Contents 1 Phenomena 2 Units of mass 3 Definitions of mass 3.1
4 Pre-Newtonian concepts 4.1
5 Newtonian mass 5.1 Newton's cannonball 5.2 Universal gravitational mass 5.3 Inertial mass 6 Atomic mass
7
7.1
8
8.1 Tachyonic particles and imaginary (complex) mass 8.2 Exotic matter and negative mass 9 See also 10 Notes 11 References 12 External links Phenomena[edit] There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other,[2] current experiments have found no difference in results regardless of how it is measured: Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationship F = ma). Active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of
an applied force. The inertia and the inertial mass describe the same
properties of physical bodies at the qualitative and quantitative
level respectively, by other words, the mass quantitatively describes
the inertia. According to
The kilogram is one of the seven
The standard
the tonne (t) (or "metric ton") is equal to 1000 kg. the electronvolt (eV) is a unit of energy, but because of the mass–energy equivalence it can easily be converted to a unit of mass, and is often used like one. In this context, the mass has units of eV/c2 (where c is the speed of light). The electronvolt and its multiples, such as the MeV (megaelectronvolt), are commonly used in particle physics. the atomic mass unit (u) is 1/12 of the mass of a carbon-12 atom, approximately 6973166000000000000♠1.66×10−27 kg.[note 2] The atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include: the slug (sl) is an Imperial unit of mass (about 14.6 kg).
the pound (lb) is a unit of both mass and force, used mainly in the
United States (about 0.45 kg or 4.5 N). In scientific
contexts where pound (force) and pound (mass) need to be
distinguished,
Definitions of mass[edit] The relation between properties of mass and their associated physical constants. Every massive object is believed to exhibit all five properties. However, due to extremely large or extremely small constants, it is generally impossible to verify more than two or three properties for any object. The
In physical science, one may distinguish conceptually between at least seven different aspects of mass, or seven physical notions that involve the concept of mass.[5] Every experiment to date has shown these seven values to be proportional, and in some cases equal, and this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or operationally defined: Inertial mass is a measure of an object's resistance to acceleration
when a force is applied. It is determined by applying a force to an
object and measuring the acceleration that results from that force. An
object with small inertial mass will accelerate more than an object
with large inertial mass when acted upon by the same force. One says
the body of greater mass has greater inertia.
Active gravitational mass[note 3] is a measure of the strength of an
object's gravitational flux (gravitational flux is equal to the
surface integral of gravitational field over an enclosing surface).
a = M m g . displaystyle a= frac M m g. This says that the ratio of gravitational to inertial mass of any
object is equal to some constant K if and only if all objects fall at
the same rate in a given gravitational field. This phenomenon is
referred to as the "universality of free-fall". In addition, the
constant K can be taken as 1 by defining our units appropriately.
The first experiments demonstrating the universality of free-fall
were—according to scientific ‘folklore’—conducted by Galileo
obtained by dropping objects from the Leaning Tower of Pisa. This is
most likely apocryphal: he is more likely to have performed his
experiments with balls rolling down nearly frictionless inclined
planes to slow the motion and increase the timing accuracy.
Increasingly precise experiments have been performed, such as those
performed by Loránd Eötvös,[8] using the torsion balance pendulum,
in 1889. As of 2008[update], no deviation from universality, and thus
from Galilean equivalence, has ever been found, at least to the
precision 10−12. More precise experimental efforts are still being
carried out.
