Mass

Mass is an intrinsic property of physical bodies that exist in 3-dimensional space. Mass is the measurement of the amount of matter a physical body possesses and is an underlying fundamental concept that governs several physical behaviors through concepts such as gravity, inertia, and rest energy.

The SI units for mass are kilograms (kg), a base unit in the International System of Units. Additional SI units utilized for mass are the tonne (1000 kg) and the amu (1.660539040×10−27 kg). In everyday life, units of force such as the pound might also be used to indicate mass because the weight of an object near the surface of the earth is directly proportional to its mass.

Defining Mass

There are many properties which depend on mass, and, accordingly, many ways to measure and define mass.¹ Below are some of these properties and their corresponding definitions. The mass of any given object should be the same regardless of the definition of mass used.

Inertial Mass

Main page: Inertia

The resistance of an object to changes in its motion (its inertia is directly proportional to its mass; that is, the acceleration an object undergoes as a result of a net force acting on it is inversely proportional to its mass. In other words, more massive objects will undergo smaller accelerations than less massive objects acted on by an equal force. The mass of an object can therefore be defined by how difficult it is to accelerate. Mass defined this way is called "inertial mass."

Gravitational Mass

Main page: Gravitational Force

The strength of an object's gravitational interactions with other objects depends on its mass. The strength of the gravitational force [math]\displaystyle{ \mathbf{F}_{grav} }[/math] between two bodies with masses [math]\displaystyle{ m_1 }[/math] and [math]\displaystyle{ m_2 }[/math] is given by

[math]\displaystyle{ |\mathbf{F}_{grav}|= G \frac{m_1 m_2}{r^2} }[/math]

where [math]\displaystyle{ G }[/math] is the universal gravitational constant ([math]\displaystyle{ 6.6740831 \times 10^{-11} {\rm \ N \ m^{2} \ kg^{-2} } }[/math]) and [math]\displaystyle{ r }[/math] is the distance between the bodies.

The equation above shows that the magnitude of the force is proportional to the mass of each body. The mass of an object can therefore be defined by how strongly its gravitational interactions with other objects are. Mass defined this way is called "gravitational mass."

Gravitational mass can be further divided into "active" and "passive" gravitational mass. Active gravitational mass is mass defined by the ability of an object to exert force on other objects (or generate a gravitational field), while passive gravitational mass is mass defined by the ability of an object to experience force as a result of other objects.

Active Gravitational Mass

Active gravitational mass is the measure of a body's ability to exert gravitational force on other bodies, which is synonymous with its ability to generate a gravitational field. The strength of the gravitational field [math]\displaystyle{ \mathbf{g} }[/math] generated by a body of mass [math]\displaystyle{ m_1 }[/math] at a distance [math]\displaystyle{ r }[/math] away is given by

[math]\displaystyle{ |\mathbf{g}|=\frac{Gm_1}{r^2} }[/math]

where [math]\displaystyle{ G }[/math] is the universal gravitational constant ([math]\displaystyle{ 6.6740831 \times 10^{-11} {\rm \ N \ m^{2} \ kg^{-2} } }[/math]).

The strength of a body's gravitational field can be measured either at an arbitrary specific distance or by the flux the field has through a closed surface that encloses the body (which does not depend on the surface's size or shape). Either way, the strength of the body's gravitational field is directly proportional to its mass, so it can be used to measure and define mass. This definition of mass is often used to describe objects that generate significant gravitational fields, such as planets, stars, and galaxies.

Passive Gravitational Mass

Passive gravitational mass is the measure of the force a body experiences in the presence of another body. In other words, it is a measure of how affected an body is by a gravitational field. The strength of the gravitational force [math]\displaystyle{ \mathbf{F} }[/math] experienced by a body with mass [math]\displaystyle{ m_2 }[/math] in the presence of a gravitational field of magnitude [math]\displaystyle{ g }[/math] is given by

[math]\displaystyle{ |\mathbf{F}| = m_2g }[/math].

Because the force experienced by the object is proportional to its mass, it can be used to measure and define mass. This definition of mass is often used to describe objects that exist in the gravitational fields of other objects but are too small to generate significant gravitational fields of their own. In fact, whenever you weigh an object to determine its mass, you are finding its passive gravitational mass because you are finding the force it experiences as a result of the gravitational field of the earth.

Rest Energy of Mass

Main page: Rest Mass Energy

The mass-energy equivalence states that there exists an intrinsic energy quantity equivalent for any quantity of mass and vice versa. That is, all objects have some amount of energy just by virtue of being comprised of matter, even if they have no additional energy of any kind (no kinetic, potential, elastic, chemical, thermal, or other energy). This energy is called rest mass energy. The following famous equation written by Albert Einstein gives the amount of rest mass energy [math]\displaystyle{ E_{rest} }[/math] an object of mass [math]\displaystyle{ m }[/math] possesses:

[math]\displaystyle{ E_{rest} = mc^2 }[/math]

where [math]\displaystyle{ c }[/math] is the speed of light (approximately [math]\displaystyle{ 3.00 \times 10^{8} {\rm \ m/s} }[/math] in a vacuum).

Because the amount of rest mass energy an object possesses is directly proportional to its mass, it can be used to measure and define mass.

