A force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N) , and force is often represented by the symbol F .
226-434: Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion . 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
452-418: A Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong , electromagnetic , weak , and gravitational . High-energy particle physics observations made during
678-432: A differential equation for S {\displaystyle S} . Bodies move over time in such a way that their trajectories are perpendicular to the surfaces of constant S {\displaystyle S} , analogously to how a light ray propagates in the direction perpendicular to its wavefront. This is simplest to express for the case of a single point mass, in which S {\displaystyle S}
904-469: A 2-dimensional harmonic oscillator. However it is solved, the result is that orbits will be conic sections , that is, ellipses (including circles), parabolas , or hyperbolas . The eccentricity of the orbit, and thus the type of conic section, is determined by the energy and the angular momentum of the orbiting body. Planets do not have sufficient energy to escape the Sun, and so their orbits are ellipses, to
1130-529: A Lagrangian for a multi-particle system, and so, Newton's third law is a theorem rather than an assumption. In Hamiltonian mechanics , the dynamics of a system are represented by a function called the Hamiltonian, which in many cases of interest is equal to the total energy of the system. The Hamiltonian is a function of the positions and the momenta of all the bodies making up the system, and it may also depend explicitly upon time. The time derivatives of
1356-443: A body add as vectors , and so the total force on a body depends upon both the magnitudes and the directions of the individual forces. When the net force on a body is equal to zero, then by Newton's second law, the body does not accelerate, and it is said to be in mechanical equilibrium . A state of mechanical equilibrium is stable if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise,
1582-411: A body moving in a circle of radius r {\displaystyle r} at a constant speed v {\displaystyle v} , its acceleration has a magnitude a = v 2 r {\displaystyle a={\frac {v^{2}}{r}}} and is directed toward the center of the circle. The force required to sustain this acceleration, called the centripetal force ,
1808-456: A constant mass m {\displaystyle m} to then have any predictive content, it must be combined with further information. Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference. 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, which ultimately do not affect how
2034-443: A constant speed in a straight line. This applies, for example, to a collision between two bodies. If the total external force is not zero, then the center of mass changes velocity as though it were a point body of mass M {\displaystyle M} . This follows from the fact that the internal forces within the collection, the forces that the objects exert upon each other, occur in balanced pairs by Newton's third law. In
2260-438: A continuum out of discrete or indivisible points or moments). Among other things, this implies that there can be no definite (indivisible) moment when a motion begins. This discussion, together with that of speed and the different behavior of the four different species of motion, eventually helps Aristotle answer the famous paradoxes of Zeno , which purport to show the absurdity of motion's existence. Book VII briefly deals with
2486-578: A different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result . If both of these pieces of information are not known for each force, the situation is ambiguous. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate
SECTION 10
#17327661451772712-479: A force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion. Some forces are consequences of
2938-458: 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: F 1 , 2 + F 2 , 1 = 0. {\displaystyle \mathbf {F} _{1,2}+\mathbf {F} _{2,1}=0.} More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but
3164-471: A force that existed intrinsically between two charges . The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction , was both attractive and repulsive (there was intrinsic polarity ), was independent of the mass of the charged objects, and followed the superposition principle . Coulomb's law unifies all these observations into one succinct statement. Subsequent mathematicians and physicists found
3390-579: 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. Static equilibrium was understood well before the invention of classical mechanics. Objects that are not accelerating have zero net force acting on them. The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on
3616-460: A function S ( q 1 , q 2 , … , t ) {\displaystyle S(\mathbf {q} _{1},\mathbf {q} _{2},\ldots ,t)} of positions q i {\displaystyle \mathbf {q} _{i}} and time t {\displaystyle t} . The Hamiltonian is incorporated into the Hamilton–Jacobi equation,
3842-484: A good approximation; because the planets pull on one another, actual orbits are not exactly conic sections. If a third mass is added, the Kepler problem becomes the three-body problem, which in general has no exact solution in closed form . That is, there is no way to start from the differential equations implied by Newton's laws and, after a finite sequence of standard mathematical operations, obtain equations that express
4068-457: A key principle of Newtonian physics. In the early 17th century, before Newton's Principia , the term "force" ( Latin : vis ) was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named vis viva (live force) by Leibniz . The modern concept of force corresponds to Newton's vis motrix (accelerating force). Sir Isaac Newton described
4294-482: 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. Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because
4520-555: A magnetic field experiences a force that is also proportional to its charge, in a direction perpendicular to both the field and the body's direction of motion. Using the vector cross product , F = q E + q v × B . {\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} .} Physics (Aristotle) The Physics ( Greek : Φυσικὴ ἀκρόασις Phusike akroasis ; Latin : Physica , or Naturales Auscultationes , possibly meaning " Lectures on nature ")
4746-410: A mechanics textbook that does not involve friction can be expressed in this way. The fact that the force can be written in this way can be understood from the conservation of energy . Without friction to dissipate a body's energy into heat, the body's energy will trade between potential and (non-thermal) kinetic forms while the total amount remains constant. Any gain of kinetic energy, which occurs when
SECTION 20
#17327661451774972-424: A person standing on the ground watching a train go past is an inertial observer. If the observer on the ground sees the train moving smoothly in a straight line at a constant speed, then a passenger sitting on the train will also be an inertial observer: the train passenger feels no motion. The principle expressed by Newton's first law is that there is no way to say which inertial observer is "really" moving and which
5198-484: A point mass is − ∂ S ∂ t = H ( q , ∇ S , t ) . {\displaystyle -{\frac {\partial S}{\partial t}}=H\left(\mathbf {q} ,\mathbf {\nabla } S,t\right).} The relation to Newton's laws can be seen by considering a point mass moving in a time-independent potential V ( q ) {\displaystyle V(\mathbf {q} )} , in which case
5424-446: A point, moving along some trajectory, and returning to the initial point — is zero. If this is the case, then the force can be written in terms of the gradient of a function called a scalar potential : F = − ∇ U . {\displaystyle \mathbf {F} =-\mathbf {\nabla } U\,.} This is true for many forces including that of gravity, but not for friction; indeed, almost any problem in
5650-405: A research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter. Overly brief paraphrases of the third law, like "action equals reaction " might have caused confusion among generations of students: the "action" and "reaction" apply to different bodies. For example, consider a book at rest on
5876-464: A selection of one continuous text, but typically gives notes stating the alternative sections of text. Determining which text is to be presented as "original" is a detailed scholarly investigation. The recension is often known by its scholarly editor's name. In reverse chronological order: A commentary differs from a note in being a distinct work analyzing the language and subsumed concepts of some other work classically notable. A note appears within
6102-414: A single number, indicating where it is relative to some chosen reference point. For example, a body might be free to slide along a track that runs left to right, and so its location can be specified by its distance from a convenient zero point, or origin , with negative numbers indicating positions to the left and positive numbers indicating positions to the right. If the body's location as a function of time
6328-450: A situation, Newton's laws can be applied to the individual pieces of matter, keeping track of which pieces belong to the object of interest over time. For instance, if a rocket of mass M ( t ) {\displaystyle M(t)} , moving at velocity v ( t ) {\displaystyle \mathbf {v} (t)} , ejects matter at a velocity u {\displaystyle \mathbf {u} } relative to
