurs in derived units like p.s.i. (pounds per square inch). It is also, of course, common in everyday usage.

If units of mass are to be used as units of weight (force), either directly ("kilogram", "pound") or indirectly ("kilogram-force", "pound-force") then, for precision work, it is necessary to specify the strength of gravity that is assumed. Usually this is so-called standard gravity.

To convert weight and mass we need to know the strength of gravity. The gravitational force exerted on an object is proportional to the mass of the object, so it is reasonable to measure the strength of gravity in terms of force per unit mass, that is, newtons per kilogram (N/kg). However, the unit N/kg resolves to m/s² (metres per second per second), which is the SI unit of acceleration, and in practice gravitational strength is usually quoted as an acceleration.

Gravitational acceleration at the earth's surface is approximately 9.8 m/s², which means that a falling object (ignoring air resistance) will increase in speed by approximately 9.8 metres per second every second. Plugging this value into Newton's second law, F = ma (force = mass × acceleration), we see that a one kilogram mass experiences a gravitational force (weight) of 1 kg × 9.8 m/s² = 9.8 newtons. In general, to convert mass in kilograms to weight in newtons (at the earth's surface), multiply by 9.8. Conversely, to convert weight to mass divide by 9.8. For different gravitational strengths, simply substitute the appropriate gravitational acceleration in place of the value 9.8.

Sensation of weight

The weight force that we actually sense is not the downward force of gravity, but the normal (upward) force exerted by the surface we stand on, which opposes gravity and prevents us falling to the center of the Earth. This normal force, called the apparent weight, is the one that is measured by a weighing scale.

For a body supported in a stationary position, the normal force exactly balances the earth's gravitational force, and so apparent weight has the same magnitude as actual weight. (Technically, things are slightly more complicated. For example, due to the earth's rotation objects are subject to a small centrifugal force, varying with latitude, which partially offsets gravity. The normal force therefore balances a force slightly less than the true force of gravity.)

If there is no contact with any surface to provide such an opposing force then there is no sensation of weight (no apparent weight). This happens in free-fall, as experienced by sky-divers and astronauts in orbit who feel "weightless" even though their bodies are still subject to the force of gravity. The experience of having no apparent weight is also known as microgravity.

A degree of reduction of apparent weight occurs, for example, in elevators. In an elevator, a spring scale will register a decrease in a person's (apparent) weight as the elevator starts to accelerate downwards. This is because the opposing force of the elevator's floor decreases as it accelerates away underneath one's feet. See under Apparent weight for a more detailed explanation of this phenomenon.

Measuring weight

Weight (or mass) may be measured indirectly with a balance, which compares the item in question to another of known weight (or mass). This comparative method is independent of the strength of gravity.

A spring scale or hydraulic or pneumatic scale measures the weight force (strictly the apparent weight force) directly. Most scales measure weight using a spring. Most household scales are calibrated in units of mass (such as kilograms) under the assumption that standard gravity will apply.

Relative weights on the earth, moon and planets

The following is a list of the weights of a mass on some of the bodies in the solar system, relative to its weight on Earth:{| Mercury