The universality of free-fall only applies to systems in which gravity
is the only acting force. All other forces, especially friction and
air resistance, must be absent or at least negligible. For example, if
a hammer and a feather are dropped from the same height through the
air on Earth, the feather will take much longer to reach the ground;
the feather is not really in free-fall because the force of air
resistance upwards against the feather is comparable to the downward
force of gravity. On the other hand, if the experiment is performed in
a vacuum, in which there is no air resistance, the hammer and the
feather should hit the ground at exactly the same time (assuming the
acceleration of both objects towards each other, and of the ground
towards both objects, for its own part, is negligible). This can
easily be done in a high school laboratory by dropping the objects in
transparent tubes that have the air removed with a vacuum pump. It is
even more dramatic when done in an environment that naturally has a
vacuum, as
Depiction of early balance scales in the
The concept of amount is very old and predates recorded history. Humans, at some early era, realized that the weight of a collection of similar objects was directly proportional to the number of objects in the collection: W n ∝ n , displaystyle W_ n propto n, where W is the weight of the collection of similar objects and n is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant ratio: W n n = W m m displaystyle frac W_ n n = frac W_ m m , or equivalently W n W m = n m . displaystyle frac W_ n W_ m = frac n m . An early use of this relationship is a balance scale, which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields. Hence, if they have similar masses then their weights will also be similar. This allows the scale, by comparing weights, to also compare masses. Consequently, historical weight standards were often defined in terms of amounts. The Romans, for example, used the carob seed (carat or siliqua) as a measurement standard. If an object's weight was equivalent to 1728 carob seeds, then the object was said to weigh one Roman pound. If, on the other hand, the object's weight was equivalent to 144 carob seeds then the object was said to weigh one Roman ounce (uncia). The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed. The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was: o u n c e p o u n d = W 144 W 1728 = 144 1728 = 1 12 . displaystyle frac mathrm ounce mathrm pound = frac W_ 144 W_ 1728 = frac 144 1728 = frac 1 12 . Planetary motion[edit]
See also: Kepler's laws of planetary motion
In 1600 AD,
Sometime prior to 1638, Galileo turned his attention to the phenomenon
of objects in free fall, attempting to characterize these motions.
Galileo was not the first to investigate Earth's gravitational field,
nor was he the first to accurately describe its fundamental
characteristics. However, Galileo's reliance on scientific
experimentation to establish physical principles would have a profound
effect on future generations of scientists. It is unclear if these
were just hypothetical experiments used to illustrate a concept, or if
they were real experiments performed by Galileo,[9] but the results
obtained from these experiments were both realistic and compelling. A
biography by Galileo's pupil
"a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results."[11] Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time: Distance ∝ Time 2 displaystyle text
Galileo had shown that objects in free fall under the influence of the Earth’s gravitational field have a constant acceleration, and Galileo’s contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun’s gravitational mass. However, Galileo’s free fall motions and Kepler’s planetary motions remained distinct during Galileo’s lifetime. Newtonian mass[edit]
Earth's Moon
Semi-major axis Sidereal orbital period 0.002 569 AU 0.074 802 sidereal year 1.2 π 2 ⋅ 10 − 5 AU 3 y 2 = 3.986 ⋅ 10 14 m 3 s 2 displaystyle 1.2pi ^ 2 cdot 10^ -5 frac text AU ^ 3 text y ^ 2 =3.986cdot 10^ 14 frac text m ^ 3 text s ^ 2 Earth's gravity Earth's radius 9.806 65 m/s2 6 375 km
g = − μ R ^
R
2 displaystyle mathbf g =-mu frac hat mathbf R mathbf R ^ 2 where g is the apparent acceleration of a body as it passes through a
region of space where gravitational fields exist, μ is the
gravitational mass (standard gravitational parameter) of the body
causing gravitational fields, and R is the radial coordinate (the
distance between the centers of the two bodies).
By finding the exact relationship between a body's gravitational mass
and its gravitational field, Newton provided a second method for
measuring gravitational mass. The mass of the
A cannon on top of a very high mountain shoots a cannonball
horizontally. If the speed is low, the cannonball quickly falls back
to
Main article: Newton's cannonball
An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center In contrast to earlier theories (e.g. celestial spheres) which stated
that the heavens were made of entirely different material, Newton's
theory of mass was groundbreaking partly because it introduced
universal gravitational mass: every object has gravitational mass, and
therefore, every object generates a gravitational field. Newton
further assumed that the strength of each object's gravitational field
would decrease according to the square of the distance to that object.
If a large collection of small objects were formed into a giant
spherical body such as the
Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper. Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice. According to Newton's theory all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere. However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the Cavendish experiment, did not occur until 1797, over a hundred years later. Cavendish found that the Earth's density was 5.448 ± 0.033 times that of water. As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures.[clarification needed] Given two objects A and B, of masses MA and MB, separated by a displacement RAB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude F AB = − G M A M B R ^ AB
R AB
2
displaystyle mathbf F _ text AB =-GM_ text A M_ text B frac hat mathbf R _ text AB mathbf R _ text AB ^ 2 , where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is F = M g displaystyle F=Mg . This is the basis by which masses are determined by weighing. In simple spring scales, for example, the force F is proportional to the displacement of the spring beneath the weighing pan, as per Hooke's law, and the scales are calibrated to take g into account, allowing the mass M to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a balance measures relative weight, giving the relative gravitation mass of each object. Inertial mass[edit] Massmeter, a device for measuring the inertial mass of an astronaut in weightlessness. The mass is calculated via the oscillation period for a spring with the astronaut attached (Tsiolkovsky State Museum of the History of Cosmonautics) Inertial mass is the mass of an object measured by its resistance to acceleration. This definition has been championed by Ernst Mach[18][19] and has since been developed into the notion of operationalism by Percy W. Bridgman.[20][21] The simple classical mechanics definition of mass is slightly different than the definition in the theory of special relativity, but the essential meaning is the same. In classical mechanics, according to Newton's second law, we say that a body has a mass m if, at any instant of time, it obeys the equation of motion F = m a , displaystyle mathbf F =mmathbf a , where F is the resultant force acting on the body and a is the
acceleration of the body's centre of mass.[note 7] For the moment, we
will put aside the question of what "force acting on the body"
actually means.