6554-476: A source of activities is more typically to the genera of natural kinds (the secondary substance ). But, contra Plato , Aristotle attempts to resolve a philosophical quandary that was well understood in the fourth century. The Eudoxian planetary model sufficed for the wandering stars , but no deduction of terrestrial substance would be forthcoming based solely on the mechanical principles of necessity, (ascribed by Aristotle to material causation in chapter 9). In
6780-430: A straight line will see it continuing to do so. 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. Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time . If
7006-745: A surface up to the limit specified by the coefficient of static friction ( μ s f {\displaystyle \mu _{\mathrm {sf} }} ) multiplied by the normal force ( F N {\displaystyle \mathbf {F} _{\text{N}}} ). In other words, the magnitude of the static friction force satisfies the inequality: 0 ≤ F s f ≤ μ s f F N . {\displaystyle 0\leq \mathbf {F} _{\mathrm {sf} }\leq \mu _{\mathrm {sf} }\mathbf {F} _{\mathrm {N} }.} The kinetic friction force ( F k f {\displaystyle F_{\mathrm {kf} }} )
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7232-482: A system can lead to the whole system behaving in a radically different way within a short time. Noteworthy examples include the three-body problem, the double pendulum , dynamical billiards , and the Fermi–Pasta–Ulam–Tsingou problem . Newton's laws can be applied to fluids by considering a fluid as composed of infinitesimal pieces, each exerting forces upon neighboring pieces. The Euler momentum equation
7458-410: A system of two bodies with one much more massive than the other, the center of mass will approximately coincide with the location of the more massive body. When Newton's laws are applied to rotating extended bodies, they lead to new quantities that are analogous to those invoked in the original laws. The analogue of mass is the moment of inertia , the counterpart of momentum is angular momentum , and
7684-406: A system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained. Some textbooks use Newton's second law as a definition of force. However, for the equation F = m a {\displaystyle \mathbf {F} =m\mathbf {a} } for
7910-471: A table. The Earth's gravity pulls down upon the book. The "reaction" to that "action" is not the support force from the table holding up the book, but the gravitational pull of the book acting on the Earth. Newton's third law relates to a more fundamental principle, the conservation of momentum . The latter remains true even in cases where Newton's statement does not, for instance when force fields as well as material bodies carry momentum, and when momentum
8136-575: Is s ( t ) {\displaystyle s(t)} , then its average velocity over the time interval from t 0 {\displaystyle t_{0}} to t 1 {\displaystyle t_{1}} is Δ s Δ t = s ( t 1 ) − s ( t 0 ) t 1 − t 0 . {\displaystyle {\frac {\Delta s}{\Delta t}}={\frac {s(t_{1})-s(t_{0})}{t_{1}-t_{0}}}.} Here,
8362-476: Is drag force , ∝ ρ v 2 {\displaystyle \propto \rho v^{2}} . In this case, the terminal velocity is C ( W ρ ) 1 / 2 {\displaystyle C\left({\frac {W}{\rho }}\right)^{1/2}} A recension is a selection of a specific text for publication. The manuscripts on a given work attributed to Aristotle offer textual variants. One recension makes
8588-412: Is looped to calculate, approximately, the bodies' trajectories. Generally speaking, the shorter the time interval, the more accurate the approximation. Newton's laws of motion allow the possibility of chaos . That is, qualitatively speaking, physical systems obeying Newton's laws can exhibit sensitive dependence upon their initial conditions: a slight change of the position or velocity of one part of
8814-418: Is "really" standing still. One observer's state of rest is another observer's state of uniform motion in a straight line, and no experiment can deem either point of view to be correct or incorrect. There is no absolute standard of rest. Newton himself believed that absolute space and time existed, but that the only measures of space or time accessible to experiment are relative. By "motion", Newton meant
9040-404: Is a force that varies randomly from instant to instant, representing the net effect of collisions with the surrounding particles. This is used to model Brownian motion . Newton's three laws can be applied to phenomena involving electricity and magnetism , though subtleties and caveats exist. Coulomb's law for the electric force between two stationary, electrically charged bodies has much
9266-520: Is a function S ( q , t ) {\displaystyle S(\mathbf {q} ,t)} , and the point mass moves in the direction along which S {\displaystyle S} changes most steeply. In other words, the momentum of the point mass is the gradient of S {\displaystyle S} : v = 1 m ∇ S . {\displaystyle \mathbf {v} ={\frac {1}{m}}\mathbf {\nabla } S.} The Hamilton–Jacobi equation for
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9492-497: Is a named text, written in ancient Greek, collated from a collection of surviving manuscripts known as the Corpus Aristotelicum , attributed to the 4th-century BC philosopher Aristotle . It is a collection of treatises or lessons that deals with the most general (philosophical) principles of natural or moving things, both living and non-living, rather than physical theories (in the modern sense) or investigations of
9718-421: Is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at
9944-402: Is an expression of Newton's second law adapted to fluid dynamics. A fluid is described by a velocity field, i.e., a function v ( x , t ) {\displaystyle \mathbf {v} (\mathbf {x} ,t)} that assigns a velocity vector to each point in space and time. A small object being carried along by the fluid flow can change velocity for two reasons: first, because
10170-959: Is another re-expression of Newton's second law. The expression in brackets is a total or material derivative as mentioned above, in which the first term indicates how the function being differentiated changes over time at a fixed location, and the second term captures how a moving particle will see different values of that function as it travels from place to place: [ ∂ ∂ t + 1 m ( ∇ S ⋅ ∇ ) ] = [ ∂ ∂ t + v ⋅ ∇ ] = d d t . {\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]=\left[{\frac {\partial }{\partial t}}+\mathbf {v} \cdot \mathbf {\nabla } \right]={\frac {d}{dt}}.} In statistical physics ,
10396-417: Is based on the idea of specifying positions using numerical coordinates. Movement is represented by these numbers changing over time: a body's trajectory is represented by a function that assigns to each value of a time variable the values of all the position coordinates. The simplest case is one-dimensional, that is, when a body is constrained to move only along a straight line. Its position can then be given by
10622-466: Is constant. Alternatively, if p {\displaystyle \mathbf {p} } is known to be constant, it follows that the forces have equal magnitude and opposite direction. Various sources have proposed elevating other ideas used in classical mechanics to the status of Newton's laws. For example, in Newtonian mechanics, the total mass of a body made by bringing together two smaller bodies
10848-471: Is defined properly, in quantum mechanics as well. In Newtonian mechanics, if two bodies have momenta p 1 {\displaystyle \mathbf {p} _{1}} and p 2 {\displaystyle \mathbf {p} _{2}} respectively, then the total momentum of the pair is p = p 1 + p 2 {\displaystyle \mathbf {p} =\mathbf {p} _{1}+\mathbf {p} _{2}} , and
11074-408: Is different from all the others in lacking matter; being pure form, it is also in an eternal actuality, not being imperfect in any respect; hence needing not to move. This is demonstrated by describing the celestial bodies thus: the first things to be moved must undergo an infinite, single and continuous movement, that is, circular. This is not caused by any contact but (integrating the view contained in
11300-521: Is different from what we mean today by this word, not only to the extent that it belongs to antiquity whereas the modern physical sciences belong to modernity, rather above all it is different by virtue of the fact that Aristotle's "physics" is philosophy, whereas modern physics is a positive science that presupposes a philosophy.... This book determines the warp and woof of the whole of Western thinking, even at that place where it, as modern thinking, appears to think at odds with ancient thinking. But opposition
11526-414: Is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of
SECTION 50
#173276614517711752-413: Is equal in magnitude to the force that q 2 {\displaystyle q_{2}} exerts upon q 1 {\displaystyle q_{1}} , and it points in the exact opposite direction. Coulomb's law is thus consistent with Newton's third law. Electromagnetism treats forces as produced by fields acting upon charges. The Lorentz force law provides an expression for
11978-591: Is inappropriate to say that something properly becomes, from not-man, man: generation and corruption are not kinesis in the full sense. Book VI discusses how a changing thing can reach the opposite state, if it has to pass through infinite intermediate stages. It investigates by rational and logical arguments the notions of continuity and division , establishing that change—and, consequently, time and place—are not divisible into indivisible parts; they are not mathematically discrete but continuous, that is, infinitely divisible (in other words, that you cannot build up
