In science and engineering , the weight of an object is a quantity associated with the gravitational force exerted on the object by other objects in its environment, although there is some variation and debate as to the exact definition.
124-414: Some standard textbooks define weight as a vector quantity, the gravitational force acting on the object. Others define weight as a scalar quantity, the magnitude of the gravitational force. Yet others define it as the magnitude of the reaction force exerted on a body by mechanisms that counteract the effects of gravity: the weight is the quantity that is measured by, for example, a spring scale. Thus, in
248-426: A {\displaystyle {\mathfrak {a}}} . Vectors are usually shown in graphs or other diagrams as arrows (directed line segments ), as illustrated in the figure. Here, the point A is called the origin , tail , base , or initial point , and the point B is called the head , tip , endpoint , terminal point or final point . The length of the arrow is proportional to the vector's magnitude , while
372-473: A z = a x i + a y j + a z k . {\displaystyle \mathbf {a} =\mathbf {a} _{x}+\mathbf {a} _{y}+\mathbf {a} _{z}=a_{x}{\mathbf {i} }+a_{y}{\mathbf {j} }+a_{z}{\mathbf {k} }.} The notation e i is compatible with the index notation and the summation convention commonly used in higher level mathematics, physics, and engineering. As explained above ,
496-854: A 1 a 2 a 3 ] = [ a 1 a 2 a 3 ] T . {\displaystyle \mathbf {a} ={\begin{bmatrix}a_{1}\\a_{2}\\a_{3}\\\end{bmatrix}}=[a_{1}\ a_{2}\ a_{3}]^{\operatorname {T} }.} Another way to represent a vector in n -dimensions is to introduce the standard basis vectors. For instance, in three dimensions, there are three of them: e 1 = ( 1 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 ) , e 3 = ( 0 , 0 , 1 ) . {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1).} These have
620-412: A 1 , a 2 , a 3 ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3}).} also written, a = ( a x , a y , a z ) . {\displaystyle \mathbf {a} =(a_{x},a_{y},a_{z}).} This can be generalised to n-dimensional Euclidean space (or R ). a = (
744-397: A 1 , a 2 , a 3 , ⋯ , a n − 1 , a n ) . {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3},\cdots ,a_{n-1},a_{n}).} These numbers are often arranged into a column vector or row vector , particularly when dealing with matrices , as follows: a = [
868-452: A 1 = b 1 , a 2 = b 2 , a 3 = b 3 . {\displaystyle a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}.\,} Two vectors are opposite if they have the same magnitude but opposite direction ; so two vectors General Conference on Weights and Measures The General Conference on Weights and Measures (abbreviated CGPM from
992-598: A 3 ( 0 , 0 , 1 ) , {\displaystyle \mathbf {a} =(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),\ } or a = a 1 + a 2 + a 3 = a 1 e 1 + a 2 e 2 + a 3 e 3 , {\displaystyle \mathbf {a} =\mathbf {a} _{1}+\mathbf {a} _{2}+\mathbf {a} _{3}=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3},} where
1116-721: A 1 , a 2 , a 3 are called the vector components (or vector projections ) of a on the basis vectors or, equivalently, on the corresponding Cartesian axes x , y , and z (see figure), while a 1 , a 2 , a 3 are the respective scalar components (or scalar projections). In introductory physics textbooks, the standard basis vectors are often denoted i , j , k {\displaystyle \mathbf {i} ,\mathbf {j} ,\mathbf {k} } instead (or x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } , in which
1240-412: A . ( Uppercase letters are typically used to represent matrices .) Other conventions include a → {\displaystyle {\vec {a}}} or a , especially in handwriting. Alternatively, some use a tilde (~) or a wavy underline drawn beneath the symbol, e.g. a ∼ {\displaystyle {\underset {^{\sim }}{a}}} , which
1364-418: A directed line segment , or arrow, in a Euclidean space . In pure mathematics , a vector is defined more generally as any element of a vector space . In this context, vectors are abstract entities which may or may not be characterized by a magnitude and a direction. This generalized definition implies that the above-mentioned geometric entities are a special kind of abstract vectors, as they are elements of
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#17328523132041488-472: 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
1612-458: 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
1736-600: A Euclidean space E is defined as a set to which is associated an inner product space of finite dimension over the reals E → , {\displaystyle {\overrightarrow {E}},} and a group action of the additive group of E → , {\displaystyle {\overrightarrow {E}},} which is free and transitive (See Affine space for details of this construction). The elements of E → {\displaystyle {\overrightarrow {E}}} are called translations . It has been proven that
1860-442: A basis does not affect the properties of a vector or its behaviour under transformations. A vector can also be broken up with respect to "non-fixed" basis vectors that change their orientation as a function of time or space. For example, a vector in three-dimensional space can be decomposed with respect to two axes, respectively normal , and tangent to a surface (see figure). Moreover, the radial and tangential components of