This equation illustrates how mass relates to the inertia of a body.
Consider two objects with different masses. If we apply an identical
force to each, the object with a bigger mass will experience a smaller
acceleration, and the object with a smaller mass will experience a
bigger acceleration. We might say that the larger mass exerts a
greater "resistance" to changing its state of motion in response to
the force.
However, this notion of applying "identical" forces to different
objects brings us back to the fact that we have not really defined
what a force is. We can sidestep this difficulty with the help of
Newton's third law, which states that if one object exerts a force on
a second object, it will experience an equal and opposite force. To be
precise, suppose we have two objects of constant inertial masses m1
and m2. We isolate the two objects from all other physical influences,
so that the only forces present are the force exerted on m1 by m2,
which we denote F12, and the force exerted on m2 by m1, which we
denote F21.
F 12 = m 1 a 1 , F 21 = m 2 a 2 , displaystyle begin aligned mathbf F_ 12 &=m_ 1 mathbf a _ 1 ,\mathbf F_ 21 &=m_ 2 mathbf a _ 2 ,end aligned where a1 and a2 are the accelerations of m1 and m2, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that F 12 = − F 21 ; displaystyle mathbf F _ 12 =-mathbf F _ 21 ; and thus m 1 = m 2
a 2
a 1
. displaystyle m_ 1 =m_ 2 frac mathbf a _ 2 mathbf a _ 1 !. If a1 is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of m1. In this case, m2 is our "reference" object, and we can define its mass m as (say) 1 kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations. Additionally, mass relates a body's momentum p to its linear velocity v: p = m v displaystyle mathbf p =mmathbf v , and the body's kinetic energy K to its velocity: K = 1 2 m
v
2 displaystyle K= dfrac 1 2 mmathbf v ^ 2 . The primary difficulty with Mach's definition of mass is that it fails
to take into account the potential energy (or binding energy) needed
to bring two masses sufficiently close to one another to perform the
measurement of mass.[19] This is most vividly demonstrated by
comparing the mass of the proton in the nucleus of deuterium, to the
mass of the proton in free space (which is greater by about 0.239% -
this is due to the binding energy of deuterium.). Thus, for example,
if the reference weight m2 is taken to be the mass of the neutron in
free space, and the relative accelerations for the proton and neutron
in deuterium are computed, then the above formula over-estimates the
mass m1 (by 0.239%) for the proton in deuterium. At best, Mach's
formula can only be used to obtain ratios of masses, that is, as m1
/m2 = a2 / a1. An additional difficulty was pointed out by Henri
Poincaré, which is that the measurement of instantaneous acceleration
is impossible: unlike the measurement of time or distance, there is no
way to measure acceleration with a single measurement; one must make
multiple measurements (of position, time, etc.) and perform a
computation to obtain the acceleration. Poincaré termed this to be an
"insurmountable flaw" in the Mach definition of mass.[22]
Atomic mass[edit]
Main article: Atomic mass unit
Typically, the mass of objects is measured in relation to that of the
kilogram, which is defined as the mass of the international prototype
kilogram (IPK), a platinum alloy cylinder stored in an
environmentally-monitored safe secured in a vault at the International
Bureau of Weights and Measures in France. However, the IPK is not
convenient for measuring the masses of atoms and particles of similar
scale, as it contains trillions of trillions of atoms, and has most
certainly lost or gained a little mass over time despite the best
efforts to prevent this. It is much easier to precisely compare an
atom's mass to that of another atom, thus scientists developed the
atomic mass unit (or Dalton). By definition, 1 u is exactly one
twelfth of the mass of a carbon-12 atom, and by extension a carbon-12
atom has a mass of exactly 12 u. This definition, however, might
be changed by the proposed redefinition of SI base units, which will
leave the Dalton very close to one, but no longer exactly equal to
it.[23][24]
m r e l a t i v e = γ ( m r e s t ) displaystyle m_ mathrm relative =gamma (m_ mathrm rest )! where γ displaystyle gamma is the Lorentz factor: γ = 1 1 − v 2 / c 2 displaystyle gamma = frac 1 sqrt 1-v^ 2 /c^ 2 The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's frame of reference. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy E and the magnitude of its momentum p by the relativistic energy-momentum equation: ( m r e s t ) c 2 = E t o t a l 2 − (
p
c ) 2 . displaystyle (m_ mathrm rest )c^ 2 = sqrt E_ mathrm total ^ 2 -(mathbf p c)^ 2 .! So long as the system is closed with respect to mass and energy, both
kinds of mass are conserved in any given frame of reference. The
conservation of mass holds even as some types of particles are
converted to others.
