The quantitative measure of the inertia of an object, that is, of its resistance to being accelerated. Thus, the more massive an object is, the harder it is to accelerate it (change its state of motion). Isaac Newton described mass as the “quantity of matter” when formulating his second law of motion. He observed that the rate of change of motion is proportional to the impressed force and in the direction of that force. By “motion” he meant the product of the quantity of matter and the velocity. In modern terms Newton's “motion” is known as momentum. The second law is expressed by Eq. (1),
where F is the applied force, t the time, and p the momentum. If the mass of the object is constant, then the rate of change of momentum with time is just the product of the mass with the rate of change of the velocity with time, the acceleration a. For this situation the second law may be written as Eq. (2).
By suitable choice of units the proportionality becomes an equality as shown in Eq. (3).
In particular, the equality holds in the International System of Units (SI), in which acceleration is measured in meters/second, force is in newtons, and the unit of mass is the kilogram (kg). See also: Acceleration; Units of measurement
The primary standard of the unit of mass is the international prototype kilogram kept at Le Bureau International des Poids et Mesures in France. The prototype kilogram is a cylinder of a platinum-iridium alloy, a material chosen for its stability. The standard of mass is kept under conditions specified initially by the first Conférence Général des Poids et Mesures in 1889. Secondary standards are maintained in laboratories around the world and are compared with the mass of the prototype by means of balances whose precision reaches 1 part in 108 or better. See also: Physical measurement
The mass that is described through the second law is known as inertial mass. The inertial mass of any object may be measured by comparing it to the inertial mass of another object. For example, two objects may be made to collide and their accelerations measured. According to Newton's third law of motion, both objects experience forces of equal magnitude. Consequently the ratio of their masses will be related to their accelerations through Eq. (4),
where a1 and a2 are the magnitudes of accelerations. Thus, if the value of one mass is known, the value of the other mass may be determined by experiment.
Masses may be compared or measured by means other than collisions. Systems that oscillate do so with a frequency that depends on their mass. For example, the frequency of a mass oscillating at the end of a spring decreases as the mass is increased. This behavior is utilized to construct inertial balances, devices that measure the inertial mass. See also: Harmonic motion
Mass and weight
The most common way of measuring mass is by weighing, a process that involves the effect of gravity. With a simple two-pan balance, two masses are compared by comparing their weights. The weight of an object on Earth is the gravitational force of attraction exerted on it by the Earth. The response of the balance to the two masses is really a response to their weights. As a result, in commercial and common use, weight is often expressed in units of kilograms instead of the correct units of newtons. See also: Gravitation; Weight
The gravitational force that leads to weight is not confined to just the interaction between the Earth and objects at its surface. Indeed, the law of universal gravitation, first formulated by Newton in 1684, describes an attractive force that exists between all objects and is proportional to the product of their masses. In effect, mass is the source of the gravitational force of attraction. Consequently, there is another interpretation of mass, one that is based on gravitational force and not on inertia. It is called gravitational mass to distinguish it from inertial mass.
Newton recognized that he had two distinct concepts of mass and devised an experimental test of the equivalence of inertial and gravitational mass. Using crude methods he was unable to detect any difference between the two types of mass. In 1890, Lorand Eötvös devised a sensitive method to test the equivalence of inertial and gravitational mass. He used objects of different materials to compare the effect of the Earth's gravitational force on them with the inertial effects of the Earth's rotation. Eötvös concluded that the two types of mass were the same. More recent experiments with even greater sensitivity continue to support the idea that inertial mass and gravitational mass are identical. The principle of equivalence is the assertion that both masses are exactly equal. Albert Einstein employed the principle of equivalence in his development of the general theory of relativity. See also: Relativity
Weightlessness and microgravity
Near the Earth's surface, when air resistance is removed, all objects fall with the same acceleration. This constant acceleration is known as the acceleration of gravity, g, and has the standard value 9.80665 m/s2. Although g is the same for all objects in a given locality, it does vary slightly with altitude and latitude. Newton's second law may be used to relate the gravitational force (the weight) to the acceleration. Thus, the weight w may be expressed by Eq. (5),
which gives the relationship between mass and weight. Weight is a force proportional to the mass of an object and g is the constant of proportionality.
When you stand on a scale, the scale reading gives the magnitude of your weight. Most of the force that comprises your weight is due to the gravitational attraction between you and the Earth. However, because the Earth rotates, your weight is slightly less than it would be if the Earth were not rotating. Consequently, a more precise definition of weight is as follows: The weight of a object in a specified reference frame is the force that, when applied to the object, would give it an acceleration equal to the local acceleration of free fall in that reference frame. On the Earth, the local acceleration of free fall is g. If an object is taken to the Moon, the force of gravity exerted on it by the Moon is less than the force of gravity on it when it was on Earth. Thus, its weight on the Moon is less than its weight on Earth, even though its mass remains unchanged.