12204-479: Is invariably comprised of a decisive, and often even perilous, dependence. Without Aristotle's Physics there would have been no Galileo . Bertrand Russell says of Physics and On the Heavens (which he believed was a continuation of Physics ) that they were: ...extremely influential, and dominated science until the time of Galileo ... The historian of philosophy, accordingly, must study them, in spite of
12430-608: Is its angle from the vertical. When the angle θ {\displaystyle \theta } is small, the sine of θ {\displaystyle \theta } is nearly equal to θ {\displaystyle \theta } (see Taylor series ), and so this expression simplifies to the equation for a simple harmonic oscillator with frequency ω = g / L {\displaystyle \omega ={\sqrt {g/L}}} . A harmonic oscillator can be damped, often by friction or viscous drag, in which case energy bleeds out of
12656-477: Is just this—the primary substratum of each thing, from which it comes to be without qualification, and which persists in the result." Matter in Aristotle's thought is, however, defined in terms of sensible reality; for example, a horse eats grass: the horse changes the grass into itself; the grass as such does not persist in the horse, but some aspect of it – its matter – does. Matter
12882-401: Is known as free fall . The speed attained during free fall is proportional to the elapsed time, and the distance traveled is proportional to the square of the elapsed time. Importantly, the acceleration is the same for all bodies, independently of their mass. This follows from combining Newton's second law of motion with his law of universal gravitation . The latter states that the magnitude of
13108-408: Is made of (for example, the wood of a house), formal cause explains the form which a thing follows to become that thing (the plans of an architect to build a house), efficient cause is the actual source of the change (the physical building of the house), and final cause is the intended purpose of the change (the final product of the house and its purpose as a shelter and home). Of particular importance
13334-463: Is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each. For instance, the Earth and the Sun can both be approximated as pointlike when considering the orbit of the former around the latter, but the Earth is not pointlike when considering activities on its surface. The mathematical description of motion, or kinematics ,
13560-717: Is not specifically described, but consists of whatever is apart from quality or quantity and that of which something may be predicated. Matter in this understanding does not exist independently (i.e. as a substance ), but exists interdependently (i.e. as a "principle") with form and only insofar as it underlies change. Matter and form are analogical terms. Book II identifies "nature" ( physis ) as "a source or cause of being moved and of being at rest in that to which it belongs primarily" (1.192b21). Thus, those entities are natural which are capable of starting to move, e.g. growing, acquiring qualities, displacing themselves, and finally being born and dying. Aristotle contrasts natural things with
13786-467: Is not the same as power or pressure , for example, and mass has a different meaning than weight . The physics concept of force makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity. Translated from Latin, Newton's first law reads, Newton's first law expresses
SECTION 60
#173276614517714012-450: Is not what has nothing outside it that is infinite, but what always has something outside it" (6.206b33-207a1-2). Book IV discusses the preconditions of motion: place ( topos , chapters 1-5), void ( kenon , chapters 6-9), and time ( khronos , chapters 10-14). The book starts by distinguishing the various ways a thing can "be in" another. He likens place to an immobile container or vessel: "the innermost motionless boundary of what contains"
14238-473: Is some function of the position, V ( q ) {\displaystyle V(q)} . The physical path that the particle will take between an initial point q i {\displaystyle q_{i}} and a final point q f {\displaystyle q_{f}} is the path for which the integral of the Lagrangian is "stationary". That is, the physical path has
14464-639: Is sometimes presented as a definition of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. In order for this to be more than a tautology — acceleration implies force, force implies acceleration — some other statement about force must also be made. For example, an equation detailing the force might be specified, like Newton's law of universal gravitation . By inserting such an expression for F {\displaystyle \mathbf {F} } into Newton's second law, an equation with predictive power can be written. Newton's second law has also been regarded as setting out
14690-480: Is taken from sea level and may vary depending on location), and points toward the center of the Earth. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of m {\displaystyle m} will experience a force: F = m g . {\displaystyle \mathbf {F} =m\mathbf {g} .} For an object in free-fall, this force
14916-396: Is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position. This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If Δ x {\displaystyle \Delta x}
15142-513: Is the coefficient of kinetic friction . The coefficient of kinetic friction is normally less than the coefficient of static friction. 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
15368-480: Is the final cause or purpose ( telos ). It is a common mistake to conceive of the four causes as additive or alternative forces pushing or pulling; in reality, all four are needed to explain (7.198a22-25). What we typically mean by cause in the modern scientific idiom is only a narrow part of what Aristotle means by efficient cause. He contrasts purpose with the way in which "nature" does not work, chance (or luck), discussed in chapters 4, 5, and 6. (Chance working in
15594-401: Is the kinematic viscosity . It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time. This unphysical behavior, known as a "noncollision singularity", depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as
15820-430: Is the mass and v {\displaystyle \mathbf {v} } is the velocity . If Newton's second law is applied to a system of constant mass , m may be moved outside the derivative operator. The equation then becomes F = m d v d t . {\displaystyle \mathbf {F} =m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}.} By substituting
16046-403: Is the density of the surrounding fluid (such as air, fire, or water), n > 0 {\displaystyle n>0} is a constant, and C {\displaystyle C} is a constant depending on the shape of the object. This is correct for the terminal velocity of falling objects in fluid in a constant gravitational field, in the case where most of the fluid resistance
16272-864: Is the density, P {\displaystyle P} is the pressure, and f {\displaystyle \mathbf {f} } stands for an external influence like a gravitational pull. Incorporating the effect of viscosity turns the Euler equation into a Navier–Stokes equation : ∂ v ∂ t + ( ∇ ⋅ v ) v = − 1 ρ ∇ P + ν ∇ 2 v + f , {\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\nu \nabla ^{2}\mathbf {v} +\mathbf {f} ,} where ν {\displaystyle \nu }
16498-410: Is the displacement, the force exerted by an ideal spring equals: F = − k Δ x , {\displaystyle \mathbf {F} =-k\Delta \mathbf {x} ,} where k {\displaystyle k} is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to
16724-409: Is the distance between the two objects' centers of mass and r ^ {\displaystyle {\hat {\mathbf {r} }}} is the unit vector pointed in the direction away from the center of the first object toward the center of the second object. This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until
16950-596: Is the distance from the center of the Earth to the body's location, which is very nearly the radius of the Earth. Setting this equal to m a {\displaystyle ma} , the body's mass m {\displaystyle m} cancels from both sides of the equation, leaving an acceleration that depends upon G {\displaystyle G} , M {\displaystyle M} , and r {\displaystyle r} , and r {\displaystyle r} can be taken to be constant. This particular value of acceleration
17176-473: Is the electromagnetic force, E {\displaystyle \mathbf {E} } is the electric field at the body's location, B {\displaystyle \mathbf {B} } is the magnetic field, and v {\displaystyle \mathbf {v} } is the velocity of the particle. The magnetic contribution to the Lorentz force is the cross product of the velocity vector with
17402-488: Is the force of body 1 on body 2 and F 2 , 1 {\displaystyle \mathbf {F} _{2,1}} that of body 2 on body 1, then F 1 , 2 = − F 2 , 1 . {\displaystyle \mathbf {F} _{1,2}=-\mathbf {F} _{2,1}.} This law is sometimes referred to as the action-reaction law , with F 1 , 2 {\displaystyle \mathbf {F} _{1,2}} called
17628-581: Is the magnitude of the hypothetical test charge. Similarly, the idea of the magnetic field was introduced to express how magnets can influence one another at a distance. The Lorentz force law gives the force upon a body with charge q {\displaystyle q} due to electric and magnetic fields: F = q ( E + v × B ) , {\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),} where F {\displaystyle \mathbf {F} }
17854-424: Is the mass of the larger body being orbited. Therefore, the mass of a body can be calculated from observations of another body orbiting around it. Newton's cannonball is a thought experiment that interpolates between projectile motion and uniform circular motion. A cannonball that is lobbed weakly off the edge of a tall cliff will hit the ground in the same amount of time as if it were dropped from rest, because