1984-728: A condition may be emphasized calling the result a bound vector . When only the magnitude and direction of the vector matter, and the particular initial or terminal points are of no importance, the vector is called a free vector . The distinction between bound and free vectors is especially relevant in mechanics, where a force applied to a body has a point of contact (see resultant force and couple ). Two arrows A B ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{AB}}} and A ′ B ′ ⟶ {\displaystyle {\stackrel {\,\longrightarrow }{A'B'}}} in space represent
2108-414: A convenient algebraic characterization of both angle (a function of the dot product between any two non-zero vectors) and length (the square root of the dot product of a vector by itself). In three dimensions, it is further possible to define the cross product , which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of
2232-436: A convenient numerical fashion. For example, the sum of the two (free) vectors (1, 2, 3) and (−2, 0, 4) is the (free) vector ( 1 , 2 , 3 ) + ( − 2 , 0 , 4 ) = ( 1 − 2 , 2 + 0 , 3 + 4 ) = ( − 1 , 2 , 7 ) . {\displaystyle (1,2,3)+(-2,0,4)=(1-2,2+0,3+4)=(-1,2,7)\,.} In
2356-425: A corresponding bound vector, in this sense, whose initial point has the coordinates of the origin O = (0, 0, 0) . It is then determined by the coordinates of that bound vector's terminal point. Thus the free vector represented by (1, 0, 0) is a vector of unit length—pointing along the direction of the positive x -axis. This coordinate representation of free vectors allows their algebraic features to be expressed in
2480-598: A different nationality. elected by the General Conference on Weights and Measures (CGPM) whose principal task is to promote worldwide uniformity in units of measurement by taking direct action or by submitting proposals to the CGPM. The CIPM meets every year (since 2011 in two sessions per year) at the Pavillon de Breteuil where, among other matters, it discusses reports presented to it by its Consultative Committees. Reports of
2604-499: A man of mass 180 pounds weighs only about 30 pounds-force when visiting the Moon. 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/s (kilograms times metres per second squared). In commercial and everyday use, the term "weight"
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#17328523132042728-478: A mass of one kilogram has a weight of about 9.8 newtons on the surface of the Earth, and about one-sixth as much on the Moon . Although weight and mass are scientifically distinct quantities, the terms are often confused with each other in everyday use (e.g. comparing and converting force weight in pounds to mass in kilograms and vice versa). Further complications in elucidating the various concepts of weight have to do with
2852-507: A member of the CIPM. Apart from the CCU, membership of a CC is open to National Metrology Institutes ( NMIs ) of Member States that are recognized internationally as most expert in the field. NMIs from Member States that are active in the field, but lack the expertise to become Members, are able to attend CC meetings as observers. These committees are: The CCU's role is to advise on matters related to
2976-417: A period of more than 200 years. About a dozen people contributed significantly to its development. In 1835, Giusto Bellavitis abstracted the basic idea when he established the concept of equipollence . Working in a Euclidean plane, he made equipollent any pair of parallel line segments of the same length and orientation. Essentially, he realized an equivalence relation on the pairs of points (bipoints) in
3100-406: 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: "the heaviness or lightness of one thing, compared to another, as measured by a balance." Operational balances (rather than definitions) had, however, been around much longer. According to Aristotle, weight
3224-405: 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
3348-425: A space with no notion of length or angle. In physics, as well as mathematics, a vector is often identified with a tuple of components, or list of numbers, that act as scalar coefficients for a set of basis vectors . When the basis is transformed, for example by rotation or stretching, then the components of any vector in terms of that basis also transform in an opposite sense. The vector itself has not changed, but
3472-479: A special kind of vector space called Euclidean space . This particular article is about vectors strictly defined as arrows in Euclidean space. When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric , spatial , or Euclidean vectors. A Euclidean vector may possess a definite initial point and terminal point ; such
3596-489: A state of free fall , the weight would be zero. In this sense of weight, terrestrial objects can be weightless: so if one ignores air resistance , one could say the legendary apple falling from the tree, on its way to meet the ground near Isaac Newton , was weightless. 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
3720-406: A vector is often described by a set of vector components that add up to form the given vector. Typically, these components are the projections of the vector on a set of mutually perpendicular reference axes (basis vectors). The vector is said to be decomposed or resolved with respect to that set. The decomposition or resolution of a vector into components is not unique, because it depends on
