Mass

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Edited by Natalie Bhalla, Fall 2016

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

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.

History

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

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.
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