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177-424: Commonly used structures can be classified into five major categories, depending on the type of primary stress that may arise in the members of the structures under major design loads. However any two or more of the basic structural types described in the following may be combined in a single structure, such as a building or a bridge in order to meet the structure's functional requirements. The structural system of

354-432: 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}

531-420: A flow of viscous liquid , the force F may not be perpendicular to S ; hence the stress across a surface must be regarded a vector quantity, not a scalar. Moreover, the direction and magnitude generally depend on the orientation of S . Thus the stress state of the material must be described by a tensor , called the (Cauchy) stress tensor ; which is a linear function that relates the normal vector n of

708-489: A high-rise building is designed to cope with vertical gravity loads as well as lateral loads caused by wind or seismic activity. The structural system consists only of the members designed to carry the loads, and all other members are referred to as non-structural. A classification for the structural system of a high-rise was introduced in 1969 by Fazlur Khan and was extended to incorporate interior and exterior structures. The primary lateral load-resisting system defines if

885-426: A linear approximation may be adequate in practice if the quantities are sufficiently small. Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . Humans have known about stress inside materials since ancient times. Until the 17th century, this understanding

1062-439: A "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of

1239-466: 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

1416-523: 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

1593-443: 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,

1770-411: 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 ,

1947-443: 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

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2124-392: A coordinate system with axes e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} , the stress tensor is a diagonal matrix, and has only the three normal components λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}

2301-444: A crumpled sponge, is subject to compressive stress and may undergo shortening. The greater the force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Stress has dimension of force per area, with SI units of newtons per square meter (N/m ) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain

2478-457: A cylindrical bar such as a shaft is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of I-beams under bending loads, due to the web constraining the end plates ("flanges"). Another simple type of stress occurs when

2655-454: A differential formula for friction forces (shear stress) in parallel laminar flow . Stress is defined as the force across a small boundary per unit area of that boundary, for all orientations of the boundary. Derived from a fundamental physical quantity (force) and a purely geometrical quantity (area), stress is also a fundamental quantity, like velocity, torque or energy , that can be quantified and analyzed without explicit consideration of

2832-460: 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,

3009-480: 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

3186-410: 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

3363-424: 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

3540-403: A piston) push against them in (Newtonian) reaction . These macroscopic forces are actually the net result of a very large number of intermolecular forces and collisions between the particles in those molecules . Stress is frequently represented by a lowercase Greek letter sigma ( σ ). Strain inside a material may arise by various mechanisms, such as stress as applied by external forces to

3717-484: 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

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3894-446: 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

4071-405: 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

4248-414: 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

4425-450: 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

4602-475: A structural system is an interior or exterior one. The following interior structures are possible: The following exterior structures are possible: Stress (mechanics) In continuum mechanics , stress is a physical quantity that describes forces present during deformation . For example, an object being pulled apart, such as a stretched elastic band, is subject to tensile stress and may undergo elongation . An object being pushed together, such as

4779-448: A surface S to the traction vector T across S . With respect to any chosen coordinate system , the Cauchy stress tensor can be represented as a symmetric matrix of 3×3 real numbers. Even within a homogeneous body, the stress tensor may vary from place to place, and may change over time; therefore, the stress within a material is, in general, a time-varying tensor field . In general,

4956-1007: A surface will always be a linear function of the surface's normal vector n {\displaystyle n} , the unit-length vector that is perpendicular to it. That is, T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} , where the function σ {\displaystyle {\boldsymbol {\sigma }}} satisfies σ ( α u + β v ) = α σ ( u ) + β σ ( v ) {\displaystyle {\boldsymbol {\sigma }}(\alpha u+\beta v)=\alpha {\boldsymbol {\sigma }}(u)+\beta {\boldsymbol {\sigma }}(v)} for any vectors u , v {\displaystyle u,v} and any real numbers α , β {\displaystyle \alpha ,\beta } . The function σ {\displaystyle {\boldsymbol {\sigma }}} , now called