18080-591: Is the momentum of object 1 and p 2 {\displaystyle \mathbf {p} _{2}} the momentum of object 2, then d p 1 d t + d p 2 d t = F 1 , 2 + F 2 , 1 = 0. {\displaystyle {\frac {\mathrm {d} \mathbf {p} _{1}}{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {p} _{2}}{\mathrm {d} t}}=\mathbf {F} _{1,2}+\mathbf {F} _{2,1}=0.} Using similar arguments, this can be generalized to
18306-435: Is the momentum of the system, and F {\displaystyle \mathbf {F} } is the net ( vector sum ) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time. In common engineering applications
18532-598: Is the primary place of a body (4.212a20). Unlike space, which is a volume co-existent with a body, place is a boundary or surface. He teaches that, contrary to the Atomists and others, a void is not only unnecessary, but leads to contradictions, e.g., making locomotion impossible. Time is a constant attribute of movements and, Aristotle thinks, does not exist on its own but is relative to the motions of things. Tony Roark describes Aristotle's view of time as follows: Aristotle defines time as "a number of motion with respect to
18758-413: Is the sum of their individual masses. Frank Wilczek has suggested calling attention to this assumption by designating it "Newton's Zeroth Law". Another candidate for a "zeroth law" is the fact that at any instant, a body reacts to the forces applied to it at that instant. Likewise, the idea that forces add like vectors (or in other words obey the superposition principle ), and the idea that forces change
18984-449: Is the velocity of the object and r {\displaystyle r} is the distance to the center of the circular path and r ^ {\displaystyle {\hat {\mathbf {r} }}} is the unit vector pointing in the radial direction outwards from the center. This means that the net force felt by the object is always directed toward the center of the curving path. Such forces act perpendicular to
19210-482: Is the volume of the object in the fluid and P {\displaystyle P} is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science , and the lift associated with aerodynamics and flight . Newton%27s laws of motion Newton's laws of motion are three physical laws that describe
19436-563: Is therefore also directed toward the center of the circle and has magnitude m v 2 / r {\displaystyle mv^{2}/r} . Many orbits , such as that of the Moon around the Earth, can be approximated by uniform circular motion. In such cases, the centripetal force is gravity, and by Newton's law of universal gravitation has magnitude G M m / r 2 {\displaystyle GMm/r^{2}} , where M {\displaystyle M}
19662-499: Is to velocity as velocity is to position: it is the derivative of the velocity with respect to time. Acceleration can likewise be defined as a limit: a = d v d t = lim Δ t → 0 v ( t + Δ t ) − v ( t ) Δ t . {\displaystyle a={\frac {dv}{dt}}=\lim _{\Delta t\to 0}{\frac {v(t+\Delta t)-v(t)}{\Delta t}}.} Consequently,
19888-549: Is typically denoted g {\displaystyle g} : g = G M r 2 ≈ 9.8 m / s 2 . {\displaystyle g={\frac {GM}{r^{2}}}\approx \mathrm {9.8~m/s^{2}} .} If the body is not released from rest but instead launched upwards and/or horizontally with nonzero velocity, then free fall becomes projectile motion . When air resistance can be neglected, projectiles follow parabola -shaped trajectories, because gravity affects
20114-437: Is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals: F k f = μ k f F N , {\displaystyle \mathbf {F} _{\mathrm {kf} }=\mu _{\mathrm {kf} }\mathbf {F} _{\mathrm {N} },} where μ k f {\displaystyle \mu _{\mathrm {kf} }}
20340-439: 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. Newton's contribution to gravitational theory
20566-519: Is useful when calculating the motion of constrained bodies, like a mass restricted to move along a curving track or on the surface of a sphere. Hamiltonian mechanics is convenient for statistical physics , leads to further insight about symmetry, and can be developed into sophisticated techniques for perturbation theory . Due to the breadth of these topics, the discussion here will be confined to concise treatments of how they reformulate Newton's laws of motion. Lagrangian mechanics differs from
20792-415: The x {\displaystyle x} axis, and suppose an equilibrium point exists at the position x = 0 {\displaystyle x=0} . That is, at x = 0 {\displaystyle x=0} , the net force upon the body is the zero vector, and by Newton's second law, the body will not accelerate. If the force upon the body is proportional to the displacement from
21018-637: The Metaphysics , bk. XII ) by love and aspiration. The works of Aristotle are typically influential to the development of Western science and philosophy . The citations below are not given as any sort of final modern judgement on the interpretation and significance of Aristotle, but are only the notable views of some moderns. Martin Heidegger writes: The Physics is a lecture in which he seeks to determine beings that arise on their own, τὰ φύσει ὄντα, with regard to their being. Aristotelian "physics"
21244-469: The Aristotelian theory of motion . He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction . Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon by Isaac Beeckman , René Descartes , and Pierre Gassendi , became
21470-424: The action and − F 2 , 1 {\displaystyle -\mathbf {F} _{2,1}} the reaction . 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. and thus that there is no such thing as a unidirectional force or
21696-473: The center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system. Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved in any closed system . In a system of two particles, if p 1 {\displaystyle \mathbf {p} _{1}}
21922-468: The kinetic theory of gases applies Newton's laws of motion to large numbers (typically on the order of the Avogadro number ) of particles. Kinetic theory can explain, for example, the pressure that a gas exerts upon the container holding it as the aggregate of many impacts of atoms, each imparting a tiny amount of momentum. The Langevin equation is a special case of Newton's second law, adapted for
22148-399: The partial derivatives of the Lagrangian gives d d t ( m q ˙ ) = − d V d q , {\displaystyle {\frac {d}{dt}}(m{\dot {q}})=-{\frac {dV}{dq}},} which is a restatement of Newton's second law. The left-hand side is the time derivative of the momentum, and the right-hand side is
22374-410: The "Newtonian" description (which itself, of course, incorporates contributions from others both before and after Newton). The physical content of these different formulations is the same as the Newtonian, but they provide different insights and facilitate different types of calculations. For example, Lagrangian mechanics helps make apparent the connection between symmetries and conservation laws, and it
22600-408: The 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines . The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for
22826-423: The 20th century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet , moon , comet , or asteroid . The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed. The electrostatic force was first described in 1784 by Coulomb as
23052-416: The 4 elements could explain most of the rising and falling motions of objects with different densities. The velocity of falling objects is equal to C ( W ρ ) n {\displaystyle C\left({\frac {W}{\rho }}\right)^{n}} , where W {\displaystyle W} is the weight of the object, ρ {\displaystyle \rho }
23278-463: The Enlightenment , centuries before modern science made good on atomist intuitions , a nominal allegiance to mechanistic materialism gained popularity despite harboring Newton's action at distance , and comprising the native habitat of teleological arguments : Machines or artifacts composed of parts lacking any intrinsic relationship to each other with their order imposed from without. Thus,
23504-463: The Greek letter Δ {\displaystyle \Delta } ( delta ) is used, per tradition, to mean "change in". A positive average velocity means that the position coordinate s {\displaystyle s} increases over the interval in question, a negative average velocity indicates a net decrease over that interval, and an average velocity of zero means that the body ends
23730-820: The Hamilton–Jacobi equation becomes − ∂ S ∂ t = 1 2 m ( ∇ S ) 2 + V ( q ) . {\displaystyle -{\frac {\partial S}{\partial t}}={\frac {1}{2m}}\left(\mathbf {\nabla } S\right)^{2}+V(\mathbf {q} ).} Taking the gradient of both sides, this becomes − ∇ ∂ S ∂ t = 1 2 m ∇ ( ∇ S ) 2 + ∇ V . {\displaystyle -\mathbf {\nabla } {\frac {\partial S}{\partial t}}={\frac {1}{2m}}\mathbf {\nabla } \left(\mathbf {\nabla } S\right)^{2}+\mathbf {\nabla } V.} Interchanging
23956-434: The Newtonian formulation by considering entire trajectories at once rather than predicting a body's motion at a single instant. It is traditional in Lagrangian mechanics to denote position with q {\displaystyle q} and velocity with q ˙ {\displaystyle {\dot {q}}} . The simplest example is a massive point particle, the Lagrangian for which can be written as
24182-435: The acceleration is the second derivative of position, often written d 2 s d t 2 {\displaystyle {\frac {d^{2}s}{dt^{2}}}} . Position, when thought of as a displacement from an origin point, is a vector : a quantity with both magnitude and direction. Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide
24408-438: 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 inverse square law . Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body. Combining these ideas gives a formula that relates the mass ( m ⊕ {\displaystyle m_{\oplus }} ) and
