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Sir Isaac Newton (1642-1727)
Sir Isaac Newton (1642-1727)

Mass is a concept used in the physical sciences to explain a number of observable behaviors, and in everyday usage, it is common to identify mass with those resulting behaviors. In particular, mass is commonly identified with weight. But according to our modern scientific understanding, the weight of an object results from the interaction of its mass with a gravitational field, so while mass is part of the explanation of weight, it is not the complete explanation.

For example, a mail carrier lifting a heavy package on earth may associate the heaviness (weight) of the package with the mass of its contents. However, if the same package were on the moon it would weigh much less and would be easy to lift. Therefore, the mass of the package is only part of the reason that the package is difficult to lift.

Also, a groundskeeper encountering two large rocks may associate the size of the rocks with their respective masses. And from this association, the groundskeeper may expect the larger rock to be heavier and more difficult to move. However, if the larger rock were composed of pumice and the smaller of granite then the smaller rock may in fact be much heavier. Mass is part of the explanation of an object’s size, but not the complete explanation. The complete explanation involves mass, structure, and composition.

The human body is equipped with physical senses through which one can experience many of the effects associated with mass. For example, one may visually observe an object to determine its size, lift it to feel its weight, and push it to feel the force of its inertial resistance to changing motion. Humans throughout history have always observed these effects. They have also observed the planets moving through the night sky under the influence of the sun’s gravity, and they have observed objects falling to the earth under the influence of the earth’s gravity. Since these effects are all part of human existence, humans have always had an intuitive understanding of these physical phenomena. This intuitive understanding, however, has only recently evolved into the modern abstract concept of mass.

Our current understanding of mass did not come from a specific type of human experience. Rather, our understanding came from a synthesis of many different types of human experience. The modern concept was introduced in, and is central to, Isaac Newton’s explanation of gravitation and inertia. Prior to Newton’s time, the various gravitational and inertial phenomena were viewed as distinct and potentially unrelated. However, Isaac Newton united these phenomena by asserting that they all stemmed from a single underlying property called mass. Since Newton’s time, this abstract concept of mass has grown to include explanations for both quantum and relativistic effects. (See the following section entitled “Summary of concepts of mass” for a brief summery of these mass related phenomena)

Units of mass

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

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

In normal situations, the weight of an object is proportional to its mass, which usually makes it unproblematic to use the same unit for both concepts. However, the distinction between mass 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 surface of the Earth, such as in space or on other planets.

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 mass instead. For example, the eV energy unit is normally used as a unit of mass (roughly 1.783×10 kg) in particle physics. A mass can sometimes also be expressed in terms of length. Here one identifies the mass of a particle with its inverse Compton wavelength (1 cm ≈ 3.52×10 kg).

Summary of concepts of mass

The above diagram illustrates five interrelated properties of mass together with the proportionality constants that relate these properties. Every sample of mass is believed to exhibit all five properties, however, due to extremely large proportionality constants, it is generally impossible to verify more than two or three properties for a specific sample of mass.

The Schwarzschild radius (rs) represents the ability of mass to cause curvature in space and time.
The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
Inertial mass (m) represents the Newtonian response of mass to forces.
Rest energy (E0) represents the ability of mass to be converted into other forms of energy.
The Compton wavelength (λ) represents the quantum response of mass to local geometry.
The above diagram illustrates five interrelated properties of mass together with the proportionality constants that relate these properties. Every sample of mass is believed to exhibit all five properties, however, due to extremely large proportionality constants, it is generally impossible to verify more than two or three properties for a specific sample of mass.
  • The Schwarzschild radius (rs) represents the ability of mass to cause curvature in space and time.
  • The standard gravitational parameter (μ) represents the ability of a massive body to exert Newtonian gravitational forces on other bodies.
  • Inertial mass (m) represents the Newtonian response of mass to forces.
  • Rest energy (E0) represents the ability of mass to be converted into other forms of energy.
  • The Compton wavelength (λ) represents the quantum response of mass to local geometry.

In physical science, one may distinguish conceptually between at least seven attributes of mass, or seven physical phenomena that can be explained using the concept of mass:

  • Inertial mass is a measure of an object's resistance to changing its state of motion 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.
  • The amount of matter in certain types of samples can be exactly determined through electrodeposition or other precise processes. The mass of an exact sample 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 mass," or mass deficit).
  • Active gravitational mass 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 inclosing surface). Gravitational field can be measured by allowing a small ‘test object’ to freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower than the same object would if it were in free-fall near the earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.
  • Passive gravitational mass is a measure of the strength of an object's interaction with a gravitational field. Passive gravitational mass is determined by dividing an object’s weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration, however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.
  • Energy also has mass according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair production and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.
  • Curvature of spacetime is a relativistic manifestation of the existence of mass. Curvature is extremely weak and difficult to measure. For this reason, curvature wasn’t discovered until after it was predicted by Einstein’s theory of general relativity. Extremely precise atomic clocks on the surface of the earth, for example, are found to measure less time (run slower) than similar clocks in space. This difference in elapsed time is a form of curvature called gravitational time dilation. Other forms of curvature have been measured using the Gravity Probe B satellite.

Inertial and gravitational mass

Although inertial mass, passive gravitational mass and active gravitational mass 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 mass has to equal the inertial mass. 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 mass 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 mass [...] remains as puzzling as ever".

Inertial mass

This section uses mathematical equations involving differential calculus.

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

To understand what the inertial mass 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 mass 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 "mass" in any essential way.

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 = \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 mass of the body in question is a constant. This assumption, known as the conservation of mass, rests on the ideas that (i) mass 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 mass 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 mass and energy in relativity always occur in exact association.

When the mass of a body is constant (neither mass 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} = m a

where a denotes the acceleration of the body.

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 A and B, with constant inertial masses mA and mB. 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 fAB, and the force exerted on B by A, which we denote fBA. As we have seen, Newton's second law states that

f_{AB} = m_B a_B \, and f_{BA} = m_A a_A \,

where aA and aB 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 aA be non-zero ensures that the fraction is well-defined.

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

Gravitational mass

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

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

|f| = {G M_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 mass 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 masses 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 mass M 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 mass; only the spring scale measures weight.

Equivalence of inertial and gravitational masses

The equivalence of inertial and gravitational masses 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 masses m and M 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{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 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 experiments with 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. 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 masses are fundamentally the same thing.

Mass and energy in relativity

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. However, the more general invariant mass (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 mass 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 mass/energy is allowed across it. In as much as energy is conserved in closed systems in relativity, the relativistic definition(s) of mass are quantities which are conserved also; they do not change over time, even as some types of particles are converted to others.

In bound systems, the binding energy must (often) be subtracted from the mass of the unbound system, simply because this energy has mass, and this mass is subtracted from the system when it is given off, at the time it is bound. Mass 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 mass than the free particles (nucleons) of which they are composed.

The term relativistic mass is also used, and this is the total quantity of energy in a body or system (divided by c). The relativistic mass (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 mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.

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

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

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Published in July 2009.

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