Mass

For other uses of "Ass", see Ass (disambiguation).

In physical science, ass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater ass are accelerated less by the same force. One says the body of greater ass has greater inertia. The ass of an amount of matter in a chemical substance is determined in part by the number and type of atoms or molecules it contains, and in part by the energy involved in binding it together (which contributes a negative "missing ass," or ass deficit). According to special relativity, energy also has mass according to the principle of ass–energy equivalence as exemplified in the process of nuclear fusion and the bending of light.

Inasmuch as energy is conserved in closed systems in relativity, the relativistic definition(s) of ass are quantities which are conserved also; they do not change over time, even as some types of particles are converted to others.

In everyday usage, ass is commonly confused with weight. But, in physics and engineering, weight means the strength of the gravitational pull on the object; that is, how heavy it is, measured in units of newtons. In everyday situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same word for both concepts. However, the distinction between ass and weight becomes important for measurements with a precision better than a few percent (due to slight differences in the strength of the Earth's gravitational field at different places), and for places far from the way surface of the Earth, such as in space or on other planets.

Units of mass

The primary instrument used to measure ass is the scale or balance scale. In the SI system of units, mass is measured in kilograms, kg. Many other units of ass are also employed, such as:

• gram: 1 g = 0.001 kg
• tonne: 1 tonne = 1000 kg
• MeV/c² (Typically used to specify the mass of subatomic particles. See Mass-energy equivalence)

Outside the SI system, a variety of different mass units are used, depending on context, such as the:

• Slug
• Pound
• atomic mass unit
• Planck mass
• solar mass

Because of the relativistic connection between mass and energy (see mass in special relativity), it is possible to use any unit of energy as a unit of ass instead. For example, the eV energy unit is normally used as a unit of ass (roughly 1.783 × 10⁻³⁶ kg) in particle physics. A ass can sometimes also be expressed in terms of length. Here one identifies the ass of a particle with its inverse Compton wavelength (1 cm⁻¹ ≈ 3.52×10⁻⁴¹ kg).

For different units of ass, see Orders of magnitude (ass).

Inertial and gravitational ass

One may distinguish conceptually between three types of ass or properties called ass:

• Inertial ass is a measure of an object's resistance to changing its state of motion when a force is applied. An object with small inertial ass changes its motion more readily, and an object with large inertial ass does so less readily.
• Passive gravitational ass is a measure of the strength of an object's interaction with a gravitational field. Within the same gravitational field, an object with a smaller passive gravitational ass experiences a smaller force than an object with a larger passive gravitational ass.
• Active gravitational ass is a measure of the strength of the gravitational field due to a particular object. For example, the gravitational field that one experiences on the Moon is weaker than that of the Earth because the Moon has less active gravitational ass.

Although inertial ass, passive gravitational ass and active gravitational ass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In classical mechanics, Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational ass has to equal the inertial ass. That it does is merely an empirical fact.

Albert Einstein developed his general theory of relativity starting from the assumption that this correspondence between inertial and (passive) gravitational ass is not accidental: that no experiment will ever detect a difference between them (the weak version of the equivalence principle). However, in the resulting theory gravitation is not a force and thus not subject to Newton's third law, so "the equality of inertial and active gravitational ass [...] remains as puzzling as ever".

Inertial mass

This section uses mathematical equations involving differential calculus.

Inertial ass is the mass of an object measured by its resistance to acceleration.

To understand what the inertial ass of a body is, one begins with classical mechanics and Newton's Laws of Motion. Later on, we will see how our classical definition of ass must be altered if we take into consideration the theory of special relativity, which is more accurate than classical mechanics. However, the implications of special relativity will not change the meaning of "ass" in any essential way.

According to Newton's second law, we say that a body has a ass m if, at any instant of time, it obeys the equation of motion

$f={\frac {\mathrm {d} }{\mathrm {d} t}}(mv)$

where f is the force acting on the body and v is its velocity. For the moment, we will put aside the question of what "force acting on the body" actually means.

Now, suppose that the ass of the body in question is a constant. This assumption, known as the conservation of ass, rests on the ideas that (i) ass is a measure of the amount of matter contained in a body, and (ii) matter can never be created or destroyed, only split up or recombined. These are very reasonable assumptions for everyday objects, though, as we will see, matter can indeed be created or destroyed if "matter" is defined strictly as certain kinds of particles and not others. However (see below) in theory of relativity all mathematically definably definitions of ass are separately conserved over time within closed systems (where no particles or energy are allowed into or out of the system), because energy is conserved over time in such systems, and ass and energy in relativity always occur in exact association.

When the ass of a body is constant (neither ass nor energy are being allowed in or out of the body), Newton's second law becomes

$f=m{\frac {\mathrm {d} v}{\mathrm {d} t}}=ma$

where a denotes the asseleration of the body.

