Force
Force | |
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Common symbols | , F, F |
kilopond | |
In SI base units | kg·m·s−2 |
Derivations from other quantities | F = ma |
Dimension |
Part of a series on |
Classical mechanics |
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In
Force plays a central role in classical mechanics, figuring in all three of Newton's laws of motion, which specify that the force on an object with an unchanging mass is equal to the product of the object's mass and the acceleration that it undergoes. Types of forces often encountered in classical mechanics include elastic, frictional, contact or "normal" forces, and gravitational. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part often applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In equilibrium these stresses cause no acceleration of the body as the forces balance one another. If these are not in equilibrium they can cause deformation of solid materials, or flow in fluids.
In
Development of the concept
Philosophers in
By the early 20th century,
Pre-Newtonian concepts
Since antiquity the concept of force has been recognized as integral to the functioning of each of the
Though
In the early 17th century, before Newton's
Newtonian mechanics
Sir Isaac Newton described the motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, Philosophiæ Naturalis Principia Mathematica.[1][14] In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day.[14] The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.[15]
First law
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line.[14] The latter follows from the former because of the principle that the laws of physics are the same for all inertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.[16]: 1–7
Second law
According to the first law, motion at constant speed in a straight line does not need a cause. It is change in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion.
A modern statement of Newton's second law is a vector equation:
where is the momentum of the system, and is the net (In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
Third law
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if is the force of body 1 on body 2 and that of body 2 on body 1, then
Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies.[18][19] and thus that there is no such thing as a unidirectional force or a force that acts on only one body.
In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
Combining Newton's Second and Third Laws, it is possible to show that the
Defining "force"
Some textbooks use Newton's second law as a definition of force.[20][21][22][23] However, for the equation for a constant mass to then have any predictive content, it must be combined with further information.[24][4]: 12-1 Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference.[5]: 59 The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways,[25][26]: vii which ultimately do not affect how the theory is used in practice.[25] Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll.[27][28]
Combining forces
Forces act in a particular
Historically, forces were first quantitatively investigated in conditions of
As well as being added, forces can also be resolved into independent components at
Equilibrium
When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium.[17]: 566 Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is in static equilibrium with respect to a frame of reference if it at rest and not accelerating, whereas a body in dynamic equilibrium is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa.[17]: 566
Static
Static equilibrium was understood well before the invention of classical mechanics. Objects that are at rest have zero net force acting on them.[31]
The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration.[1]
Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by
A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as
Dynamic
Dynamic equilibrium was first described by
Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with
Examples of forces in classical mechanics
Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered a rigid body.[citation needed]
Gravitational
What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the
For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward.[4]: ch.12 [5]
Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a
Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an
In this equation, a dimensional constant is used to describe the relative strength of gravity. This constant has come to be known as the
This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of
Electromagnetic
The
Subsequent mathematicians and physicists found the construct of the
The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs.[41] These "Maxwell's equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum.[42]
Normal
When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects.[36]: 264 The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface.[4]: ch.12 [5]
Friction
Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces:
The static friction force () will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the
The kinetic friction force () is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals:
where is the
Tension
Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal pulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object.[43] By connecting the same string multiple times to the same object through the use of a configuration that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. Such machines allow a mechanical advantage for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine.[4]: ch.12 [5][44]
Spring
A simple elastic force acts to return a
Centripetal
For an object in
