Weight

Weight is commonly defined in physics as the force exerted on a body by gravity. Unlike mass, which characterizes a body's inertia, weight depends on the gravitational environment. A related quantity, apparent weight, is the force registered by a supporting scale or suspension. These quantities often have the same magnitude for an object at rest, but differ during accelerated motion and free fall. In everyday language, “weight” also commonly means mass, as when a person's weight is stated in kilograms. (openstax.org)

Definition and relation to mass

In classical mechanics, gravitational weight is expressed as

where $m$ is the body's mass and $\mathbf{g}$ is the local gravitational acceleration. Weight is a vector: its direction follows the gravitational field. Its magnitude is

Near Earth's surface, $g$ is approximately $9.8\ \mathrm{m\,s^{-2}}$, so an object of mass $1\ \mathrm{kg}$ has a gravitational weight of approximately $9.8\ \mathrm{N}$. (openstax.org)

Mass and weight describe different physical properties. Mass determines the acceleration produced by a given net force through Newton's second law, whereas weight describes a gravitational force. Moving an unchanged object from Earth to the Moon does not change its mass, but reduces its weight because the Moon's surface gravitational acceleration is approximately $1.62\ \mathrm{m\,s^{-2}}$. (openstax.org)

The equation $W=mg$ does not imply that weight is always the net force. A stationary object on a table experiences both downward weight and an upward normal force. These forces balance in a simple horizontal arrangement. They are not a Newton's-third-law pair: both act on the same object, while third-law partners act on different bodies. (openstax.org)

Units and standard gravity

The unit of weight in the International System of Units is the newton, symbol $\mathrm{N}$:

The kilogram is a unit of mass, not force. Thus, a statement such as “the object weighs 10 kilograms” normally reports its mass in ordinary language rather than its gravitational weight in strict physical terminology. (bipm.org)

Standard gravity is a conventional reference acceleration,

adopted by the General Conference on Weights and Measures in 1901. It is not a claim that gravitational acceleration has this value everywhere on Earth. (bipm.org)

The non-SI kilogram-force is the force corresponding to a mass of one kilogram under standard gravity:

Distinguishing kilograms from kilogram-force prevents the same numerical label from being used ambiguously for mass and force. (nvlpubs.nist.gov)

Dependence on location

For a small body outside an ideal spherically symmetric gravitating body, Newtonian gravitation gives

where $G$ is the gravitational constant, $M$ is the attracting body's mass, and $r$ is the distance from its center. Consequently,

Weight therefore decreases with increasing distance from the attracting body's center, rather than simply with height above its surface. These expressions apply directly to spherical mass distributions outside their boundaries; irregular bodies require a more detailed calculation of the gravitational field. (openstax.org)

Weight also varies across Earth's surface. Earth is not perfectly spherical, and its rotation affects the support force required to remain stationary relative to the ground. At the equator, part of the gravitational force provides the acceleration associated with Earth's rotation, reducing the scale reading. Earth's greater equatorial radius also reduces gravitational attraction relative to the poles. (openstax.org)

A distinction is therefore necessary between gravitational attraction considered in an inertial reference frame and effective gravity in an Earth-fixed rotating frame. In the latter description, the centrifugal contribution is incorporated into the effective acceleration used for local weighing. Precise calculations must specify which meaning of $g$, and consequently of weight, is intended. (openstax.org)

Apparent weight and accelerated motion

Apparent weight is operationally associated with the force exerted by a support or suspension. For a person standing on a horizontal scale, it is the magnitude of the scale's normal force. In a simple stationary situation, with other forces neglected, this equals $mg$. It need not equal gravitational weight when the support accelerates. (nist.gov)

For vertical motion, taking upward as positive and treating $g$ as constant, Newton's second law gives

and hence

where $N$ is the scale's supporting force and $a$ is the vertical acceleration. This yields three important cases:

• Upward acceleration: $N>mg$; the scale registers a greater apparent weight.
• Downward acceleration: $N<mg$, while contact is maintained.
• Constant velocity: $a=0$, so $N=mg$, regardless of whether the motion is upward or downward.

These are consequences of the force balance, not changes in the person's mass or in the local gravitational attraction. (openstax.org)

Weightlessness and orbital motion

During ideal free fall, gravity is the only force acting on a body. A freely falling scale and the object resting on it accelerate together, so the scale provides no supporting force and the object's apparent weight is zero. Under the gravitational-force definition, however, its weight remains nonzero. “Weightlessness” usually denotes the absence of support force, not the absence of gravity. (openstax.org)

Astronauts float in an orbiting spacecraft because the spacecraft, its occupants, and unsecured objects undergo approximately the same gravitational acceleration. Their orbital motion is a continuing fall around Earth. At typical space-station altitudes, gravitational attraction is still about 90 percent of its surface value; it is strong enough to maintain the orbit. (nasa.gov)

The term microgravity describes the near-weightless conditions obtained in orbital spacecraft and other free-fall environments. It should not be interpreted as meaning that Earth's gravitational field has become negligible. (nasa.gov)

Measurement and weighing

A force-sensitive scale responds to the force applied to its sensing mechanism. When used to report mass, its response must be related to known mass standards under appropriate conditions. In metrology, the objects called “weights” are often calibrated mass standards rather than standards of an invariant gravitational force. (nvlpubs.nist.gov)

An equal-arm balance compares the forces associated with an unknown object and a reference mass. Because both are subject to substantially the same local gravitational acceleration, that common factor cancels in the ideal comparison. This permits determination of mass without assuming that local gravity equals standard gravity. High-precision weighing nevertheless requires corrections for environmental and instrumental effects. (nvlpubs.nist.gov)

One important correction concerns buoyancy. An object weighed in a fluid experiences an upward buoyant force. For a stationary, fully immersed object supported from below, with other forces neglected,

where $\rho_{\mathrm f}$ is the fluid's density and $V$ is the displaced volume. This reduces apparent weight without reducing mass. Air produces the same effect on a smaller scale: objects with different volumes can require different buoyancy corrections even when their masses are equal. Precision mass comparisons therefore use information about air density and the volumes or densities of the objects being compared. (nvlpubs.nist.gov)

Terminology and standardization

The modern distinction between mass and weight was explicitly addressed by the General Conference on Weights and Measures in 1901. Its declaration identified the kilogram as a unit of mass, defined weight as a force equal to mass multiplied by gravitational acceleration, and established the conventional value of standard gravity. The declaration responded to persistent ambiguity in ordinary usage. (bipm.org)

That ambiguity remains relevant in scientific communication. A mass value, a calculated gravitational force, and a measured support force answer different questions, although everyday weighing often makes them appear interchangeable. Stating the quantity, unit, gravitational conditions, and measurement arrangement identifies which meaning is being used. (openstax.org)

References

1. 4 Mass and Weight — University Physics Volume 1openstax.org
2. 3 Newton's Second Law of Motion — Physicsopenstax.org
3. 5 Normal, Tension, and Other Examples of Forces — College Physics 2eopenstax.org
4. SI Brochure — Brochure sur le SIbipm.org
5. Declaration 2 of the 3rd CGPM (1901)bipm.org
6. Gravity Measurements and the Standards Laboratorynvlpubs.nist.gov
7. 5 Newton’s Universal Law of Gravitation — College Physics 2eopenstax.org
8. 2 Gravitation Near Earth's Surface — University Physics Volume 1openstax.org
9. How Do You Measure the Strength of Gravity?nist.gov
10. SI Units — Massnist.gov
11. What Is Microgravity? (Grades 5–8)nasa.gov
12. What Is Microgravity?nasa.gov