24634-443: The actions of humans is tuche and in unreasoning agents automaton .) Something happens by chance when all the lines of causality converge without that convergence being purposefully chosen, and produce a result similar to the teleologically caused one. In chapters 7 through 9, Aristotle returns to the discussion of nature. With the enrichment of the preceding four chapters, he concludes that nature acts for an end, and he discusses
24860-423: The actually infinite in any form, including infinite bodies, substances, and voids. Aristotle here says the only type of infinity that exists is the potentially infinite. Aristotle characterizes this as that which serves as "the matter for the completion of a magnitude and is potentially (but not actually) the completed whole" (207a22-23). The infinite, lacking any form, is thereby unknowable. Aristotle writes, "it
25086-461: 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 modern physics , which includes relativity and quantum mechanics , the laws governing motion are revised to rely on fundamental interactions as
25312-544: The annotated work on the same page or in a separate list. Commentaries are typically arranged by lemmas, or quotes from the notable work, followed by an analysis of the author of the commentary. The commentaries on every work of Aristotle are a vast and mainly unpublished topic. They extend continuously from the death of the philosopher, representing the entire history of Graeco-Roman philosophy. There are thousands of commentators and commentaries known wholly or more typically in fragments of manuscripts. The latter especially occupy
25538-399: The applied force is opposed by static friction , generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and
25764-426: The applied load. For an object in uniform circular motion , the net force acting on the object equals: F = − m v 2 r r ^ , {\displaystyle \mathbf {F} =-{\frac {mv^{2}}{r}}{\hat {\mathbf {r} }},} where m {\displaystyle m} is the mass of the object, v {\displaystyle v}
25990-519: The artificial: artificial things can move also, but they move according to what they are made of, not according to what they are. For example, if a wooden bed were buried and somehow sprouted as a tree, it would be according to what it is made of, not what it is. Aristotle contrasts two senses of nature: nature as matter and nature as form or definition. By "nature", Aristotle means the natures of particular things and would perhaps be better translated "a nature." In Book II, however, his appeal to "nature" as
26216-502: 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: F V = − ∇ P , {\displaystyle {\frac {\mathbf {F} }{V}}=-\mathbf {\nabla } P,} where V {\displaystyle V}
26442-535: The attracting force is proportional to the product of their masses, and inversely proportional to the square of the distance between them. Finding the shape of the orbits that an inverse-square force law will produce is known as the Kepler problem . The Kepler problem can be solved in multiple ways, including by demonstrating that the Laplace–Runge–Lenz vector is constant, or by applying a duality transformation to
26668-433: The before and after" ( Phys. 219b1–2), by which he intends to denote motion's susceptibility to division into undetached parts of arbitrary length, a property that it possesses both by virtue of its intrinsic nature and also by virtue of the capacities and activities of percipient souls. Motion is intrinsically indeterminate, but perceptually determinable, with respect to its length. Acts of perception function as determiners;
26894-661: The body's center of mass and movement around the center of mass. Significant aspects of the motion of an extended body can be understood by imagining the mass of that body concentrated to a single point, known as the center of mass. The location of a body's center of mass depends upon how that body's material is distributed. For a collection of pointlike objects with masses m 1 , … , m N {\displaystyle m_{1},\ldots ,m_{N}} at positions r 1 , … , r N {\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{N}} ,
27120-744: The body's momentum, the Hamiltonian is H ( p , q ) = p 2 2 m + V ( q ) . {\displaystyle {\mathcal {H}}(p,q)={\frac {p^{2}}{2m}}+V(q).} In this example, Hamilton's equations are d q d t = ∂ H ∂ p {\displaystyle {\frac {dq}{dt}}={\frac {\partial {\mathcal {H}}}{\partial p}}} and d p d t = − ∂ H ∂ q . {\displaystyle {\frac {dp}{dt}}=-{\frac {\partial {\mathcal {H}}}{\partial q}}.} Evaluating these partial derivatives,
27346-435: The body's vertical motion and not its horizontal. At the peak of the projectile's trajectory, its vertical velocity is zero, but its acceleration is g {\displaystyle g} downwards, as it is at all times. Setting the wrong vector equal to zero is a common confusion among physics students. When a body is in uniform circular motion, the force on it changes the direction of its motion but not its speed. For
27572-430: The case of describing a small object bombarded stochastically by even smaller ones. It can be written m a = − γ v + ξ {\displaystyle m\mathbf {a} =-\gamma \mathbf {v} +\mathbf {\xi } \,} where γ {\displaystyle \gamma } is a drag coefficient and ξ {\displaystyle \mathbf {\xi } }
27798-413: The center of mass is located at R = ∑ i = 1 N m i r i M , {\displaystyle \mathbf {R} =\sum _{i=1}^{N}{\frac {m_{i}\mathbf {r} _{i}}{M}},} where M {\displaystyle M} is the total mass of the collection. In the absence of a net external force, the center of mass moves at
28024-410: The concept of energy after Newton's time, but it has become an inseparable part of what is considered "Newtonian" physics. Energy can broadly be classified into kinetic , due to a body's motion, and potential , due to a body's position relative to others. Thermal energy , the energy carried by heat flow, is a type of kinetic energy not associated with the macroscopic motion of objects but instead with
28250-710: The concept of a limit . A function f ( t ) {\displaystyle f(t)} has a limit of L {\displaystyle L} at a given input value t 0 {\displaystyle t_{0}} if the difference between f {\displaystyle f} and L {\displaystyle L} can be made arbitrarily small by choosing an input sufficiently close to t 0 {\displaystyle t_{0}} . One writes, lim t → t 0 f ( t ) = L . {\displaystyle \lim _{t\to t_{0}}f(t)=L.} Instantaneous velocity can be defined as
28476-429: The concept of energy before that of force, essentially "introductory Hamiltonian mechanics". The Hamilton–Jacobi equation provides yet another formulation of classical mechanics, one which makes it mathematically analogous to wave optics . This formulation also uses Hamiltonian functions, but in a different way than the formulation described above. The paths taken by bodies or collections of bodies are deduced from
28702-428: The constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles , such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries
28928-405: The constant forward velocity. 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 kinetic friction . In such a situation,
29154-513: The construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical " test charge " anywhere in space and then using Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as E = F q , {\displaystyle \mathbf {E} ={\mathbf {F} \over {q}},} where q {\displaystyle q}
29380-400: The contributions from each of the points. This provides a means to characterize a body's rotation about an axis, by adding up the angular momenta of its individual pieces. The result depends on the chosen axis, the shape of the body, and the rate of rotation. Newton's law of universal gravitation states that any body attracts any other body along the straight line connecting them. The size of
29606-558: The counterpart of force is torque . Angular momentum is calculated with respect to a reference point. If the displacement vector from a reference point to a body is r {\displaystyle \mathbf {r} } and the body has momentum p {\displaystyle \mathbf {p} } , then the body's angular momentum with respect to that point is, using the vector cross product , L = r × p . {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} .} Taking
29832-407: The crucial properties that forces are additive vector quantities : they have magnitude and direction. When two forces act on a point particle , the resulting force, the resultant (also called the net force ), can be determined by following the parallelogram rule of vector addition : the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that
30058-441: The curvature of the Earth becomes significant: the ground itself will curve away from the falling cannonball. A very fast cannonball will fall away from the inertial straight-line trajectory at the same rate that the Earth curves away beneath it; in other words, it will be in orbit (imagining that it is not slowed by air resistance or obstacles). Consider a body of mass m {\displaystyle m} able to move along
30284-417: The definition of acceleration , the algebraic version of Newton's second law is derived: F = m a . {\displaystyle \mathbf {F} =m\mathbf {a} .} Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if F 1 , 2 {\displaystyle \mathbf {F} _{1,2}}
30510-420: The definition of change based on Aristotle's notions of potentiality and actuality . Change, he says, is the actualization of a thing's ability insofar as it is able. The rest of the book (chapters 4-8) discusses the infinite ( apeiron , the unlimited). He distinguishes between the infinite by addition and the infinite by division, and between the actually infinite and potentially infinite. He argues against
