In mathematics , the origin of a Euclidean space is a special point , usually denoted by the letter O , used as a fixed point of reference for the geometry of the surrounding space.
113-506: (Redirected from Third Law ) Third law may refer to: Newton's third law of motion, one of Newton's laws of motion Third law of thermodynamics Kepler's Third law of planetary motion Mendel's third law, or the Law of Dominance Third Law (album) , 2016 album by Roly Porter See also [ edit ] Rule of thirds Rule of three (disambiguation) Topics referred to by
226-432: 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}
339-469: 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
452-529: 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
565-443: 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,
678-411: 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 ,
791-443: 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
904-460: 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,
1017-484: 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
1130-421: 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} .} Origin (mathematics) In physical problems,
1243-410: 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
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#17328014360401356-424: 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
1469-484: 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
1582-446: 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
1695-413: A positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates —that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. In a polar coordinate system , the origin may also be called
1808-405: 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
1921-414: 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
2034-450: 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
2147-482: 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
2260-410: 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
2373-523: 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
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#17328014360402486-575: 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,
2599-412: 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
2712-418: 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
2825-404: 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
2938-520: 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
3051-402: 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
3164-959: 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 ,
3277-417: 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
3390-466: 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
3503-471: 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
Third law - Misplaced Pages Continue
3616-622: Is different from Wikidata All article disambiguation pages All disambiguation pages Newton%27s laws of motion Newton's laws of motion are three physical laws that describe 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
3729-413: 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
3842-608: 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
3955-401: 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
4068-463: 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 ,
4181-467: 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
4294-473: 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
4407-639: 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
4520-401: 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
4633-864: 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 }
Third law - Misplaced Pages Continue
4746-596: 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
4859-424: 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
4972-413: 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
5085-563: 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}
5198-499: 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,
5311-549: 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
5424-519: 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
5537-415: 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
5650-468: 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
5763-399: 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
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#17328014360405876-410: 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
5989-463: 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
6102-820: 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
6215-434: 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
6328-435: 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
6441-535: 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
6554-661: 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}} ,
6667-744: 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,
6780-435: 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
6893-430: 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 } }
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#17328014360407006-413: 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
7119-421: The choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry . In a Cartesian coordinate system , the origin is the point where the axes of the system intersect. The origin divides each of these axes into two halves,
7232-410: 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
7345-710: 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
7458-429: 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
7571-400: 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
7684-558: 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
7797-441: 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
7910-404: 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
8023-418: 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
8136-423: 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
8249-446: 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
8362-416: 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} }
8475-670: 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 }
8588-415: 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
8701-459: 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
8814-436: 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
8927-498: 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
9040-444: 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}
9153-455: 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
9266-424: 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
9379-514: 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
9492-408: 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
9605-416: 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
9718-405: 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
9831-522: 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
9944-409: 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
10057-399: 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
10170-425: 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
10283-586: 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
10396-412: 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
10509-462: 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 }
10622-406: The pole. It does not itself have well-defined polar coordinates, because the polar coordinates of a point include the angle made by the positive x -axis and the ray from the origin to the point, and this ray is not well-defined for the origin itself. In Euclidean geometry , the origin may be chosen freely as any convenient point of reference. The origin of the complex plane can be referred as
10735-410: 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
10848-502: 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
10961-416: 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,
11074-712: 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
11187-467: 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
11300-402: 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,
11413-399: 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
11526-403: 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
11639-400: 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
11752-414: 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}}
11865-415: The same term [REDACTED] This disambiguation page lists articles associated with the title Third law . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Third_law&oldid=1233170427 " Category : Disambiguation pages Hidden categories: Short description
11978-532: 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,
12091-510: 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
12204-452: 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
12317-732: 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
12430-401: 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
12543-546: 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
12656-444: 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
12769-530: 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
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