E r e s t = ( m r e s t ) c 2 displaystyle E_ mathrm rest =(m_ mathrm rest )c^ 2 ! E t o t a l = ( m r e l a t i v e ) c 2 displaystyle E_ mathrm total =(m_ mathrm relative )c^ 2 ! The "relativistic" mass and energy concepts are related to their
"rest" counterparts, but they do not have the same value as their rest
counterparts in systems where there is a net momentum. Because the
relativistic mass is proportional to the energy, it has gradually
fallen into disuse among physicists.[26] There is disagreement over
whether the concept remains useful pedagogically.[27][28][29]
In bound systems, the binding energy must often be subtracted from the
mass of the unbound system, because binding energy commonly leaves the
system at the time it is bound. The mass of the system changes in this
process merely because the system was not closed during the binding
process, so the energy escaped. For example, the binding energy of
atomic nuclei is often lost in the form of gamma rays when the nuclei
are formed, leaving nuclides which have less mass than the free
particles (nucleons) of which they are composed.
d d t
( ∂ L ∂ x ˙ i ) = m x ¨ i displaystyle frac mathrm d mathrm d t left(, frac partial L partial dot x _ i ,right) = m, ddot x _ i . After quantization, replacing the position vector x with a wave function, the parameter m appears in the kinetic energy operator: i ℏ ∂ ∂ t Ψ ( r , t ) = ( − ℏ 2 2 m ∇ 2 + V ( r ) ) Ψ ( r , t ) displaystyle ihbar frac partial partial t Psi (mathbf r ,,t)=left(- frac hbar ^ 2 2m nabla ^ 2 +V(mathbf r )right)Psi (mathbf r ,,t) . In the ostensibly covariant (relativistically invariant) Dirac equation, and in natural units, this becomes: ( − i γ μ ∂ μ + m ) ψ = 0 displaystyle (-igamma ^ mu partial _ mu +m)psi =0, where the "mass" parameter m is now simply a constant associated with
the quantum described by the wave function ψ.
In the
G ψ ψ ¯ ϕ ψ displaystyle G_ psi overline psi phi psi . This shifts the explanandum of the value for the mass of each
elementary particle to the value of the unknown couplings Gψ.
Tachyonic particles and imaginary (complex) mass[edit]
Main articles:
E 2 = p 2 c 2 + m 2 c 4 displaystyle E^ 2 =p^ 2 c^ 2 +m^ 2 c^ 4 ; (where p is the relativistic momentum of the bradyon and m is its rest mass) should still apply, along with the formula for the total energy of a particle: E = m c 2 1 − v 2 c 2 . displaystyle E= frac mc^ 2 sqrt 1- frac v^ 2 c^ 2 . This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy. When v is larger than c, the denominator in the equation for the energy is "imaginary", as the value under the radical is negative. Because the total energy must be real, the numerator must also be imaginary: i.e. the rest mass m must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number. Exotic matter and negative mass[edit] Main article: negative mass The negative mass exists in the model to describe dark energy (phantom energy) and radiation in negative-index metamaterial in a unified way.[42] In this way, the negative mass is associated with negative momentum, negative pressure, negative kinetic energy and FTL (faster-than-light). See also[edit]
Notes[edit] ^ When a distinction is necessary, M is used to denote the active
gravitational mass and m the passive gravitational mass.