30736-404: The difference between its kinetic and potential energies: L ( q , q ˙ ) = T − V , {\displaystyle L(q,{\dot {q}})=T-V,} where the kinetic energy is T = 1 2 m q ˙ 2 {\displaystyle T={\frac {1}{2}}m{\dot {q}}^{2}} and the potential energy
30962-406: The early 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and inertia . With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised
31188-537: The edition is the Academy's, they do match any line counts. Book I introduces Aristotle's approach to nature, which is to be based on principles, causes, and elements. Before offering his particular views, he engages previous theories, such as those offered by Melissus and Parmenides. Aristotle's own view comes out in Ch. 7 where he identifies three principles: substances, opposites, and privation. Chapters 3 and 4 are among
31414-427: The elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as
31640-418: The energy of a body, have both been described as a "fourth law". The study of the behavior of massive bodies using Newton's laws is known as Newtonian mechanics. Some example problems in Newtonian mechanics are particularly noteworthy for conceptual or historical reasons. If a body falls from rest near the surface of the Earth, then in the absence of air resistance, it will accelerate at a constant rate. This
31866-423: The equilibrium is unstable. A common visual representation of forces acting in concert is the free body diagram , which schematically portrays a body of interest and the forces applied to it by outside influences. For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, "normal" force , friction, and string tension. Newton's second law
32092-446: The equilibrium point, and directed to the equilibrium point, then the body will perform simple harmonic motion . Writing the force as F = − k x {\displaystyle F=-kx} , Newton's second law becomes m d 2 x d t 2 = − k x . {\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-kx\,.} This differential equation has
32318-403: The equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment
32544-418: The fact that hardly a sentence in either can be accepted in the light of modern science. Italian theoretical physicist Carlo Rovelli considers Aristotle's physics as a correct and non-intuitive special case of Newtonian physics for the motion of matter in fluid after it has reached terminal velocity (steady state). His theory disregards the initial phase of acceleration, which is too short to be observed by
32770-416: The first term is the total force upon the first body, and the second term is the total force upon the second body. If the two bodies are isolated from outside influences, the only force upon the first body can be that from the second, and vice versa. By Newton's third law, these forces have equal magnitude but opposite direction, so they cancel when added, and p {\displaystyle \mathbf {p} }
32996-670: The fluid density , and there is a net force upon it if the fluid pressure varies from one side of it to another. Accordingly, a = F / m {\displaystyle \mathbf {a} =\mathbf {F} /m} becomes ∂ v ∂ t + ( ∇ ⋅ v ) v = − 1 ρ ∇ P + f , {\displaystyle {\frac {\partial v}{\partial t}}+(\mathbf {\nabla } \cdot \mathbf {v} )\mathbf {v} =-{\frac {1}{\rho }}\mathbf {\nabla } P+\mathbf {f} ,} where ρ {\displaystyle \rho }
33222-422: 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. 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
33448-413: The force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever ; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion . Dynamic equilibrium
33674-415: The force of gravity only affects the cannonball's momentum in the downward direction, and its effect is not diminished by horizontal movement. If the cannonball is launched with a greater initial horizontal velocity, then it will travel farther before it hits the ground, but it will still hit the ground in the same amount of time. However, if the cannonball is launched with an even larger initial velocity, then
33900-404: The force upon a charged body that can be plugged into Newton's second law in order to calculate its acceleration. According to the Lorentz force law, a charged body in an electric field experiences a force in the direction of that field, a force proportional to its charge q {\displaystyle q} and to the strength of the electric field. In addition, a moving charged body in
34126-399: The force vectors preserved so that graphical vector addition can be done to determine the net force. As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields
34352-436: The force, represented in terms of the potential energy. Landau and Lifshitz argue that the Lagrangian formulation makes the conceptual content of classical mechanics more clear than starting with Newton's laws. Lagrangian mechanics provides a convenient framework in which to prove Noether's theorem , which relates symmetries and conservation laws. The conservation of momentum can be derived by applying Noether's theorem to
34578-498: The former equation becomes d q d t = p m , {\displaystyle {\frac {dq}{dt}}={\frac {p}{m}},} which reproduces the familiar statement that a body's momentum is the product of its mass and velocity. The time derivative of the momentum is d p d t = − d V d q , {\displaystyle {\frac {dp}{dt}}=-{\frac {dV}{dq}},} which, upon identifying
34804-400: 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 . 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
35030-444: The gravitational force from the Earth upon the body is F = G M m r 2 , {\displaystyle F={\frac {GMm}{r^{2}}},} where m {\displaystyle m} is the mass of the falling body, M {\displaystyle M} is the mass of the Earth, G {\displaystyle G} is Newton's constant, and r {\displaystyle r}
35256-455: The horizontal axis and 4 metres per second along the vertical axis. The same motion described in a different coordinate system will be represented by different numbers, and vector algebra can be used to translate between these alternatives. The study of mechanics is complicated by the fact that household words like energy are used with a technical meaning. Moreover, words which are synonymous in everyday speech are not so in physics: force
35482-424: The instantaneous velocity is the derivative of the position with respect to time. It can roughly be thought of as the ratio between an infinitesimally small change in position d s {\displaystyle ds} to the infinitesimally small time interval d t {\displaystyle dt} over which it occurs. More carefully, the velocity and all other derivatives can be defined using
35708-409: 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 . When objects are in contact,
35934-514: The lack of a relativistic speed limit in Newtonian physics. It is not yet known whether or not the Euler and Navier–Stokes equations exhibit the analogous behavior of initially smooth solutions "blowing up" in finite time. The question of existence and smoothness of Navier–Stokes solutions is one of the Millennium Prize Problems . Classical mechanics can be mathematically formulated in multiple different ways, other than
36160-408: The limit of the average velocity as the time interval shrinks to zero: d s d t = lim Δ t → 0 s ( t + Δ t ) − s ( t ) Δ t . {\displaystyle {\frac {ds}{dt}}=\lim _{\Delta t\to 0}{\frac {s(t+\Delta t)-s(t)}{\Delta t}}.} Acceleration
36386-400: 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. A simple elastic force acts to return a spring to its natural length. An ideal spring
36612-565: The long series of physical, cosmological and biological treatises, whose ancient Greek title, τὰ φυσικά, means "the [writings] on nature" or " natural philosophy ". The Physics is composed of eight books, which are further divided into chapters. This system is of ancient origin, now obscure. In modern languages, books are referenced with Roman numerals, standing for ancient Greek capital letters (the Greeks represented numbers with letters, e.g. A for 1). Chapters are identified by Arabic numerals, but
36838-423: The magnetic field. 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 . These " Maxwell's equations " fully described the sources of the fields as being stationary and moving charges, and
37064-416: The magnitude of the vector indicated by the length of the arrow. Numerically, a vector can be represented as a list; for example, a body's velocity vector might be v = ( 3 m / s , 4 m / s ) {\displaystyle \mathbf {v} =(\mathrm {3~m/s} ,\mathrm {4~m/s} )} , indicating that it is moving at 3 metres per second along
37290-432: The magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to
37516-440: The mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum, F = d p d t = d ( m v ) d t , {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} \left(m\mathbf {v} \right)}{\mathrm {d} t}},} where m
37742-549: The mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object. A modern statement of Newton's second law is a vector equation: F = d p d t , {\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},} where p {\displaystyle \mathbf {p} }
37968-405: The means to describe motion in two, three or more dimensions. Vectors are often denoted with an arrow, as in s → {\displaystyle {\vec {s}}} , or in bold typeface, such as s {\displaystyle {\bf {s}}} . Often, vectors are represented visually as arrows, with the direction of the vector being the direction of the arrow, and