3844-726: A vector is one type of tensor . In pure mathematics , a vector is any element of a vector space over some field and is often represented as a coordinate vector . The vectors described in this article are a very special case of this general definition, because they are contravariant with respect to the ambient space. Contravariance captures the physical intuition behind the idea that a vector has "magnitude and direction". Vectors are usually denoted in lowercase boldface, as in u {\displaystyle \mathbf {u} } , v {\displaystyle \mathbf {v} } and w {\displaystyle \mathbf {w} } , or in lowercase italic boldface, as in
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3968-494: A vector pointing into and behind the diagram. These can be thought of as viewing the tip of an arrow head on and viewing the flights of an arrow from the back. In order to calculate with vectors, the graphical representation may be too cumbersome. Vectors in an n -dimensional Euclidean space can be represented as coordinate vectors in a Cartesian coordinate system . The endpoint of a vector can be identified with an ordered list of n real numbers ( n - tuple ). These numbers are
4092-848: A vector relate to the radius of rotation of an object. The former is parallel to the radius and the latter is orthogonal to it. In these cases, each of the components may be in turn decomposed with respect to a fixed coordinate system or basis set (e.g., a global coordinate system, or inertial reference frame ). The following section uses the Cartesian coordinate system with basis vectors e 1 = ( 1 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 ) , e 3 = ( 0 , 0 , 1 ) {\displaystyle {\mathbf {e} }_{1}=(1,0,0),\ {\mathbf {e} }_{2}=(0,1,0),\ {\mathbf {e} }_{3}=(0,0,1)} and assumes that all vectors have
4216-412: A vector, since force is a vector quantity. However, some textbooks also take weight to be a scalar by defining: The weight W of a body is equal to the magnitude F g of the gravitational force on the body. The gravitational acceleration varies from place to place. Sometimes, it is simply taken to have a standard value of 9.80665 m/s , which gives the standard weight . The force whose magnitude
4340-462: 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, not the speed of motion as supposed by the Aristotelean view of physics. The introduction of Newton's laws of motion and the development of Newton's law of universal gravitation led to considerable further development of
4464-471: Is Minkowski space (which is important to our understanding of special relativity ). However, it is not always possible or desirable to define the length of a vector. This more general type of spatial vector is the subject of vector spaces (for free vectors) and affine spaces (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from thermodynamics , where many quantities of interest can be considered vectors in
4588-416: Is −15 N. In either case, the magnitude of the vector is 15 N. Likewise, the vector representation of a displacement Δ s of 4 meters would be 4 m or −4 m, depending on its direction, and its magnitude would be 4 m regardless. Vectors are fundamental in the physical sciences. They can be used to represent any quantity that has magnitude, has direction, and which adheres to
4712-417: Is a convention for indicating boldface type. If the vector represents a directed distance or displacement from a point A to a point B (see figure), it can also be denoted as A B ⟶ {\displaystyle {\stackrel {\longrightarrow }{AB}}} or AB . In German literature, it was especially common to represent vectors with small fraktur letters such as
4836-407: Is a vector-valued physical quantity , including units of measurement and possibly a support , formulated as a directed line segment . A vector is frequently depicted graphically as an arrow connecting an initial point A with a terminal point B , and denoted by A B ⟶ . {\textstyle {\stackrel {\longrightarrow }{AB}}.} A vector
4960-453: 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 distinction between mass and weight
5084-418: Is equal to mg newtons is also known as the m kilogram weight (which term is abbreviated to kg-wt ) 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 . Since W is the downward force on the body by the centre of earth and there is no acceleration in the body, there exists an opposite and equal force by
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5208-458: Is essentially the modern system of vector analysis. In 1901, Edwin Bidwell Wilson published Vector Analysis , adapted from Gibbs's lectures, which banished any mention of quaternions in the development of vector calculus. In physics and engineering , a vector is typically regarded as a geometric entity characterized by a magnitude and a relative direction . It is formally defined as
5332-488: Is immersed in a fluid the displacement of the fluid will cause an upward force on the object, making it appear lighter when weighed on a scale. The apparent weight may be similarly affected by levitation and mechanical suspension. When the gravitational definition of weight is used, the operational weight measured by an accelerating scale is often also referred to as the apparent weight. In modern scientific usage, weight and mass are fundamentally different quantities: mass
5456-472: 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