5133-434: A surface with normal vector n {\displaystyle n} (which is covariant - "row; horizontal" - vector) with coordinates n 1 , n 2 , n 3 {\displaystyle n_{1},n_{2},n_{3}} is then a matrix product T = n ⋅ σ {\displaystyle T=n\cdot {\boldsymbol {\sigma }}} (where T in upper index

5310-482: 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

5487-413: A system must be balanced by internal reaction forces, which are almost always surface contact forces between adjacent particles — that is, as stress. Since every particle needs to be in equilibrium, this reaction stress will generally propagate from particle to particle, creating a stress distribution throughout the body. The typical problem in stress analysis is to determine these internal stresses, given

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5664-434: A system of partial differential equations involving the stress tensor field and the strain tensor field, as unknown functions to be determined. The external body forces appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. The basic stress analysis problem is therefore a boundary-value problem . Stress analysis for elastic structures

5841-410: 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

6018-519: 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

6195-489: A two-dimensional one, and/or replace the general stress and strain tensors by simpler models like uniaxial tension/compression, simple shear, etc. Still, for two- or three-dimensional cases one must solve a partial differential equation problem. Analytical or closed-form solutions to the differential equations can be obtained when the geometry, constitutive relations, and boundary conditions are simple enough. Otherwise one must generally resort to numerical approximations such as

6372-575: 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,

6549-412: 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

6726-1092: Is transposition , and as a result we get covariant (row) vector) (look on Cauchy stress tensor ), that is [ T 1 T 2 T 3 ] = [ n 1 n 2 n 3 ] ⋅ [ σ 11 σ 21 σ 31 σ 12 σ 22 σ 32 σ 13 σ 23 σ 33 ] {\displaystyle {\begin{bmatrix}T_{1}&T_{2}&T_{3}\end{bmatrix}}={\begin{bmatrix}n_{1}&n_{2}&n_{3}\end{bmatrix}}\cdot {\begin{bmatrix}\sigma _{11}&\sigma _{21}&\sigma _{31}\\\sigma _{12}&\sigma _{22}&\sigma _{32}\\\sigma _{13}&\sigma _{23}&\sigma _{33}\end{bmatrix}}} The linear relation between T {\displaystyle T} and n {\displaystyle n} follows from

6903-418: 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

7080-404: 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

7257-520: 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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7434-410: Is actually the average of a very large number of atomic forces between their molecules; and physical quantities like mass, velocity, and forces that act through the bulk of three-dimensional bodies, like gravity, are assumed to be smoothly distributed over them. Depending on the context, one may also assume that the particles are large enough to allow the averaging out of other microscopic features, like

7611-583: Is an essential tool in engineering for the study and design of structures such as tunnels, dams, mechanical parts, and structural frames, under prescribed or expected loads. It is also important in many other disciplines; for example, in geology, to study phenomena like plate tectonics , vulcanism and avalanches ; and in biology, to understand the anatomy of living beings. Stress analysis is generally concerned with objects and structures that can be assumed to be in macroscopic static equilibrium . By Newton's laws of motion , any external forces being applied to such

7788-402: 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

7965-959: 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 ,

8142-406: Is assumed fixed, the normal component can be expressed by a single number, the dot product T · n . This number will be positive if P is "pulling" on Q (tensile stress), and negative if P is "pushing" against Q (compressive stress). The shear component is then the vector T − ( T · n ) n . The dimension of stress is that of pressure , and therefore its coordinates are measured in

8319-478: Is based on the theory of elasticity and infinitesimal strain theory . When the applied loads cause permanent deformation, one must use more complicated constitutive equations, that can account for the physical processes involved ( plastic flow , fracture , phase change , etc.). Engineered structures are usually designed so the maximum expected stresses are well within the range of linear elasticity (the generalization of Hooke's law for continuous media); that is,