Deformation of Spacetime

Main page: Einstein's Theory of Special Relativity

The deformation of spacetime is a relativistic phenomenon that is the result of the existence of mass².

Gravitational time dilation is one way the deformation of spacetime can be observed. According to the idea of gravitational time dilation, time passes more slowly near massive objects. In popular culture, Christopher Nolan's science fiction film Interstellar depicted this phenomenon when astronauts Joe Cooper, Amelia Brand, and Dr. Doyle approach the supermassive black hole Gargantua, while scientist Dr. Romilly remains further from the black hole's spacetime deformation. As a result, in the movie, for every hour the characters Cooper, Brand, and Doyle remain close to the black hole's huge mass and deformation of spacetime, Romilly observes the passage of 23 years of time.

Because the effects of spacetime deformation are proportional to the mass of the body causing it, they can be used to measure and define mass.

Differentiating between Mass and Weight

Main page: Weight

In everyday usage, the terms "mass" and "weight" are often interchanged incorrectly. For example, one may state that he or she weighs 80 kg, even though the kilogram is a unit of mass, not weight. However, mass and weight have different definitions: while mass is a measure of the amount of matter within an object, weight is the magnitude of the gravitational force acting on it. Near the surface of the earth, the magnitude of the earth's gravitational field is nearly constant, so the weight of an object is proportional to its mass (meaning every weight corresponds to a specific mass and vice versa). Because we humans and our common everyday objects exist on the surface of the earth, the distinction between mass and weight can be overlooked in everyday life. However, it becomes important to differentiate between the two properties when objects in differing gravitational fields are compared. For example, an object on the surface of the moon would weigh less than an object of the same mass on the surface of the earth.

Calculating Center of Mass

Main page: Center of Mass

The center of mass of a system is a point in space that represents the average position of all of the matter in that system. The center of mass of a system is a useful quantity for several reasons, such as for modeling systems as point particles or for determining the axis of rotation of a free-floating body.

Atomic Mass

The atomic mass of an atom is its mass, which is the sum of the masses of its constituent protons, nucleons, and electrons. It is typically measured in atomic mass units (amu). 1 amu is defined as 1/12 the weight of a carbon-12 atom, which is 1.660539040×10−27 kg. The mass each proton and neutron (together referred to as "nucleons") is about 1 amu, while the mass of each electron is negligible and therefore considered 0 amu. The atomic mass of an atom depends on which element it is (elements with larger atomic numbers generally have larger atomic masses) and which isotope of that element is (different isotopes have different numbers of neutrons, affecting mass but not chemical properties).

Average Atomic Mass

The average atomic mass of an element is the average atomic mass of its different isotopes weighted by the relative abundance of those isotopes on earth. These are the atomic mass values that appear on periodic tables. They are often used to convert samples of an element between moles and mass because isotope ratios in a typical sample of most elements reflect their relative abundances on earth. This means that in a natural sample of an element, one can treat every atom as though it has the average atomic mass of that element for the sake of converting between moles and mass.

Connectedness

In addition to being a value of interest in and of itself, properties such as Linear Momentum, Angular Momentum, Moments of Inertia, Gravitational Force, and Kinetic Energy depend on the masses of objects.

The use of mass in calculations has a wide range of industrial applications including measuring the quantity of a substance, determining the energy necessary to move an object, and calculating the inertia of moving machines such as vehicles to determine adequate braking force.

History

A basic understanding of the idea of mass was commonplace well before the common era, as evidenced by the use of scales to measure quantities of substances such as grain. The invention and use of the scale required knowledge that the weight of a sample of a substance is directly proportional to the amount of that substance⁷. The active gravitational properties of mass were investigated in the 17th century by Galileo Galilei, Robert Hooke, and Isaac Newton, who discovered that the gravitational force between two objects was inversely proportional to the square of the distance between them. Around the same time, Ernst Mach and Newton discovered the direct relationship between mass and inertia. The role of mass in relativity was discovered by Albert Einstein in the early 20th century. In 1964, Peter Higgs and his lab proposed that a particle called the Higgs boson endows particles with mass through a quantum interaction, an idea that was supported by observational results generated by the Large Hadron Collider in 2013⁸.

References

W. Rindler (2006). Relativity: Special, General, And Cosmological. Oxford University Press. pp. 16–18. ISBN 0-19-856731-6.
A. Einstein, "Relativity : the Special and General Theory by Albert Einstein." Project Gutenberg. <https://www.gutenberg.org/etext/5001.>
Emery, Katrina Y. "Mass vs Weight." NASA. NASA, n.d. Web. 27 Nov. 2016.
Helmenstein, Anne Marie. "3 Ways To Calculate Atomic Mass." About.com Education. N.p., 02 Dec. 2015. Web. 27 Nov. 2016.
"Mass and Weight." Mass, Weight, Density. N.p., n.d. Web. 27 Nov. 2016.
"The Motion of the Center of Mass." 183_notes:center_of_mass [Projects & Practices in Physics]. (2015, September 27). Retrieved April 09, 2017, from http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes%3Acenter_of_mass
https://en.wikipedia.org/wiki/Mass#Pre-Newtonian_concepts
https://en.wikipedia.org/wiki/Higgs_mechanism