38194-531: The most difficult in all of Aristotle's works and involve subtle refutations of the thought of Parmenides, Melissus and Anaxagoras. In chapter 5, he continues his review of his predecessors, particularly how many first principles there are. Chapter 6 narrows down the number of principles to two or three. He presents his own account of the subject in chapter 7, where he first introduces the word matter (Greek: hyle ) to designate fundamental essence (ousia). He defines matter in chapter 9: "For my definition of matter
38420-434: The motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, Philosophiæ Naturalis Principia Mathematica . In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day. The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches. Newton's first law of motion states that
38646-522: The motion of many physical objects and systems. In the time since Newton, new insights, especially around the concept of energy, built the field of classical mechanics on his foundations. Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds ( special relativity ), are very massive ( general relativity ), or are very small ( quantum mechanics ). Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume
38872-409: The movements of the atoms and molecules of which they are made. According to the work-energy theorem , when a force acts upon a body while that body moves along the line of the force, the force does work upon the body, and the amount of work done is equal to the change in the body's kinetic energy. In many cases of interest, the net work done by a force when a body moves in a closed loop — starting at
39098-568: The naked eye. Galileo 's inclined plane experiment bypasses the issue, as it slows down acceleration enough to allow observing the initial phase of acceleration by the naked eye. The five elements explain forms of observed motions. Ether explains circular motion in the sky, earth and water explains downward motion, and fire and air explains upward motion. To explain downward motion, instead of postulating one element, he proposed two, because wood moves up in water but down in air, while earth moves down in both water and air. The complex interaction between
39324-416: The name of matter, and the changes in it. Both causes must be stated by the physicist, but especially the end; for that is the cause of the matter, not vice versa; and the end is 'that for the sake of which', and the beginning starts from the definition or essence… In chapter 3, Aristotle presents his theory of the four causes (material, efficient, formal, and final ). Material cause explains what something
39550-643: 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. 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
39776-399: The negative derivative of the potential with the force, is just Newton's second law once again. As in the Lagrangian formulation, in Hamiltonian mechanics the conservation of momentum can be derived using Noether's theorem, making Newton's third law an idea that is deduced rather than assumed. Among the proposals to reform the standard introductory-physics curriculum is one that teaches
40002-425: The net force on the body accelerates it to a higher speed, must be accompanied by a loss of potential energy. So, the net force upon the body is determined by the manner in which the potential energy decreases. A rigid body is an object whose size is too large to neglect and which maintains the same shape over time. In Newtonian mechanics, the motion of a rigid body is often understood by separating it into movement of
40228-561: The normal force in action is the impact force on an object crashing into an immobile surface. 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: static friction and kinetic friction . The static friction force ( F s f {\displaystyle \mathbf {F} _{\mathrm {sf} }} ) will exactly oppose forces applied to an object parallel to
40454-463: The object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction. 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
40680-445: The object. A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances . For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that
40906-586: The order of the partial derivatives on the left-hand side, and using the power and chain rules on the first term on the right-hand side, − ∂ ∂ t ∇ S = 1 m ( ∇ S ⋅ ∇ ) ∇ S + ∇ V . {\displaystyle -{\frac {\partial }{\partial t}}\mathbf {\nabla } S={\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\mathbf {\nabla } S+\mathbf {\nabla } V.} Gathering together
41132-480: The original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on
41358-412: The oscillator and the amplitude of the oscillations decreases over time. Also, a harmonic oscillator can be driven by an applied force, which can lead to the phenomenon of resonance . Newtonian physics treats matter as being neither created nor destroyed, though it may be rearranged. It can be the case that an object of interest gains or loses mass because matter is added to or removed from it. In such
41584-404: The other two. When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium . 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
41810-518: The others can be reduced. Book VII.1-3 also exist in an alternative version, not included in the Bekker edition . Book VIII (which occupies almost a fourth of the entire Physics , and probably constituted originally an independent course of lessons) discusses two main topics, though with a wide deployment of arguments: the time limits of the universe , and the existence of a Prime Mover — eternal, indivisible, without parts and without magnitude. Isn't
42036-421: The particular contents of the universe. The chief purpose of the work is to discover the principles and causes of (and not merely to describe) change, or movement, or motion (κίνησις kinesis ), especially that of natural wholes (mostly living things, but also inanimate wholes like the cosmos). In the conventional Andronicean ordering of Aristotle's works, it stands at the head of, as well as being foundational to,
42262-462: The pivot, the force upon the pendulum is gravity, and Newton's second law becomes d 2 θ d t 2 = − g L sin θ , {\displaystyle {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{L}}\sin \theta ,} where L {\displaystyle L} is the length of the pendulum and θ {\displaystyle \theta }
42488-410: The position and momentum variables are given by partial derivatives of the Hamiltonian, via Hamilton's equations . The simplest example is a point mass m {\displaystyle m} constrained to move in a straight line, under the effect of a potential. Writing q {\displaystyle q} for the position coordinate and p {\displaystyle p} for
42714-502: The position and velocity the body has at a given time, like t = 0 {\displaystyle t=0} . One reason that the harmonic oscillator is a conceptually important example is that it is good approximation for many systems near a stable mechanical equilibrium. For example, a pendulum has a stable equilibrium in the vertical position: if motionless there, it will remain there, and if pushed slightly, it will swing back and forth. Neglecting air resistance and friction in
42940-416: The principle of inertia : the natural behavior of a body is to move in a straight line at constant speed. A body's motion preserves the status quo, but external forces can perturb this. The modern understanding of Newton's first law is that no inertial observer is privileged over any other. The concept of an inertial observer makes quantitative the everyday idea of feeling no effects of motion. For example,
43166-500: The projectile to its target. This explanation requires a continuous medium such as air to sustain the motion. Though Aristotelian physics was criticized as early as the 6th century, its shortcomings would not be corrected until the 17th century work of Galileo Galilei , who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus . Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove
43392-712: The property that small perturbations of it will, to a first approximation, not change the integral of the Lagrangian. Calculus of variations provides the mathematical tools for finding this path. Applying the calculus of variations to the task of finding the path yields the Euler–Lagrange equation for the particle, d d t ( ∂ L ∂ q ˙ ) = ∂ L ∂ q . {\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}}}\right)={\frac {\partial L}{\partial q}}.} Evaluating
43618-467: The quantity now called momentum , which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving. In modern notation, the momentum of a body is the product of its mass and its velocity: p = m v , {\displaystyle \mathbf {p} =m\mathbf {v} \,,} where all three quantities can change over time. Newton's second law, in modern form, states that
43844-414: The radius ( R ⊕ {\displaystyle R_{\oplus }} ) of the Earth to the gravitational acceleration: g = − G m ⊕ R ⊕ 2 r ^ , {\displaystyle \mathbf {g} =-{\frac {Gm_{\oplus }}{{R_{\oplus }}^{2}}}{\hat {\mathbf {r} }},} where
44070-402: The rate of change of p {\displaystyle \mathbf {p} } is d p d t = d p 1 d t + d p 2 d t . {\displaystyle {\frac {d\mathbf {p} }{dt}}={\frac {d\mathbf {p} _{1}}{dt}}+{\frac {d\mathbf {p} _{2}}{dt}}.} By Newton's second law,
44296-399: The reference point ( r = 0 {\displaystyle \mathbf {r} =0} ) or if the force F {\displaystyle \mathbf {F} } and the displacement vector r {\displaystyle \mathbf {r} } are directed along the same line. The angular momentum of a collection of point masses, and thus of an extended body, is found by adding