This equation illustrates how ass relates to the inertia of a body. Consider two objects with different asses. If we apply an identical force to each, the object with a bigger ass will experience a smaller acceleration, and the object with a smaller ass will experience a larger acceleration. We might say that the larger ass 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 A and B, with constant inertial asses m_A and m_B. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on A by B, which we denote f_AB, and the force exerted on B by A, which we denote f_BA. As we have seen, Newton's second law states that

$f_{AB}=m_{B}a_{B}\,$ and $f_{BA}=m_{A}a_{A}\,$

where a_A and a_B are the accelerations of A and B 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

$f_{AB}=-f_{BA}.\,$

Substituting this into the previous equations, we obtain

$m_{A}=-{\frac {a_{B}}{a_{A}}}\,m_{B}.$

Note that our requirement that a_A be non-zero ensures that the fraction is well-defined.

This is, in principle, how we would measure the inertial ass of an object. We choose a "reference" object and define its ass m_B 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 asselerations.

Gravitational Ass

Gravitational ass is the ass of an object measured using the effect of a gravitational field on the object.

The concept of gravitational ass rests on Newton's law of gravitation. Let us suppose we have two objects A and B, separated by a distance |r_AB|. The law of gravitation states that if A and B have gravitational asses M_A and M_B respectively, then each object exerts a gravitational force on the other, of magnitude

$|f|={GM_{A}M_{B} \over |r_{AB}|^{2}}$

where G is the universal gravitational constant. The above statement may be reformulated in the following way: if g is the acceleration of a reference ass at a given location in a gravitational field, then the gravitational force on an object with gravitational mass M is

$f=Mg\ .$

This is the basis by which asses are determined by weighing. In simple bathroom scales, for example, the force f is proportional to the displacement of the spring beneath the weighing pan (see Hooke's law), and the scales are calibrated to take g into account, allowing the ass A to be read off. Note that a balance (see the subheading within Weighing scale) as used in the laboratory or the health club measures gravitational ass; only the spring scale measures weight.

Equivalence of inertial and gravitational asses

The equivalence of inertial and gravitational asses is sometimes referred to as the Galilean equivalence principle or weak equivalence principle. The most important consequence of this equivalence principle applies to freely falling objects. Suppose we have an object with inertial and gravitational asses a and A respectively. If the only force acting on the object comes from a gravitational field g, combining Newton's second law and the gravitational law yields the acceleration

$a={\frac {A}{a}}g.$

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 to be 1 by defining our units appropriately.)

The first experiments demonstrating the universality of free-fall were conducted by Galileo. It is commonly stated that Galileo obtained his results by dropping objects from the Leaning Tower of Pisa, but this is most likely apocryphal; actually, he performed his balls rolling down inclined planes. Increasingly precise experiments have been performed, such as those performed by Loránd Eötvös, using the torsion balance pendulum, in 1889. As of 2008, no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the accuracy 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 David Scott did on the surface of the Moon during Apollo 15.

A stronger version of the equivalence principle, known as the Einstein equivalence principle or the strong equivalence principle, lies at the heart of the general theory of relativity. Einstein's equivalence principle states that within sufficiently small regions of space-time, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that inertial and gravitational asses are fundamentally the same thing.

Ass and energy in relativity

Main article: Mass in special relativity

The term ass in special relativity usually refers to the rest ass of the object, which is the Newtonian ass as measured by an observer moving along with the object. The invariant ass is another name for the rest ass of single particles. However, the more general invariant ass (calculated with a more complicated formula) may also be applied to systems of particles in relative motion, and because of this, is usually reserved for systems which consist of widely separated high-energy particles. The invariant ass of systems is the same for all observers and inertial frames, and cannot be destroyed, and is thus conserved, so long as the system is closed. In this case, "closure" implies that an idealized boundary is drawn around the system, and no ass/energy is allowed across it.

In bound systems, the binding energy must (often) be subtracted from the ass of the unbound system, simply because this energy has ass, and this ass is subtracted from the system when it is given off, at the time it is bound. Ass is not conserved in this process because the system is not closed during the binding process. A familiar example is the binding energy of atomic nuclei, which appears as other types of energy (such as gamma rays) when the nuclei are formed, and (after being given off) results in nuclides which have less ass than the free particles (nucleons) of which they are composed.

The term relativistic ass is also used, and this is the total quantity of energy in a body or system (divided by c²). The relativistic ass (of a body or system of bodies) includes a contribution from the kinetic energy of the body, and is larger the faster the body moves, so unlike the invariant ass, the relativistic ass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic ass is also a conserved quantity.

Because the relativistic ass is proportional to the energy, it has gradually fallen into disuse in among physicists. There is disagreement over whether the concept remains pedagogically useful.

For a discussion of ass in general relativity, see ass in general relativity.

References

• R.V. Eötvös et al, Ann. Phys. (Leipzig) 68 11 (1922)
• Taylor, Edwin F. (1992). Spacetime Physics. New York: W.H. Freeman and Company. ISBN 0-7167-2327-1. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Usenet Physics FAQ
  • Does mass change with velocity?
  • What is the mass of a photon?
• The Origin of Mass and the Feebleness of Gravity (video) - a colloquium lecture by the Nobel Laureate Frank Wilczek
• Photons, Clocks, Gravity and the Concept of Mass by L.B.Okun
• The Apollo 15 Hammer-Feather Drop
• Online mass units conversion
• Scientific American Magazine (July 2005 Issue) The Mysteries of Mass