5580-461: Is often expressed in the formula W = mg , where W is the weight, m the mass of the object, and g gravitational acceleration . In 1901, the 3rd General Conference on Weights and Measures (CGPM) established this as their official definition of weight : 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. This resolution defines weight as
5704-431: Is often presented as the standard Euclidean space of dimension n . This is motivated by the fact that every Euclidean space of dimension n is isomorphic to the Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} More precisely, given such a Euclidean space, one may choose any point O as an origin . By Gram–Schmidt process , one may also find an orthonormal basis of
5828-410: 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. 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
5952-484: 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. In many real world situations the act of weighing may produce a result that differs from the ideal value provided by the definition used. This is usually referred to as the apparent weight of the object. A common example of this is the effect of buoyancy , when an object
6076-568: Is thus an equivalence class of directed segments with the same magnitude (e.g., the length of the line segment ( A , B ) ) and same direction (e.g., the direction from A to B ). In physics, Euclidean vectors are used to represent physical quantities that have both magnitude and direction, but are not located at a specific place, in contrast to scalars , which have no direction. For example, velocity , forces and acceleration are represented by vectors. In modern geometry, Euclidean spaces are often defined from linear algebra . More precisely,
6200-518: Is unimportant for many practical purposes because the strength of gravity does not vary too much 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
6324-400: 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). 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
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#17328523132046448-610: Is what is needed to "carry" the point A to the point B ; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B . Many algebraic operations on real numbers such as addition , subtraction , multiplication , and negation have close analogues for vectors, operations which obey
6572-413: The n -tuple of its Cartesian coordinates, and every vector to its coordinate vector . Since the physicist's concept of force has a direction and a magnitude, it may be seen as a vector. As an example, consider a rightward force F of 15 newtons . If the positive axis is also directed rightward, then F is represented by the vector 15 N, and if positive points leftward, then the vector for F
6696-574: The Convention du Mètre ( Metre Convention ) was signed by 17 states. This treaty established an international organisation, the Bureau international des poids et mesures (BIPM), which has two governing organs: The organization has a permanent laboratory and secretariat function (sometimes referred to as the Headquarters), the activities of which include the establishment of the basic standards and scales of
6820-584: The French : Conférence générale des poids et mesures ) is the supreme authority of the International Bureau of Weights and Measures (BIPM), the intergovernmental organization established in 1875 under the terms of the Metre Convention through which member states act together on matters related to measurement science and measurement standards . The CGPM is made up of delegates of the governments of
6944-502: The International system of units . The brochure is produced by the CCU in conjunction with a number of other international organisations. Initially the brochure was only in French – the official language of the metre convention, but recent versions have been published simultaneously in both English and French, with the French text being the official text. The 6th edition was published in 1991,
7068-407: 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
7192-487: The coordinates of the endpoint of the vector, with respect to a given Cartesian coordinate system , and are typically called the scalar components (or scalar projections ) of the vector on the axes of the coordinate system. As an example in two dimensions (see figure), the vector from the origin O = (0, 0) to the point A = (2, 3) is simply written as a = ( 2 , 3 ) . {\displaystyle \mathbf {a} =(2,3).} The notion that
7316-417: The dot product . This makes sense, as the addition in such a vector space acts freely and transitively on the vector space itself. That is, R n {\displaystyle \mathbb {R} ^{n}} is a Euclidean space, with itself as an associated vector space, and the dot product as an inner product. The Euclidean space R n {\displaystyle \mathbb {R} ^{n}}
7440-417: The electric and magnetic field , are represented as a system of vectors at each point of a physical space; that is, a vector field . Examples of quantities that have magnitude and direction, but fail to follow the rules of vector addition, are angular displacement and electric current. Consequently, these are not vectors. In the Cartesian coordinate system , a bound vector can be represented by identifying
7564-396: The hat symbol ^ {\displaystyle \mathbf {\hat {}} } typically denotes unit vectors ). In this case, the scalar and vector components are denoted respectively a x , a y , a z , and a x , a y , a z (note the difference in boldface). Thus, a = a x + a y +