44522-419: The relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics , can be paraphrased as follows: The three laws of motion were first stated by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica ( Mathematical Principles of Natural Philosophy ), originally published in 1687. Newton used them to investigate and explain
44748-404: The relationship of the moved to his mover, which Aristotle describes in substantial divergence with Plato ' s theory of the soul as capable of setting itself in motion ( Laws book X, Phaedrus , Phaedo ). Everything which moves is moved by another. He then tries to correlate the species of motion and their speeds, with the local change (locomotion, phorà ) as the most fundamental to which
44974-523: The result is determinate units of kinetic length, which is precisely what a temporal unit is. Books V and VI deal with how motion occurs. Book V classifies four species of movement, depending on where the opposites are located. Movement categories include quantity (e.g. a change in dimensions, from great to small), quality (as for colors: from pale to dark), place (local movements generally go from up downwards and vice versa), or, more controversially, substance. In fact, substances do not have opposites, so it
45200-403: The rocket, then F = M d v d t − u d M d t {\displaystyle \mathbf {F} =M{\frac {d\mathbf {v} }{dt}}-\mathbf {u} {\frac {dM}{dt}}\,} where F {\displaystyle \mathbf {F} } is the net external force (e.g., a planet's gravitational pull). Physicists developed
45426-621: The same laws of motion , his law of gravity had to be universal. Succinctly stated, Newton's law of gravitation states that the force on a spherical object of mass m 1 {\displaystyle m_{1}} due to the gravitational pull of mass m 2 {\displaystyle m_{2}} is F = − G m 1 m 2 r 2 r ^ , {\displaystyle \mathbf {F} =-{\frac {Gm_{1}m_{2}}{r^{2}}}{\hat {\mathbf {r} }},} where r {\displaystyle r}
45652-546: The same amount of work . Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids . Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology . In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of
45878-400: The same direction. The remaining term is the torque, τ = r × F . {\displaystyle \mathbf {\tau } =\mathbf {r} \times \mathbf {F} .} When the torque is zero, the angular momentum is constant, just as when the force is zero, the momentum is constant. The torque can vanish even when the force is non-zero, if the body is located at
46104-414: The same mathematical form as Newton's law of universal gravitation: the force is proportional to the product of the charges, inversely proportional to the square of the distance between them, and directed along the straight line between them. The Coulomb force that a charge q 1 {\displaystyle q_{1}} exerts upon a charge q 2 {\displaystyle q_{2}}
46330-532: The solution x ( t ) = A cos ω t + B sin ω t {\displaystyle x(t)=A\cos \omega t+B\sin \omega t\,} where the frequency ω {\displaystyle \omega } is equal to k / m {\displaystyle {\sqrt {k/m}}} , and the constants A {\displaystyle A} and B {\displaystyle B} can be calculated knowing, for example,
46556-477: The sometimes non-obvious force of friction and a consequently inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton . With his mathematical insight, Newton formulated laws of motion that were not improved for over two hundred years. By
46782-406: The source of an apparent thing's activities is not the whole itself, but its parts. While Aristotle asserts that the matter (and parts) are a necessary cause of things – the material cause – he says that nature is primarily the essence or formal cause (1.193b6), that is, the information, the whole species itself. The necessary in nature, then, is plainly what we call by
47008-399: The string by the first object is accompanied by a force directed along the string in the opposite direction by the second object. 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
47234-510: The terms that depend upon the gradient of S {\displaystyle S} , [ ∂ ∂ t + 1 m ( ∇ S ⋅ ∇ ) ] ∇ S = − ∇ V . {\displaystyle \left[{\frac {\partial }{\partial t}}+{\frac {1}{m}}\left(\mathbf {\nabla } S\cdot \mathbf {\nabla } \right)\right]\mathbf {\nabla } S=-\mathbf {\nabla } V.} This
47460-407: The theory is used in practice. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll . Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as " vector quantities ". This means that forces follow
47686-452: The three bodies' motions over time. Numerical methods can be applied to obtain useful, albeit approximate, results for the three-body problem. The positions and velocities of the bodies can be stored in variables within a computer's memory; Newton's laws are used to calculate how the velocities will change over a short interval of time, and knowing the velocities, the changes of position over that time interval can be computed. This process
47912-732: The time derivative of the angular momentum gives d L d t = ( d r d t ) × p + r × d p d t = v × m v + r × F . {\displaystyle {\frac {d\mathbf {L} }{dt}}=\left({\frac {d\mathbf {r} }{dt}}\right)\times \mathbf {p} +\mathbf {r} \times {\frac {d\mathbf {p} }{dt}}=\mathbf {v} \times m\mathbf {v} +\mathbf {r} \times \mathbf {F} .} The first term vanishes because v {\displaystyle \mathbf {v} } and m v {\displaystyle m\mathbf {v} } point in
48138-401: The time derivative of the momentum is the force: F = d p d t . {\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.} If the mass m {\displaystyle m} does not change with time, then the derivative acts only upon the velocity, and so the force equals the product of the mass and the time derivative of
48364-546: The time interval in the same place as it began. Calculus gives the means to define an instantaneous velocity, a measure of a body's speed and direction of movement at a single moment of time, rather than over an interval. One notation for the instantaneous velocity is to replace Δ {\displaystyle \Delta } with the symbol d {\displaystyle d} , for example, v = d s d t . {\displaystyle v={\frac {ds}{dt}}.} This denotes that
48590-418: The ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes. Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines , but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of
48816-468: The universe eternal, has it had a beginning, will it ever end? Aristotle's response, as a Greek, could hardly be affirmative, never having been told of a creatio ex nihilo, but he also has philosophical reasons for denying that motion had not always existed, on the grounds of the theory presented in the earlier books of the Physics . Eternity of motion is also confirmed by the existence of a substance which
49042-564: The use of the English word "chapter" is strictly conventional. Ancient "chapters" (capita) are generally very short, often less than a page. Additionally, the Bekker numbers give the page and column (a or b) used in the Prussian Academy of Sciences' edition of Aristotle's works, instigated and managed by Bekker himself. These are evident in the 1831 2-volume edition. Bekker's line numbers may be given. These are often given, but unless
49268-491: The vector direction is given by r ^ {\displaystyle {\hat {\mathbf {r} }}} , is the unit vector directed outward from the center of the Earth. In this equation, a dimensional constant G {\displaystyle G} is used to describe the relative strength of gravity. This constant has come to be known as the Newtonian constant of gravitation , though its value
49494-444: The velocity field at its position is changing over time, and second, because it moves to a new location where the velocity field has a different value. Consequently, when Newton's second law is applied to an infinitesimal portion of fluid, the acceleration a {\displaystyle \mathbf {a} } has two terms, a combination known as a total or material derivative . The mass of an infinitesimal portion depends upon
49720-412: The velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates
49946-530: The velocity, which is the acceleration: F = m d v d t = m a . {\displaystyle \mathbf {F} =m{\frac {d\mathbf {v} }{dt}}=m\mathbf {a} \,.} As the acceleration is the second derivative of position with respect to time, this can also be written F = m d 2 s d t 2 . {\displaystyle \mathbf {F} =m{\frac {d^{2}\mathbf {s} }{dt^{2}}}.} The forces acting on
50172-423: The way that necessity is present in natural things. For Aristotle, the motion of natural things is determined from within them, while in the modern empirical sciences, motion is determined from without (more properly speaking: there is nothing to have an inside). In order to understand "nature" as defined in the previous book, one must understand the terms of the definition. To understand motion, book III begins with
50398-521: Was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic . Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame " did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of
50624-436: Was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as g {\displaystyle \mathbf {g} } and has a magnitude of about 9.81 meters per second squared (this measurement
50850-458: 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 law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion . Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that
51076-503: Was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of G {\displaystyle G} using a torsion balance ; this was widely reported in the press as a measurement of the mass of the Earth since knowing G {\displaystyle G} could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed
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