Continuum mechanics
Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows:
where is the volume of the object in the fluid and is the
A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:
where:- is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
- is the velocity of the object.[4]: ch.12 [5]
More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as
where is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the : 38-1–38-11Fictitious
There are forces that are
In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.[48]
Concepts derived from force
Rotation and torque
Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force is defined relative to an arbitrary reference point as the cross product:
where is theTorque is the rotation equivalent of force in the same way that
- is the moment of inertia of the body
- is the angular acceleration of the body.[17]: 502
This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the
Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:[49]
Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques,
Yank
The yank is defined as the rate of change of force[51]: 131
The term is used in biomechanical analysis,[52] athletic assessment[53] and robotic control.[54] The second (called "tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.[51]
Kinematic integrals
Forces can be used to define a number of physical concepts by
Similarly, integrating with respect to position gives a definition for the work done by a force:[4]: 13-3
which is equivalent to changes inPower P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change in a time interval dt:[4]: 13-2
Potential energy
Instead of a force, often the mathematically related concept of a
Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not.[4]: ch.12 [5]
Conservation
A conservative force that acts on a
Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector emanating from
For gravity:
For electrostatic forces:
For spring forces:
For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of
The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases.[4]: ch.12 [5]
Units
The
The gravitational
The pound-force has a metric counterpart, less commonly used than the newton: the
newton | dyne | kilogram-force, kilopond |
pound-force | poundal | |
---|---|---|---|---|---|
1 N | ≡ 1 kg⋅m/s2 | = 105 dyn | ≈ 0.10197 kp | ≈ 0.22481 lbf | ≈ 7.2330 pdl |
1 dyn | = 10–5 N | ≡ 1 g⋅cm/s2 | ≈ 1.0197×10−6 kp | ≈ 2.2481×10−6 lbf | ≈ 7.2330×10−5 pdl |
1 kp | = 9.80665 N | = 980665 dyn | ≡ gn × 1 kg | ≈ 2.2046 lbf | ≈ 70.932 pdl |
1 lbf | ≈ 4.448222 N | ≈ 444822 dyn | ≈ 0.45359 kp | ≡ gn × 1 lb | ≈ 32.174 pdl |
1 pdl | ≈ 0.138255 N | ≈ 13825 dyn | ≈ 0.014098 kp | ≈ 0.031081 lbf | ≡ 1 lb⋅ft/s2 |
The value of gn as used in the official definition of the kilogram-force (9.80665 m/s2) is used here for all gravitational units. |
- See also Ton-force.
Revisions of the force concept
At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly.
Special theory of relativity
In the
The expression relating force and acceleration for a particle with constant non-zero
The general theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below.
Quantum mechanics
Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object has, one considers what result might appear when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result.[61][62] The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.[63]
In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance.[64][65]
Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution[66] and the Pauli exclusion principle prevents electrons from sharing the same probability distribution.[67] This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability.[68]
Quantum field theory
In modern
While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of
Fundamental interactions
All of the known forces of the universe are classified into four
: 359The fundamental theories for forces developed from the
Property/Interaction | Gravitation | Weak | Electromagnetic | Strong | |
---|---|---|---|---|---|
(Electroweak) | Fundamental | Residual | |||
Acts on: | Mass - Energy | Flavor | Electric charge | Color charge | Atomic nuclei |
Particles experiencing: | All | Quarks, leptons | Electrically charged | Quarks, Gluons | Hadrons |
Particles mediating: | Graviton (not yet observed) |
W+ W− Z0 | γ | Gluons | Mesons |
Strength in the scale of quarks: | 10−41 | 10−4 | 1 | 60 | Not applicable to quarks |
Strength in the scale of protons/neutrons: |
10−36 | 10−7 | 1 | Not applicable to hadrons |
20 |
Gravitational
Newton's law of gravitation is an example of action at a distance: one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance is instantaneous. According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time.
Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in
Electromagnetic
Maxwell's equations and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects.[77] Understanding quantized electromagnetic interactions between elementary particles requires quantum electrodynamics (or QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force.[78]
Strong nuclear
There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The
The strong force is today understood to represent the
Weak nuclear
Unique among the fundamental interactions, the weak nuclear force creates no bound states.
See also
- Contact force – Force between two objects that are in physical contact
- Force control – Force control is given by the machine
- Force gauge – Instrument for measuring force
- Orders of magnitude (force)
- Parallel force system – Situation in mechanical engineering
- Rigid body – Physical object which does not deform when forces or moments are exerted on it
- Specific force – Concept in physics
References
- ^ ISBN 0-201-07199-1.