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#17328523132047688-570: The metric system . In 1960 the 11th CGPM approved the title International System of Units , usually known as "SI". The General Conference receives the report of the CIPM on work accomplished; it discusses and examines the arrangements required to ensure the propagation and improvement of the International System of Units (SI); it endorses the results of new fundamental metrological determinations and various scientific resolutions of international scope; and it decides all major issues concerning
7812-448: The real line , Hamilton considered the vector v to be the imaginary part of a quaternion: The algebraically imaginary part, being geometrically constructed by a straight line, or radius vector, which has, in general, for each determined quaternion, a determined length and determined direction in space, may be called the vector part, or simply the vector of the quaternion. Several other mathematicians developed vector-like systems in
7936-436: The theory of relativity according to which gravity is modeled as a consequence of the curvature of spacetime . In the teaching community, a considerable debate has existed for over half a century on how to define weight for their students. The current situation is that a multiple set of concepts co-exist and find use in their various contexts. Discussion of the concepts of heaviness (weight) and lightness (levity) date back to
8060-408: 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
8184-491: The 19th century) as equivalence classes under equipollence , of ordered pairs of points; two pairs ( A , B ) and ( C , D ) being equipollent if the points A , B , D , C , in this order, form a parallelogram . Such an equivalence class is called a vector , more precisely, a Euclidean vector. The equivalence class of ( A , B ) is often denoted A B → . {\displaystyle {\overrightarrow {AB}}.} A Euclidean vector
8308-1110: The 21st meeting of the CGPM in October 1999, the category of "associate" was created for states not yet BIPM members and for economic unions . Albania (2007) Azerbaijan (2015) Bangladesh (2010) Bolivia (2008) Bosnia and Herzegovina (2011) Botswana (2012) Cambodia (2021) Caribbean Community (2005) Chinese Taipei (2002) Ethiopia (2018) Georgia (2008) Ghana (2009) Hong Kong (2000) Jamaica (2003) Kuwait (2018) Latvia (2001) Luxembourg (2014) Malta (2001) Mauritius (2010) Moldova (2007) Mongolia (2013) Namibia (2012) North Macedonia (2006) Oman (2012) Panama (2003) Paraguay (2009) Peru (2009) Philippines (2002) Qatar (2016) Sri Lanka (2007) Syria (2012) Tanzania (2018) Uzbekistan (2018) Vietnam (2003) Zambia (2010) Zimbabwe (2010–2020, 2022) Cuba (2000–2021) Seychelles (2010–2021) Sudan (2014–2021) The International Committee for Weights and Measures consists of eighteen persons, each of
8432-1857: The British Government signed the convention on behalf of the United Kingdom. This number grew to 21 in 1900, 32 in 1950, and 49 in 2001. As of 18 November 2022 , there are 64 Member States and 36 Associate States and Economies of the General Conference (with year of partnership in parentheses): Argentina (1877) Australia (1947) Austria (1875) Belarus (2020) Belgium (1875) Brazil (1921) Bulgaria (1911) Canada (1907) Chile (1908) China (1977) Colombia (2012) Costa Rica (2022) Croatia (2008) Czech Republic (1922) Denmark (1875) Ecuador (2019) Egypt (1962) Estonia (2021) Finland (1913) France (1875) Germany (1875) Greece (2001) Hungary (1925) India (1880) Indonesia (1960) Iran (1975) Iraq (2013) Ireland (1925) Israel (1985) Italy (1875) Japan (1885) Kazakhstan (2008) Kenya (2010) Lithuania (2015) Malaysia (2001) Mexico (1890) Montenegro (2018) Morocco (2019) Netherlands (1929) New Zealand (1991) Norway (1875) Pakistan (1973) Poland (1925) Portugal (1876) Romania (1884) Russia (1875) Saudi Arabia (2011) Serbia (2001) Singapore (1994) Slovakia (1922) Slovenia (2016) South Africa (1964) South Korea (1959) Spain (1875) Sweden (1875) Switzerland (1875) Thailand (1912) Tunisia (2012) Turkey (1875) Ukraine (2018) United Arab Emirates (2015) United Kingdom (1884) United States (1878) Uruguay (1908) Cameroon (1970–2012) Dominican Republic (1954–2015) North Korea (1982–2012) Peru (1875–1956) Venezuela (1879–1907, 1960–2018) At
8556-506: The CIPM include: From time to time the CIPM has been charged by the CGPM to undertake major investigations related to activities affecting the CGPM or the BIPM. Reports produced include: The Blevin Report , published in 1998, examined the state of worldwide metrology. The report originated from a resolution passed at the 20th CGPM (October 1995) which committed the CIPM to study and report on
8680-602: The Conference of the Metre in 1875, representatives of seventeen signed the convention on 20 May 1875. In April 1884, H. J. Chaney, Warden of Standards in London unofficially contacted the BIPM inquiring whether the BIPM would calibrate some metre standards that had been manufactured in the United Kingdom. Broch , director of the BIPM replied that he was not authorised to perform any such calibrations for non-member states. On 17 September 1884,
8804-425: 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