An astronaut inside an orbiting spacecraft experiences a condition called weightlessness. In orbit, both the spacecraft and the astronaut are in free fall. They are both being accelerated by gravity at the same rate and there is no noticeable attraction between them. The astronaut floats around freely within the spacecraft. If the astronaut were to stand on a scale attached to the spacecraft, the scale would record essentially zero weight. If an object (including the astronaut) is placed at a location a meter or so from the center of mass of the spacecraft, it will actually drift slightly relative to the center of mass. The drift occurs because the object and the spacecraft experience slightly different accelerations due to the Earth's gravity because of the slight difference in radial distance from the Earth and the corresponding slight difference in orbital path. For the astronaut in a low Earth orbit, 300 km (186 mi) above the surface, this relative motion would seem to be due to a very small force. An object displaced a few meters from the center of mass of the spacecraft would have an acceleration relative to the spacecraft of approximately one millionth the value of g at the Earth's surface. Consequently, this condition of near weightlessness is known as microgravity. See also: Free fall; Weightlessness
In monitoring the health of the crew in sustained orbit, the body mass of the astronauts provides important information. On Earth, their weight would be measured using scales. In the microgravity of orbit, their weight is meaningless. Instead, astronauts measure their mass with an inertial balance that consists of a chair that moves relative to the spacecraft with a simple back-and-forth motion similar to that of a porch glider. The mass is computed from measurements of the frequency of the chair's motion.
Mass and energy
For many years it was thought that mass was a conserved quantity; that is, that it can neither be created nor destroyed. During the eighteenth and nineteenth centuries much of the development of chemistry was based on the conservation of mass. Careful measurements of the mass of constituents of chemical reactions and their products showed that the quantity of matter was unchanged by chemical reactions. However, in the twentieth century, observations and measurements in nuclear and particle physics demonstrated conclusively that mass is not a conserved quantity. For example, radioactive polonium-210 decays into lead-206 by the emission of an energetic alpha particle (a helium nucleus). The combined mass of the lead nucleus and the alpha particle is less than the mass of the original polonium nucleus. The difference in the masses appears as the kinetic energy of the alpha and the lead nucleus. This correspondence between mass and energy was shown by Einstein in Eq. (6)
. In the case of the alpha decay of polonium, the loss of mass during the decay results an increase in the kinetic energy of the decay products. See also: Conservation of mass
There are many other examples that demonstrate the nonconservation of mass. One of these is the annihilation process that occurs when matter and antimatter combine. An electron can combine with its antiparticle, a positron, and the two annihilate each other. The two massive particles disappear and a pair of massless photons (the quanta of light) emerge. In a coordinate system in which the electron-positron pair is at rest, the two photons are emitted in opposite directions with identical energies as required by momentum conservation. These two photons carry away the energy that was present in the form of the mass of the electron and the positron. As a result, each photon has an energy equivalent to the mass of a single electron. That energy may be found from Eq. (6). Inserting m = 9.11 × 10−31 kg and c = 2.997 × 108 m/s gives an energy value of 8.19 × 10−14 joules. In units of electronvolts (eV) the photon energy is 5.11 × 105 eV or 0.511 MeV. See also: Antimatter; Electronvolt
The energies released in chemical reactions are typically a fraction of an electronvolt and the resulting mass changes are thus approximately a millionth of the mass of an electron. Because the electron mass is only a small fraction of the atomic mass, the change in mass-energy in chemical reactions is only at the level of parts per billion. The practical result is that on the scale of chemical reactions mass appears to be conserved. At much higher energies, however, the changes in mass-energy become significant. It is the energy release in the fission of heavy elements such as uranium and plutonium that makes nuclear reactors practical sources of energy. See also: Nuclear fission
In the present model for understanding fundamental particles and their interactions, that is, the standard model, the masses of particles arise from the energy of the force fields of their constituents. The masses of the nuclear particles, the protons and neutrons, are thought to arise from the interactions of subnuclear particles called quarks. The mass of a proton, for example, comes at least in part from the energy of the force fields of the quarks.
When a particle is accelerated it gains kinetic energy as its speed and momentum increase. As its speed approaches the speed of light, its momentum increases without limit. Further increases in kinetic energy result in increases in momentum but the speed changes more slowly as it approaches the limiting speed, the speed of light. A change of kinetic energy (or work done on the particle) at high speeds does not produce the same change of speed that would result from the identical increment of energy when the particle is moving at a lower speed. In the early twentieth century, the particle was generally regarded as becoming more massive as its speed approached the speed of light. However, the current practice is to take the view that the mass is invariant, unlike the m used in Eq. (6), and is related to the energy through Eq. (7).
Here, E represents the total energy of the particle of mass m moving with momentum p. As the energy, and hence the speed, of the particle increases, the momentum changes according to Eq. (8).
In this view, the mass m is the invariant mass and is a constant. Then at speeds near the speed of light, changes in energy result in negligible changes in speed even though the changes in momentum are large. See also: Flight; Momentum