Mass
Claimed and edited by Nisha Shah (Spring 2017)
Mass is one of the intrinsic properties 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 other physical science 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). Another common unit also used is the pound (.45 kg or 4.5 N), which is used for both weight and mass.
Defining Mass
One may differentiate between at least seven different aspects of mass, or seven distinct physical approaches to relating mass.1 However, there exists some constant that unifies all widely accepted concepts related to mass. Below are some of these concepts.
Inertial Mass
Main article: Inertia
Inertial mass is the measure of some physical body's resistance to changes in motion (the definition of inertia). A physical body's motional resistance is inversely proportional to its inertial mass. Put more simply, under the same force [math]\displaystyle{ F }[/math], a body with mass [math]\displaystyle{ m }[/math] will experience greater acceleration than that of a body with mass [math]\displaystyle{ M }[/math], when [math]\displaystyle{ m \lt M }[/math]. This is because more force is needed to move more matter due to the Law of Inertia, which states matter's natural resistance to movement.
Gravitational Mass
See also: Gravitational Force
Active Gravitational Mass
Active gravitational mass is the measure of the magnitude of a body's gravitational field at corresponding distances. 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
- [math]\displaystyle{ \mathbf{g}=\frac{\mathbf{F}}{m}=-\frac{{\rm d}^2\mathbf{r}}{{\rm d}t^2}=-Gm\frac{\mathbf{\hat{r}}}{|\mathbf{r}|^2}, }[/math]
- 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
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.
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
- 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.