^ Since the
μ = 4 π 2 distance 3 time 2 ∝ gravitational mass displaystyle mu =4pi ^ 2 frac text distance ^ 3 text time ^ 2 propto text gravitational mass ^ At the time when Viviani asserts that the experiment took place,
Galileo had not yet formulated the final version of his law of free
fall. He had, however, formulated an earlier version which predicted
that bodies of the same material falling through the same medium would
fall at the same speed. See Drake, S. (1978). Galileo at Work.
University of Chicago Press. pp. 19–20.
ISBN 0-226-16226-5.
^ These two properties are very useful, as they allow spherical
collections of objects to be treated exactly like large individual
objects.
^ In its original form,
References[edit] ^ http://dictionary.reference.com/browse/mass
^ "New
English
^ Hecht, Eugene (2006). "There Is No Really Good Definition of Mass"
(PDF). Phys. Teach. 44: 40–45. Bibcode:2006PhTea..44...40H.
doi:10.1119/1.2150758.
^ Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W.
H. Freeman. p. 466. ISBN 978-0-7167-0344-0.
^ a b c Lisa Randall, Warped Passages: Unraveling the Mysteries of the
Universe's Hidden Dimensions, p.286: "People initially thought of
tachyons as particles travelling faster than the speed of light...But
we now know that a tachyon indicates an instability in a theory that
contains it. Regrettably for science fiction fans, tachyons are not
real physical particles that appear in nature."
^ Tipler, Paul A.; Llewellyn, Ralph A. (2008). Modern
External links[edit] Wikimedia Commons has media related to
Francisco Flores (6 Feb 2012). "The Equivalence of
v t e SI base quantities Base quantity Quantity SI unit Name Symbol Dimension symbol Unit name (symbol) Example length l, x, r, (etc.) L metre (m) r = 10 m mass m M kilogram (kg) m = 10 kg time, duration t T second (s) t = 10 s electric current I , i I ampere (A) I = 10 A thermodynamic temperature T Θ kelvin (K) T = 10 K amount of substance n N mole (mol) n = 10 mol luminous intensity Iv J candela (cd) Iv = 10 cd Specification The quantity (not the unit) can have a specification: Tmax = 300 K Derived quantity Definition A quantity Q is expressed in the base quantities: Q = f ( l , m , t , I , T , n , I v ) displaystyle Q=fleft( mathit l,m,t,I,T,n,I mathrm _ v right) Derived dimension dim Q = La · Mb · Tc · Id · Θe · Nf · Jg (Superscripts a–g are algebraic exponents, usually a positive, negative or zero integer.) Example Quantity acceleration = l1 · t−2, dim acceleration = L1 · T−2 possible units: m1 · s−2, km1 · Ms−2, etc. See also History of the metric system International System of Quantities Proposed redefinitions Systems of measurement Book Category Outline v t e
Linear/translational quantities Angular/rotational quantities Dimensions 1 L L2 Dimensions 1 1 1 T time: t s absement: A m s T time: t s 1 distance: d, position: r, s, x, displacement m area: A m2 1 angle: θ, angular displacement: θ rad solid angle: Ω rad2, sr T−1 frequency: f s−1, Hz speed: v, velocity: v m s−1 kinematic viscosity: ν, specific angular momentum: h m2 s−1 T−1 frequency: f s−1, Hz angular speed: ω, angular velocity: ω rad s−1 T−2 acceleration: a m s−2 T−2 angular acceleration: α rad s−2 T−3 jerk: j m s−3 T−3 angular jerk: ζ rad s−3 M mass: m kg ML2 moment of inertia: I kg m2 MT−1 momentum: p, impulse: J kg m s−1, N s action: 𝒮, actergy: ℵ kg m2 s−1, J s ML2T−1 angular momentum: L, angular impulse: ΔL kg m2 s−1 action: 𝒮, actergy: ℵ kg m2 s−1, J s MT−2 force: F, weight: Fg kg m s−2, N energy: E, work: W kg m2 s−2, J ML2T−2 torque: τ, moment: M kg m2 s−2, N m energy: E, work: W kg m2 s−2, J MT−3 yank: Y kg m s−3, N s−1 power: P kg m2 s−3, W ML2T−3 rotatum: P kg m2 s−3, N m s−1 power: P kg m2 s−3, W Authority control LCCN: sh85081853 GND: 4169025-4 SUDOC: 027767027 BNF: cb11973906k (data) NDL: 0057 |