8496-417: 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

8673-461: 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

8850-471: 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

9027-413: 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

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9204-637: Is given in the article on viscosity . The same for normal viscous stresses can be found in Sharma (2019). The relation between stress and its effects and causes, including deformation and rate of change of deformation, can be quite complicated (although a linear approximation may be adequate in practice if the quantities are small enough). Stress that exceeds certain strength limits of the material will result in permanent deformation (such as plastic flow , fracture , cavitation ) or even change its crystal structure and chemical composition . In some situations,

9381-428: Is important, for example, in prestressed concrete and tempered glass . Stress may also be imposed on a material without the application of net forces , for example by changes in temperature or chemical composition, or by external electromagnetic fields (as in piezoelectric and magnetostrictive materials). The relation between mechanical stress, strain, and the strain rate can be quite complicated, although

9558-608: 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

9735-401: 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

9912-463: 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 ,

10089-467: 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

10266-507: Is often used for safety certification and monitoring. Most stress is analysed by mathematical methods, especially during design. The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum ) and the Euler-Cauchy stress principle , together with the appropriate constitutive equations. Thus one obtains

10443-408: Is perpendicular to the layer, the net internal force across S , and hence the stress, will be zero. As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F / A will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes. Shear stress is observed also when

10620-467: 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

10797-639: 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

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10974-412: Is subjected to tension by opposite forces of magnitude F {\displaystyle F} along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, F with continuity through the full cross-sectional area , A . Therefore,

11151-401: 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

11328-860: 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 }

11505-596: 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

11682-424: 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

11859-403: Is the measure of the relative deformation of the material. For example, when a solid vertical bar is supporting an overhead weight , each particle in the bar pushes on the particles immediately below it. When a liquid is in a closed container under pressure , each particle gets pushed against by all the surrounding particles. The container walls and the pressure -inducing surface (such as

12036-413: 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

12213-437: Is then reduced to a scalar (tension or compression of the bar), but one must take into account also a bending stress (that tries to change the bar's curvature, in some direction perpendicular to the axis) and a torsional stress (that tries to twist or un-twist it about its axis). Stress analysis is a branch of applied physics that covers the determination of the internal distribution of internal forces in solid objects. It

12390-559: 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}

12567-499: 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,

12744-576: Is too small to be detected. In a solid material, such strain will in turn generate an internal elastic stress, analogous to the reaction force of a stretched spring , tending to restore the material to its original undeformed state. Fluid materials (liquids, gases and plasmas ) by definition can only oppose deformations that would change their volume. If the deformation changes with time, even in fluids there will usually be some viscous stress, opposing that change. Such stresses can be either shear or normal in nature. Molecular origin of shear stresses in fluids

12921-549: 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

13098-519: 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

13275-415: 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

13452-505: The (Cauchy) stress tensor , completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor , reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus , σ {\displaystyle {\boldsymbol {\sigma }}} is classified as a second-order tensor of type (0,2) or (1,1) depending on convention. Like any linear map between vectors,

13629-610: The capitals , arches , cupolas , trusses and the flying buttresses of Gothic cathedrals . Ancient and medieval architects did develop some geometrical methods and simple formulas to compute the proper sizes of pillars and beams, but the scientific understanding of stress became possible only after the necessary tools were invented in the 17th and 18th centuries: Galileo Galilei 's rigorous experimental method , René Descartes 's coordinates and analytic geometry , and Newton 's laws of motion and equilibrium and calculus of infinitesimals . With those tools, Augustin-Louis Cauchy

13806-498: The finite element method , the finite difference method , and the boundary element method . Other useful stress measures include the first and second Piola–Kirchhoff stress tensors , the Biot stress tensor , and the Kirchhoff stress tensor . Newton%27s laws of motion#Newton's second law Newton's laws of motion are three physical laws that describe the relationship between

13983-463: 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

14160-447: 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 the motion of many physical objects and systems. In

