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A spring scale measures the weight of an object

In the physical sciences, the weight of an object is the magnitude, W, of the force that must be applied to an object in order to support it (i.e. hold it at rest) in a gravitational field. The weight of an object equals the magnitude of the gravitational force acting on the object, less the effect of its buoyancy in any fluid in which it might be immersed. Near the surface of the Earth, the acceleration due to gravity is approximately constant; this means that an object's weight is roughly proportional to its mass.

Weight and mass

In practical or everyday applications, weight means the same as mass as that term is used in physics. In modern scientific usage, however, 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 gravity pulls on that matter.

However, the recognition of this difference is historically a relatively recent development and in many everyday situations the word "weight" continues to be 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 distinction between mass and weight is unimportant for many practical purposes because the strength of gravity is very similar everywhere on the surface of the Earth. In such 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 (for conversion formulas see below).

Nevertheless, the Earth's gravitational field 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. To eliminate this variation, when the weight of objects is used in commerce, the value given is what the objects would weigh at a nominal standard gravitational acceleration of 9.80665 m/s (approx. 32.174 ft/s). 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.

The 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.

The difference between mass and weight force becomes clear when, for example:

• objects are compared in different gravitational fields, such as 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 (as mass is an intrinsic property of the object) but the downward force due to gravity is only one-sixth of what the object would experience on Earth.

• locating the center of gravity of an object (although if the gravitation field is uniform, the center of gravity will coincide with the center of mass).

• an object is immersed in a fluid. For instance, a brick weighs less when placed in water, even though its mass is unchanged, and a helium balloon in the atmosphere appears to have negative weight, even though its mass is positive.

Units

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

SI units

In most modern scientific work, physical quantities are measured in SI units. The SI unit of mass (and hence weight in some everyday senses) is the kilogram. The SI unit of force (and hence weight in the mechanics sense) is the newton (N) – which can also be expressed in SI base units as kg·m/s² (kilograms times meters per second squared).

The gravitational force exerted on an object is proportional to the mass of the object, so it is reasonable to think of the strength of gravity as measured 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.

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 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 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.

Conversion between weight (force) and mass

To convert between weight (force) and mass, Newton's second law of "force = mass × acceleration" (F = ma) is used. Here, F is the force (weight) due to gravity, m is the mass of the object in question, and a is the acceleration due to gravity, on Earth approximately 9.8 m/s² or 32.2 ft/s². In this context the same equation is often written as W = mg, with W standing for weight, and g for the acceleration due to gravity.

For example, using SI units, a mass of 1 kg experiences a weight force of 1 kg × 9.8 m/s² = 9.8 newtons (note that the newton is equivalent to kg m/s²).

Sensation of weight

The weight force that one actually senses is not the downward force of gravity, but the normal force exerted by the surface one stands on, which opposes gravity and prevents one from falling to the center of the Earth. This normal force, called the apparent weight, is the one that is measured by a spring scale. In most cases, it has the same magnitude as actual weight.

There are situations, however, that other forces need to be taken in account. For example, the apparent weight of an object immersed in water is smaller than in air; this is due to buoyancy, which opposes the gravitational force and therefore generates a smaller normal.

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 (until they approach terminal velocity) and astronauts in orbit, who feel "weightlessness" even though their bodies are still subject to the force of gravity: they're just no longer resisting it. 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.

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/s (approx. 32.174 ft/s). 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 recalibrated 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. 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, and Pluto. 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.

See also

• Apparent weight
• Atomic weight
• Body mass index
• Curb weight
• Human weight
• Hydrostatic weighting
• Systems of measurement
• Weights and measures

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Published in July 2009.

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