- ^ Cohen, Michael. "Classical Mechanics: a Critical Introduction" (PDF). University of Pennsylvania. Archived (PDF) from the original on July 3, 2022. Retrieved January 9, 2024.
- ^ a b Heath, Thomas L. (1897). The Works of Archimedes. Retrieved 2007-10-14 – via Internet Archive.
- ^ ISBN 978-0465024933.
- ^ ISBN 978-0521198110.
- ^ ISBN 978-0-679-74408-5.
- ISBN 978-0521624534.
- ISBN 978-0-8153-1085-3.
- OCLC 878730683.
- OCLC 495305340.
- ^ ISBN 0-226-16226-5.
- OCLC 182818133.
- OCLC 16404140.
- ^ ISBN 978-0-520-08817-7. This is a recent translation into English by I. Bernard Cohenand Anne Whitman, with help from Julia Budenz.
- ISBN 978-0387280592.
- ISBN 978-0-691-21877-9.
- ^ ISBN 978-1-947-17220-3.
- S2CID 250891975.
Quoting Newton in the Principia: It is not one action by which the Sun attracts Jupiter, and another by which Jupiter attracts the Sun; but it is one action by which the Sun and Jupiter mutually endeavour to come nearer together.
- ISBN 978-0-471-32057-9.
Any single force is only one aspect of a mutual interaction between two bodies.
- ISBN 978-0-08-003304-4. Translated by: J. B. Sykes, A. D. Petford, and C. L. Petford.LCCN 67--30260. In section 7, pp. 12–14, this book defines force as dp/dt.
- ISBN 1860944248. According to page 12, "[Force] can of course be introduced, by defining it through Newton's second law".
- ISBN 978-0-19-958252-5. According to page 3, "[Newton's second law of motion] can be regarded as defining force".
- OCLC 857769535.
- OCLC 227002144.
- ^ ISBN 0-534-40896-6.
- ^ ISBN 978-0-080-06466-6.
- ISBN 978-0486406893.
- ^ Noll, Walter (April 2007). "On the Concept of Force" (PDF). Carnegie Mellon University. Retrieved 28 October 2013.
- ^ "Introduction to Free Body Diagrams". Physics Tutorial Menu. University of Guelph. Archived from the original on 2008-01-16. Retrieved 2008-01-02.
- ^ Henderson, Tom (2004). "The Physics Classroom". The Physics Classroom and Mathsoft Engineering & Education, Inc. Archived from the original on 2008-01-01. Retrieved 2008-01-02.
- ^ "Static Equilibrium". Physics Static Equilibrium (forces and torques). University of the Virgin Islands. Archived from the original on October 19, 2007. Retrieved 2008-01-02.
- S2CID 4242827.
- ^ a b Young, Hugh; Freedman, Roger; Sears, Francis; and Zemansky, Mark (1949) University Physics. Pearson Education. pp. 59–82.
- ^ Watkins, Thayer. "Perturbation Analysis, Regular and Singular". Department of Economics. San José State University. Archived from the original on 2011-02-10. Retrieved 2008-01-05.
- ^ Kollerstrom, Nick (2001). "Neptune's Discovery. The British Case for Co-Prediction". University College London. Archived from the original on 2005-11-11. Retrieved 2007-03-19.
- ^ ISBN 978-0-471-44895-2.
- ^ Coulomb, Charles (1784). "Recherches théoriques et expérimentales sur la force de torsion et sur l'élasticité des fils de metal". Histoire de l'Académie Royale des Sciences: 229–269.
- ^ ISBN 978-0465024940.
- OCLC 844001.
- ISBN 978-1-947-17221-0.
- ISBN 978-0-471-74064-3.
- ^
Duffin, William (1980). Electricity and Magnetism (3rd ed.). McGraw-Hill. pp. 364–383. ISBN 978-0-07-084111-6.
- ^ "Tension Force". Non-Calculus Based Physics I. Archived from the original on 2007-12-27. Retrieved 2008-01-04.