14337-989: The orthogonal shear stresses . The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle of stress distribution. As a symmetric 3×3 real matrix, the stress tensor σ {\displaystyle {\boldsymbol {\sigma }}} has three mutually orthogonal unit-length eigenvectors e 1 , e 2 , e 3 {\displaystyle e_{1},e_{2},e_{3}} and three real eigenvalues λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}} , such that σ e i = λ i e i {\displaystyle {\boldsymbol {\sigma }}e_{i}=\lambda _{i}e_{i}} . Therefore, in

14514-399: 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

14691-457: The principal stresses . If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame. In general, stress is not uniformly distributed over a material body, and may vary with time. Therefore, the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of

14868-410: 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

15045-459: 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

15222-820: 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

15399-434: 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

15576-435: 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

15753-531: 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

15930-661: 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}} ,

16107-744: 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,

16284-435: 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

16461-422: The bulk material (like gravity ) or to its surface (like contact forces , external pressure, or friction ). Any strain (deformation) of a solid material generates an internal elastic stress , analogous to the reaction force of a spring , that tends to restore the material to its original non-deformed state. In liquids and gases , only deformations that change the volume generate persistent elastic stress. If

16638-449: The bulk of the material, varying continuously with position and time. Other agents (like external loads and friction, ambient pressure, and contact forces) may create stresses and forces that are concentrated on certain surfaces, lines or points; and possibly also on very short time intervals (as in the impulses due to collisions). In active matter , self-propulsion of microscopic particles generates macroscopic stress profiles. In general,

16815-430: 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 } }

16992-413: 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

17169-410: 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

17346-710: 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

17523-429: 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

17700-400: 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

17877-558: 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

18054-498: The cross-section), but will vary over the cross section: the outer part will be under tensile stress, while the inner part will be compressed. Another variant of normal stress is the hoop stress that occurs on the walls of a cylindrical pipe or vessel filled with pressurized fluid. Another simple type of stress occurs when a uniformly thick layer of elastic material like glue or rubber is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to

18231-441: 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

18408-425: The deformation changes gradually with time, even in fluids there will usually be some viscous stress , opposing that change. Elastic and viscous stresses are usually combined under the name mechanical stress . Significant stress may exist even when deformation is negligible or non-existent (a common assumption when modeling the flow of water). Stress may exist in the absence of external forces; such built-in stress

18585-402: The deformations caused by internal stresses are linearly related to them. In this case the differential equations that define the stress tensor are linear, and the problem becomes much easier. For one thing, the stress at any point will be a linear function of the loads, too. For small enough stresses, even non-linear systems can usually be assumed to be linear. Stress analysis is simplified when

18762-404: 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

18939-709: The effect of gravity and other external forces can be neglected. In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic ; if it is compressive, it is called hydrostatic pressure or just pressure . Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see Z-tube . Parts with rotational symmetry , such as wheels, axles, pipes, and pillars, are very common in engineering. Often

19116-434: The elements σ x , σ y , σ z {\displaystyle \sigma _{x},\sigma _{y},\sigma _{z}} are called the orthogonal normal stresses (relative to the chosen coordinate system), and τ x y , τ x z , τ y z {\displaystyle \tau _{xy},\tau _{xz},\tau _{yz}}

19293-418: 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

19470-423: 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

19647-446: 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

19824-424: The external forces that are acting on the system. The latter may be body forces (such as gravity or magnetic attraction), that act throughout the volume of a material; or concentrated loads (such as friction between an axle and a bearing , or the weight of a train wheel on a rail), that are imagined to act over a two-dimensional area, or along a line, or at single point. In stress analysis one normally disregards

20001-416: 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} }

20178-670: 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 }

20355-415: 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

20532-454: 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

20709-436: 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

20886-498: 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

21063-412: The fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations ( Cauchy's equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that

21240-457: The grains of a metal rod or the fibers of a piece of wood . Quantitatively, the stress is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material across an imaginary separating surface S , divided by the area of S . In a fluid at rest the force is perpendicular to the surface, and is the familiar pressure . In a solid , or in

21417-444: 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}

21594-455: 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

21771-424: 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

21948-503: 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

22125-408: The layer; or a section of a soft metal bar that is being cut by the jaws of a scissors-like tool . Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F . Assuming that the direction of the forces is known, the stress across M can be expressed simply by