8928-402: The International System of Units (SI), approves the budget for the BIPM (over €13 million in 2018) and it decides all major issues concerning the organization and development of the BIPM. The structure is analogous to that of a stock corporation . The BIPM is the organisation, the CGPM is the general meeting of the shareholders, the CIPM is the board of directors appointed by the CGPM, and
9052-495: The acceleration due to gravity", thus distinguishing it from mass for official usage. In the 20th century, the Newtonian concepts of absolute time and space were challenged by relativity. Einstein's equivalence principle 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 an accelerating elevator cannot be distinguished from
9176-536: 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
9300-403: The actual gravity that would be experienced near the poles. Euclidean vector In mathematics , physics , and engineering , a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector ) is a geometric object that has magnitude (or length ) and direction . Euclidean vectors can be added and scaled to form a vector space . A vector quantity
9424-534: The associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point P of the space, as the coordinates on this basis of the vector O P → . {\displaystyle {\overrightarrow {OP}}.} These choices define an isomorphism of the given Euclidean space onto R n , {\displaystyle \mathbb {R} ^{n},} by mapping any point to
9548-455: 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: Definition Remarks 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
9672-433: The basis has, so the components of the vector must change to compensate. The vector is called covariant or contravariant , depending on how the transformation of the vector's components is related to the transformation of the basis. In general, contravariant vectors are "regular vectors" with units of distance (such as a displacement), or distance times some other unit (such as velocity or acceleration); covariant vectors, on
9796-442: The choice of the axes on which the vector is projected. Moreover, the use of Cartesian unit vectors such as x ^ , y ^ , z ^ {\displaystyle \mathbf {\hat {x}} ,\mathbf {\hat {y}} ,\mathbf {\hat {z}} } as a basis in which to represent a vector is not mandated. Vectors can also be expressed in terms of an arbitrary basis, including
9920-441: The complete quaternion product. This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs , who was exposed to quaternions through James Clerk Maxwell 's Treatise on Electricity and Magnetism , separated off their vector part for independent treatment. The first half of Gibbs's Elements of Vector Analysis , published in 1881, presents what
10044-407: 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
10168-470: The convention organisations and national governments is handled by the member state's ambassador to France, it is implicit that member states must have diplomatic relations with France, though during both world wars, nations that were at war with France retained their membership of the CGPM. CGPM meetings are chaired by the Président de l'Académie des Sciences de Paris . Of the twenty countries that attended
10292-410: The coordinates of its initial and terminal point. For instance, the points A = (1, 0, 0) and B = (0, 1, 0) in space determine the bound vector A B → {\displaystyle {\overrightarrow {AB}}} pointing from the point x = 1 on the x -axis to the point y = 1 on the y -axis. In Cartesian coordinates, a free vector may be thought of in terms of
10416-461: The development of the SI and the preparation of the SI brochure. It has liaison with other international bodies such as International Organization for Standardization (ISO) , International Astronomical Union (IAU) , International Union of Pure and Applied Chemistry (IUPAC) , International Union of Pure and Applied Physics (IUPAP) and International Commission on Illumination (CIE) . Official reports of
10540-433: The direction in which the arrow points indicates the vector's direction. On a two-dimensional diagram, a vector perpendicular to the plane of the diagram is sometimes desired. These vectors are commonly shown as small circles. A circle with a dot at its centre (Unicode U+2299 ⊙) indicates a vector pointing out of the front of the diagram, toward the viewer. A circle with a cross inscribed in it (Unicode U+2297 ⊗) indicates
10664-491: The effect of the centrifugal force from the Earth's rotation. 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 or water. As a result, a floating balloon or an object floating in water might be said to have zero weight. In the ISO International standard ISO 80000-4:2006, describing
10788-407: 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
10912-652: The familiar algebraic laws of commutativity , associativity , and distributivity . These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space . Vectors play an important role in physics : the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement ), their magnitude and direction can still be represented by
11036-471: The force necessary to accelerate an object of one-pound mass at 1 ft/s, 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/s 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