- ^ Fitzpatrick, Richard (2006-02-02). "Strings, pulleys, and inclines". Retrieved 2008-01-04.
- ^ Nave, Carl Rod. "Elasticity". HyperPhysics. University of Guelph. Retrieved 2013-10-28.
- ^ Nave, Carl Rod. "Centripetal Force". HyperPhysics. University of Guelph. Retrieved 2013-10-28.
- ^ Mallette, Vincent (1982–2008). "The Coriolis Force". Publications in Science and Mathematics, Computing and the Humanities. Inwit Publishing, Inc. Retrieved 2008-01-04.
- OCLC 317496332.
- ^ Nave, Carl Rod. "Newton's 2nd Law: Rotation". HyperPhysics. University of Guelph. Retrieved 2013-10-28.
- ^ Fitzpatrick, Richard (2007-01-07). "Newton's third law of motion". Retrieved 2008-01-04.
- ^ ISBN 978-0-470-39835-7.
- PMID 31515280.
- ISSN 1476-3141.
- .
- ISBN 978-0-13-607791-6.
- ^ Singh, Sunil Kumar (2007-08-25). "Conservative force". Connexions. Retrieved 2008-01-04.
- ^ Davis, Doug. "Conservation of Energy". General physics. Retrieved 2008-01-04.
- ^ ISBN 978-0-7844-0070-8.
- ^ ISBN 978-0-17-771075-9.
- ^ Wilson, John B. "Four-Vectors (4-Vectors) of Special Relativity: A Study of Elegant Physics". The Science Realm: John's Virtual Sci-Tech Universe. Archived from the original on 26 June 2009. Retrieved 2008-01-04.
- S2CID 119546199.
It is a fundamental quantum doctrine that a measurement does not, in general, reveal a pre-existing value of the measured property.
- S2CID 182656563.
- S2CID 126311599.
- ISBN 0-471-16433-X.
- OCLC 28854083.
- ISSN 0034-6861.
the fact that if one tries to compress a wave function anywhere then the kinetic energy will increase. This principle was provided by Sobolev (1938)...
- ^ ISSN 0273-0979.
bulk matter is stable, and has a volume proportional to the number of particles, because of the Pauli exclusion principle for fermions (Le., the electrons). Effectively the electrons behave like a fluid with energy density , and this limits the compression caused by the attractive electrostatic forces.
- ISBN 0131244051.
- ^ ISBN 978-981-02-2639-8.
- ^ "Fermions & Bosons". The Particle Adventure. Archived from the original on 2007-12-18. Retrieved 2008-01-04.
- ^ Jarlskog, Cecilia (1999-10-12). "Additional background material on the Nobel Prize in Physics 1999". Nobel Prize. Retrieved 2023-07-26.
- ^ "Standard model of particles and interactions". Contemporary Physics Education Project. 2000. Archived from the original on 2 January 2017. Retrieved 2 January 2017.
- ^ "Powerful New Black Hole Probe Arrives at Paranal". Retrieved 13 August 2015.
- ^
ISBN 978-0-7167-0344-0.
- ^
OCLC 317496332.
- ^ Siegel, Ethan (20 May 2016). "When Did Isaac Newton Finally Fail?". Forbes. Retrieved 3 January 2017.
- ISBN 978-0-486-43924-2.
- ISBN 978-0-691-14034-6.
- doi:10.1093/OED/1058721983. (Subscription or participating institution membershiprequired.)
- ^ Stevens, Tab (10 July 2003). "Quantum-Chromodynamics: A Definition – Science Articles". Archived from the original on 2011-10-16. Retrieved 2008-01-04.
- ISBN 978-0-691-16759-6.
- ^ ISBN 978-3-540-87842-1.
- ISBN 978-0-521-84704-9.
External links
- "Classical Mechanics, Week 2: Newton's Laws". MIT OpenCourseWare. Retrieved 2023-08-09.
- "Fundamentals of Physics I, Lecture 3: Newton's Laws of Motion". Open Yale Courses. Retrieved 2023-08-09.