22302-408: 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

22479-416: 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

22656-429: The material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that

22833-405: 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

23010-519: The medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. Human-made objects are often made from stock plates of various materials by operations that do not change their essentially two-dimensional character, like cutting, drilling, gentle bending and welding along the edges. The description of stress in such bodies can be simplified by modeling those parts as two-dimensional surfaces rather than three-dimensional bodies. In that view, one redefines

23187-448: The most general case, called triaxial stress , the stress is nonzero across every surface element. Combined stresses cannot be described by a single vector. Even if the material is stressed in the same way throughout the volume of the body, the stress across any imaginary surface will depend on the orientation of that surface, in a non-trivial way. Cauchy observed that the stress vector T {\displaystyle T} across

23364-409: 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

23541-420: The nature of the material or of its physical causes. Following the basic premises of continuum mechanics, stress is a macroscopic concept. Namely, the particles considered in its definition and analysis should be just small enough to be treated as homogeneous in composition and state, but still large enough to ignore quantum effects and the detailed motions of molecules. Thus, the force between two particles

23718-399: 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

23895-425: 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

24072-586: 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

24249-412: 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

24426-452: The physical causes of the forces or the precise nature of the materials. Instead, one assumes that the stresses are related to deformation (and, in non-static problems, to the rate of deformation) of the material by known constitutive equations . Stress analysis may be carried out experimentally, by applying loads to the actual artifact or to scale model, and measuring the resulting stresses, by any of several available methods. This approach

24603-424: The physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of trusses, for example, the stress field may be assumed to be uniform and uniaxial over each member. Then the differential equations reduce to a finite set of equations (usually linear) with finitely many unknowns. In other contexts one may be able to reduce the three-dimensional problem to

24780-462: 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 }

24957-445: The plate). The analysis of stress can be considerably simplified also for thin bars, beams or wires of uniform (or smoothly varying) composition and cross-section that are subjected to moderate bending and twisting. For those bodies, one may consider only cross-sections that are perpendicular to the bar's axis, and redefine a "particle" as being a piece of wire with infinitesimal length between two such cross sections. The ordinary stress

25134-410: 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

25311-502: 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

25488-416: 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,

25665-712: 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

25842-467: 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

26019-402: 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,

26196-399: 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

26373-403: 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

26550-400: 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

26727-414: 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}}

26904-657: The same units as pressure: namely, pascals (Pa, that is, newtons per square metre ) in the International System , or pounds per square inch (psi) in the Imperial system . Because mechanical stresses easily exceed a million Pascals, MPa, which stands for megapascal, is a common unit of stress. Stress in a material body may be due to multiple physical causes, including external influences and internal physical processes. Some of these agents (like gravity, changes in temperature and phase , and electromagnetic fields) act on

27081-424: The single number τ {\displaystyle \tau } , calculated simply with the magnitude of those forces, F and the cross sectional area, A . τ = F A {\displaystyle \tau ={\frac {F}{A}}} Unlike normal stress, this simple shear stress is directed parallel to the cross-section considered, rather than perpendicular to it. For any plane S that

27258-532: 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,

27435-407: The stress T that a particle P applies on another particle Q across a surface S can have any direction relative to S . The vector T may be regarded as the sum of two components: the normal stress ( compression or tension ) perpendicular to the surface, and the shear stress that is parallel to the surface. If the normal unit vector n of the surface (pointing from Q towards P )

27612-501: The stress can be assumed to be uniformly distributed over any cross-section that is more than a few times D from both ends. (This observation is known as the Saint-Venant's principle ). Normal stress occurs in many other situations besides axial tension and compression. If an elastic bar with uniform and symmetric cross-section is bent in one of its planes of symmetry, the resulting bending stress will still be normal (perpendicular to