11160-411: The geometrical and physical settings, it is sometimes possible to associate, in a natural way, a length or magnitude and a direction to vectors. In addition, the notion of direction is strictly associated with the notion of an angle between two vectors. If the dot product of two vectors is defined—a scalar-valued product of two vectors—then it is also possible to define a length; the dot product gives
11284-435: The intuitive interpretation as vectors of unit length pointing up the x -, y -, and z -axis of a Cartesian coordinate system , respectively. In terms of these, any vector a in R can be expressed in the form: a = ( a 1 , a 2 , a 3 ) = a 1 ( 1 , 0 , 0 ) + a 2 ( 0 , 1 , 0 ) +
11408-406: The length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors . The vector concept, as it is known today, is the result of a gradual development over
11532-401: 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
11656-630: The long-term national and international needs relating to metrology, the appropriate international collaborations and the unique role of the BIPM to meet these needs, and the financial and other commitments that will be required from the Member States in the coming decades. The report identified, amongst other things, a need for closer cooperation between the BIPM and other organisations such as International Organization of Legal Metrology (OIML) and International Laboratory Accreditation Cooperation (ILAC) with clearly defined boundaries and interfaces between
11780-590: The meetings of the CGPM, the CIPM, and all the Consultative Committees, are published by the BIPM. The secretariat is based in Saint-Cloud , Hauts-de-Seine , France . In 1999, the CIPM has established the CIPM Arrangement de reconnaissance mutuelle (Mutual Recognition Arrangement, MRA), which serves as the framework for the mutual acceptance of national measurement standards and for recognition of
11904-698: The member states and observers from the Associates of the CGPM. It elects the International Committee for Weights and Measures (abbreviated CIPM from the Comité international des poids et mesures ) as the supervisory board of the BIPM to direct and supervise it. Initially the work of the BIPM concerned the kilogram and the metre , but in 1921 the scope of the Metre Convention was extended to accommodate all physical measurements and hence all aspects of
12028-513: The middle of the nineteenth century, including Augustin Cauchy , Hermann Grassmann , August Möbius , Comte de Saint-Venant , and Matthew O'Brien . Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis that is similar to today's system, and had ideas corresponding to the cross product, scalar product and vector differentiation. Grassmann's work
12152-437: 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
12276-415: The organisations. Another major finding was the need for cooperation between accreditation laboratories and the need to involve developing countries in the world of metrology. The Kaarls Report published in 2003 examined the role of the BIPM in the evolving needs for metrology in trade, industry and society. The CIPM has responsibility for commissioning the SI brochure, which is the formal definition of
12400-521: The organization and development of the BIPM, including its financial endowment. The CGPM meets in Paris, usually once every four years. The 25th meeting of the CGPM took place from 18 to 20 November 2014, the 26th meeting of the CGPM took place in Versailles from 13 to 16 November 2018, and the 27th meeting of the CGPM took place from 15 to 18 November 2022. On 20 May 1875 an international treaty known as
12524-405: The origin as a common base point. A vector a will be written as a = a 1 e 1 + a 2 e 2 + a 3 e 3 . {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}.} Two vectors are said to be equal if they have
12648-488: The other hand, have units of one-over-distance such as gradient . If you change units (a special case of a change of basis ) from meters to millimeters, a scale factor of 1/1000, a displacement of 1 m becomes 1000 mm—a contravariant change in numerical value. In contrast, a gradient of 1 K /m becomes 0.001 K/mm—a covariant change in value (for more, see covariance and contravariance of vectors ). Tensors are another type of quantity that behave in this way;
12772-408: The parallelogram). In any dimension (and, in particular, higher dimensions), it is possible to define the exterior product , which (among other things) supplies an algebraic characterization of the area and orientation in space of the n -dimensional parallelotope defined by n vectors. In a pseudo-Euclidean space , a vector's squared length can be positive, negative, or zero. An important example
12896-400: 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. 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
13020-434: The plane, and thus erected the first space of vectors in the plane. The term vector was introduced by William Rowan Hamilton as part of a quaternion , which is a sum q = s + v of a real number s (also called scalar ) and a 3-dimensional vector . Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments. As complex numbers use an imaginary unit to complement
13144-399: The principal physical quantities and maintenance of the international prototype standards. The CGPM acts on behalf of the governments of its members. In so doing, it elects members to the CIPM, receives reports from the CIPM which it passes on to the governments and national laboratories on member states, examines and where appropriate approves proposals from the CIPM in respect of changes to
