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 different definitions of mass should agree with each other numerically for any given object.

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] is given by

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

When other bodies of mass are involved, active gravitational mass may be defined as the gravitational force that other bodies experience at corresponding distances. For surfaces, active gravitational mass may be more formally defined as the measure of a body's gravitational flux. Qualitatively speaking, this just means active gravitational mass determines how strong a body's gravitational field is. A body's active gravitational mass can be demonstrated by allowing a second, smaller test body to free-fall and then measuring the acceleration that the second body experiences. In classical mechanics, this can formally be shown as

where

g is the gravitational acceleration caused by active gravitational mass's resulting gravitational field

F is the gravitational force on a test body

m is the mass of a test body

r is the direction vector from the body being measured to the test body

t is time

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

Passive Gravitational Mass

Passive gravitational mass is the measure of how affected an body is by a gravitational field. When the sole force acting on a physical body is a result of its interaction with a gravitational field, passive gravitational mass of a body can be calculated by the formula

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

Algebraically solving for m gives:

[math]\displaystyle{ m = \frac{\mathbf{F}}{a} }[/math]

where

• F is the body's weight in the given
• m is the body's passive gravitational mass
• a is the free-fall acceleration of the body.

Combining the Gravitational Masses

The differentiation between active and passive gravitational masses can be bridged by combining the two equations derived above and Newton's Third Law of Motion, which results in the general gravitational force equation

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

or in vector form

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

Rest Energy of Mass

'Main article: Rest Mass Energy The mass-energy equivalence states that there exists an intrinsic energy quantity equivalent for any quantity of mass, even when the body of mass has no other form of energy (no kinetic, potential, elastic, chemical, thermal, or otherwise) and vice versa. This was made famous by Albert Einstein's equation

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

where

[math]\displaystyle{ E_{rest} }[/math] is the rest energy of a body of mass

m is the mass of the body

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

This phenomenon can be observed in many processes, including nuclear fusion (the Sun) and the gravitational bending of light.

Deformation of Spacetime

See also: Einstein's Theory of Special Relativity

A 3-D visualization of the planet Earth deforming spacetime. Credit to NASA for the image.

The deformation of spacetime is a relativistic phenomenon that is the result of the existence of mass.2 The manifestation of the deformation of spacetime can be seen with gravitational time dilation. For example, given two hypothetical, isolated bodies of mass Small and Large where the masses [math]\displaystyle{ M_{Small} \lt \lt M_{Large} }[/math], an observer near Small will observe the passage of time much slower relative to an observer near Large. 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.

Atomic Mass

Atomic mass is the measure of mass for an atom, adding up all of the masses of the protons, neutrons, and electrons in the atom. It is measured in atomic mass units (amu), symbolized by u. 1 amu is equal to the approximate mass of a single neutron or proton, which is 1.660539040×10−27 kg. For example, a carbon-12 atom, comprised of 6 neutrons and 6 protons, has an atomic mass of 12 amu.

Relative Atomic Mass and Standard Atomic Weight

Atomic mass is different from relative atomic mass. Relative atomic mass is the average mass of all isotopes of an element found in a particular sample. It is very useful when dealing with atomic particles not under standard conditions.

Standard atomic weight is a weighted average of the masses of the isotopes found on Earth. These are the numbers commonly seen on periodic tables, that are used in many calculations.

Differentiating between Mass and Weight

In everyday usage, the terms "mass" and "weight" are often interchanged incorrectly. For example, one may state that he or she weighs 100 kg, even though a kilogram is a unit of mass, not weight. Because the majority of humans exist on Earth, where the gravitational field is essentially constant, mass and weight are proportional, so the distinction can be overlooked. However, inconsistencies occur when the gravitational fields are difference. For instance, the mass of a person on both Earth and the Moon will be the same, whereas the weight of a person on Earth and the Moon will be different. This is because weight is actually a measurement of force (typically gravitational) exerted on a body of mass. The equation [math]\displaystyle{ \mathbf{F} = ma }[/math] reappears again to describe weight, where F is an object's weight, m is the object's mass, and a is the body's free-fall acceleration.

Weight

Weight is a measurement of force. On earth, the following equation is used to determine weight.

[math]\displaystyle{ W = mg }[/math]

In this equation, g is the acceleration on earth, approximately 9.8 m/s^2. This equation is used a lot when analyzing projectile motion and forces on earth. However, when dealing with physics problems that happen in space where there is no gravity, the force of weight is not included in calculations.

Calculating Center of Mass

The center of mass is useful when considering the motion of a collection of objects rather than just one.

File:Center of Mass.png

Center of Mass formula

Connectedness

How is this topic connected to something that you are interested in? Although mass is not directly related in something I am interested in, I appreciate the importance of the idea of mass. It is an essential component to consider in almost every physics problem or calculation conducted. Understanding it and understanding how it plays a role in different situations can help one understand what exactly is going on in a certain situation better. For example, we can see that mass does not affect acceleration of a falling object.

F=mg

F=ma

mg=ma

g=a

If you imagine an object with a mass of 2M and an object with a mass of M falling, and imagine the first object as two objects stuck next to each other of mass M, then you can see more clearly how they would fall with the same acceleration.

While this idea is not directly connected to my major, understanding a problem down to its most basic components (like the function of mass in a physics problem) is certainly connected to industrial engineering. Doing so is essential to doing something like root cause analysis, where you need to understand how every detail of every component works to be able to diagnose where exactly the cause of some problem lies. Whether mass is constant, not constant, whether one is looking at mass in components, total mass, or even if one must look at the center of mass, understanding its importance at the theoretical level, not just mathematical or algebraical in an equation, can significantly help solve a problem.

Mass transfer operations are an especially interesting industrial application of the idea of mass. This is used extensively in companies that rely heavily on chemical engineering. Mass transfer is basically about transporting different masses in different forms as various parts of a process. The goal is to do so in the most efficient way possible, which can mean something from getting rid of as much toxic waste as possible to conserving as much liquid as possible and minimizing evaporation.

History

Origin of Mass

A question in many people's minds is "where did mass first come from?" The Higgs Boson is the most accepted explanation as of today. This is a particle that is said to be responsible for giving other particles mass. These particles make up a field that slow other particles down enough to allow them to stick together.

Pre-Newtonian Concepts

The idea about the "amount" of something and its relationship to weight predates recorded history. Humans, at some early prehistoric time, recognized the weight of a group of objects and its direct proportionality to the number of objects in the group. The most direct and widely supported evidence of this is the discovery of weighing scales in early civilization trade. However, there exists no evidence that any of these civilizations recognized the distinction between mass and weight, since the effects of Earth's gravity near the surface ensures that the weight and mass of an object are directly proportional.

See also

• Kinds of Matter
• Gravitational Force
• Inertia
• Rest Mass Energy
• Sir Isaac Newton
• Einstein's Theory of Special Relativity

References

1. W. Rindler (2006). Relativity: Special, General, And Cosmological. Oxford University Press. pp. 16–18. ISBN 0-19-856731-6.
2. A. Einstein, "Relativity : the Special and General Theory by Albert Einstein." Project Gutenberg. <https://www.gutenberg.org/etext/5001.>
3. Emery, Katrina Y. "Mass vs Weight." NASA. NASA, n.d. Web. 27 Nov. 2016.
4. Helmenstein, Anne Marie. "3 Ways To Calculate Atomic Mass." About.com Education. N.p., 02 Dec. 2015. Web. 27 Nov. 2016.
5. "Mass and Weight." Mass, Weight, Density. N.p., n.d. Web. 27 Nov. 2016.
6. "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