27789-411: The stress distribution in a body is expressed as a piecewise continuous function of space and time. Conversely, stress is usually correlated with various effects on the material, possibly including changes in physical properties like birefringence , polarization , and permeability . The imposition of stress by an external agent usually creates some strain (deformation) in the material, even if it

27966-475: The stress is evenly distributed over the entire cross-section. In practice, depending on how the bar is attached at the ends and how it was manufactured, this assumption may not be valid. In that case, the value σ {\displaystyle \sigma } = F / A will be only the average stress, called engineering stress or nominal stress . If the bar's length L is many times its diameter D , and it has no gross defects or built-in stress , then

28143-424: The stress is maximum for surfaces that are perpendicular to a certain direction d {\displaystyle d} , and zero across any surfaces that are parallel to d {\displaystyle d} . When the shear stress is zero only across surfaces that are perpendicular to one particular direction, the stress is called biaxial , and can be viewed as the sum of two normal or shear stresses. In

28320-399: The stress patterns that occur in such parts have rotational or even cylindrical symmetry . The analysis of such cylinder stresses can take advantage of the symmetry to reduce the dimension of the domain and/or of the stress tensor. Often, mechanical bodies experience more than one type of stress at the same time; this is called combined stress . In normal and shear stress, the magnitude of

28497-684: The stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written [ σ x τ x y τ x z τ x y σ y τ y z τ x z τ y z σ z ] {\displaystyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{xy}&\sigma _{y}&\tau _{yz}\\\tau _{xz}&\tau _{yz}&\sigma _{z}\end{bmatrix}}} where

28674-411: The stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the curvature of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of

28851-1620: The stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered x 1 , x 2 , x 3 {\displaystyle x_{1},x_{2},x_{3}} or named x , y , z {\displaystyle x,y,z} , the matrix may be written as [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] {\displaystyle {\begin{bmatrix}\sigma _{11}&\sigma _{12}&\sigma _{13}\\\sigma _{21}&\sigma _{22}&\sigma _{23}\\\sigma _{31}&\sigma _{32}&\sigma _{33}\end{bmatrix}}} or [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] {\displaystyle {\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\\\end{bmatrix}}} The stress vector T = σ ( n ) {\displaystyle T={\boldsymbol {\sigma }}(n)} across

29028-431: The stress tensor is symmetric , that is σ 12 = σ 21 {\displaystyle \sigma _{12}=\sigma _{21}} , σ 13 = σ 31 {\displaystyle \sigma _{13}=\sigma _{31}} , and σ 23 = σ 32 {\displaystyle \sigma _{23}=\sigma _{32}} . Therefore,

29205-423: The stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are the uniaxial normal stress , the simple shear stress , and the isotropic normal stress . A common situation with a simple stress pattern is when a straight rod, with uniform material and cross section,

29382-440: The stress σ throughout the bar, across any horizontal surface, can be expressed simply by the single number σ, calculated simply with the magnitude of those forces, F , and cross sectional area, A . σ = F A {\displaystyle \sigma ={\frac {F}{A}}} On the other hand, if one imagines the bar being cut along its length, parallel to the axis, there will be no force (hence no stress) between

29559-510: 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

29736-452: 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

29913-732: 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

30090-401: 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

30267-546: 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

30444-469: 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

30621-440: The two halves across the cut. This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress σ {\displaystyle \sigma } change sign, and the stress is called compressive stress. This analysis assumes

30798-444: 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

30975-530: 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

31152-400: Was able to give the first rigorous and general mathematical model of a deformed elastic body by introducing the notions of stress and strain. Cauchy observed that the force across an imaginary surface was a linear function of its normal vector; and, moreover, that it must be a symmetric function (with zero total momentum). The understanding of stress in liquids started with Newton, who provided

31329-408: Was largely intuitive and empirical, though this did not prevent the development of relatively advanced technologies like the composite bow and glass blowing . Over several millennia, architects and builders in particular, learned how to put together carefully shaped wood beams and stone blocks to withstand, transmit, and distribute stress in the most effective manner, with ingenious devices such as

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