13268-596: The rules of vector addition. An example is velocity , the magnitude of which is speed . For instance, the velocity 5 meters per second upward could be represented by the vector (0, 5) (in 2 dimensions with the positive y -axis as 'up'). Another quantity represented by a vector is force , since it has a magnitude and direction and follows the rules of vector addition. Vectors also describe many other physical quantities, such as linear displacement, displacement , linear acceleration, angular acceleration , linear momentum , and angular momentum . Other physical vectors, such as
13392-563: The same free vector if they have the same magnitude and direction: that is, they are equipollent if the quadrilateral ABB′A′ is a parallelogram . If the Euclidean space is equipped with a choice of origin , then a free vector is equivalent to the bound vector of the same magnitude and direction whose initial point is the origin. The term vector also has generalizations to higher dimensions, and to more formal approaches with much wider applications. In classical Euclidean geometry (i.e., synthetic geometry ), vectors were introduced (during
13516-699: The same magnitude and direction. Equivalently they will be equal if their coordinates are equal. So two vectors a = a 1 e 1 + a 2 e 2 + a 3 e 3 {\displaystyle {\mathbf {a} }=a_{1}{\mathbf {e} }_{1}+a_{2}{\mathbf {e} }_{2}+a_{3}{\mathbf {e} }_{3}} and b = b 1 e 1 + b 2 e 2 + b 3 e 3 {\displaystyle {\mathbf {b} }=b_{1}{\mathbf {e} }_{1}+b_{2}{\mathbf {e} }_{2}+b_{3}{\mathbf {e} }_{3}} are equal if
13640-510: The staff at the site in Saint-Cloud perform the day-to-day work. The CGPM recognises two classes of membership – full membership for those states that wish to participate in the activities of the BIPM and associate membership for those countries or economies that only wish to participate in the CIPM MRA program. Associate members have observer status at the CGPM. Since all formal liaison between
13764-421: The support on the body. Also it is equal to the force exerted by the body on its support because action and reaction have same numerical value and opposite direction. 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
13888-446: 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 giant planets (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 giant planets) and therefore, generally speaking, are similar to
14012-422: 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
14136-412: The tail of the vector coincides with the origin is implicit and easily understood. Thus, the more explicit notation O A → {\displaystyle {\overrightarrow {OA}}} is usually deemed not necessary (and is indeed rarely used). In three dimensional Euclidean space (or R ), vectors are identified with triples of scalar components: a = (
14260-420: 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. Several definitions exist for weight , not all of which are equivalent. The most common definition of weight found in introductory physics textbooks defines weight as the force exerted on a body by gravity. This
14384-487: The two definitions of Euclidean spaces are equivalent, and that the equivalence classes under equipollence may be identified with translations. Sometimes, Euclidean vectors are considered without reference to a Euclidean space. In this case, a Euclidean vector is an element of a normed vector space of finite dimension over the reals, or, typically, an element of the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} equipped with
14508-702: The unit vectors of a cylindrical coordinate system ( ρ ^ , ϕ ^ , z ^ {\displaystyle {\boldsymbol {\hat {\rho }}},{\boldsymbol {\hat {\phi }}},\mathbf {\hat {z}} } ) or spherical coordinate system ( r ^ , θ ^ , ϕ ^ {\displaystyle \mathbf {\hat {r}} ,{\boldsymbol {\hat {\theta }}},{\boldsymbol {\hat {\phi }}}} ). The latter two choices are more convenient for solving problems which possess cylindrical or spherical symmetry, respectively. The choice of
14632-413: 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 are 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
14756-405: The validity of calibration and measurement certificates issued by national metrology institutes. A recent focus area of the CIPM has been the revision of the SI . The CIPM has set up a number of consultative committees (CC) to assist it in its work. These committees are under the authority of the CIPM. The president of each committee, who is expected to take the chair at CC meetings, is usually
14880-430: The variation of acceleration due to gravity (and hence the variation of weight) at various locations on the Earth's surface. The historical 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
15004-405: The web. Gross weight is a term that is generally 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. The table below shows comparative gravitational accelerations at
15128-402: 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
15252-407: Was largely neglected until the 1870s. Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇. In 1878, Elements of Dynamic was published by William Kingdon Clifford . Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from
15376-435: 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 a 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
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