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50-429: A conservative force depends only on the position of the object. If a force is conservative, it is possible to assign a numerical value for the potential at any point and conversely, when an object moves from one location to another, the force changes the potential energy of the object by an amount that does not depend on the path taken, contributing to the mechanical energy and the overall conservation of energy . If

100-400: A b F ( r ( t ) ) ⋅ r ′ ( t ) d t , {\displaystyle V(\mathbf {r} )=-\int _{C}\mathbf {F} (\mathbf {r} )\cdot \,d\mathbf {r} =-\int _{a}^{b}\mathbf {F} (\mathbf {r} (t))\cdot \mathbf {r} '(t)\,dt,} where C is a parametrized path from r 0 to r , r ( t ) ,

150-429: A ≤ t ≤ b , r ( a ) = r 0 , r ( b ) = r . {\displaystyle \mathbf {r} (t),a\leq t\leq b,\mathbf {r} (a)=\mathbf {r_{0}} ,\mathbf {r} (b)=\mathbf {r} .} The fact that the line integral depends on the path C only through its terminal points r 0 and r is, in essence, the path independence property of

200-475: A bow wake is created when a watercraft moves through the medium; as the medium cannot be compressed, it must be displaced instead, resulting in a wave. As with all wave forms , it spreads outward from the source until its energy is overcome or lost, usually by friction or dispersion . The non-dimensional parameter of interest is the Froude number . Waterfowl and boats moving across the surface of water produce

250-400: A contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of U always points straight downwards in

300-451: A wake may either be: The wake is the region of disturbed flow (often turbulent ) downstream of a solid body moving through a fluid, caused by the flow of the fluid around the body. For a blunt body in subsonic external flow, for example the Apollo or Orion capsules during descent and landing, the wake is massively separated and behind the body is a reverse flow region where the flow

350-400: A conservative vector field. The fundamental theorem of line integrals implies that if V is defined in this way, then F = –∇ V , so that V is a scalar potential of the conservative vector field F . Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral,

400-475: A contour map, the gradient is inversely proportional to Δ x , which is not similar to force F P : altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map. In fluid mechanics , a fluid in equilibrium, but in

450-405: A moving boat converts the boat's mechanical energy into not only heat and sound energy, but also wave energy at the edges of its wake . These and other energy losses are irreversible because of the second law of thermodynamics . A direct consequence of the closed path test is that the work done by a conservative force on a particle moving between any two points does not depend on the path taken by

500-515: A scalar potential only in the special case when it is a Laplacian field . Certain aspects of the nuclear force can be described by a Yukawa potential . The potential play a prominent role in the Lagrangian and Hamiltonian formulations of classical mechanics . Further, the scalar potential is the fundamental quantity in quantum mechanics . Not every vector field has a scalar potential. Those that do are called conservative , corresponding to

550-512: A time-independent magnetic field, see Faraday's law of induction for details) are examples of conservative forces, while friction and air drag are classical examples of non-conservative forces. For non-conservative forces, the mechanical energy that is lost (not conserved) has to go somewhere else, by conservation of energy . Usually the energy is turned into heat , for example the heat generated by friction. In addition to heat, friction also often produces some sound energy. The water drag on

SECTION 10

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600-513: A wake pattern, first explained mathematically by Lord Kelvin and known today as the Kelvin wake pattern . The above describes an ideal wake, where the body's means of propulsion has no other effect on the water. In practice the wave pattern between the V-shaped wavefronts is usually mixed with the effects of propeller backwash and eddying behind the boat's (usually square-ended) stern. The Kelvin angle

650-460: Is continuous and vanishes asymptotically to zero towards infinity, decaying faster than 1/ r and if the divergence of E likewise vanishes towards infinity, decaying faster than 1/ r . Written another way, let Γ ( r ) = 1 4 π 1 ‖ r ‖ {\displaystyle \Gamma (\mathbf {r} )={\frac {1}{4\pi }}{\frac {1}{\|\mathbf {r} \|}}} be

700-742: Is a fundamental concept in vector analysis and physics (the adjective scalar is frequently omitted if there is no danger of confusion with vector potential ). The scalar potential is an example of a scalar field . Given a vector field F , the scalar potential P is defined such that: F = − ∇ P = − ( ∂ P ∂ x , ∂ P ∂ y , ∂ P ∂ z ) , {\displaystyle \mathbf {F} =-\nabla P=-\left({\frac {\partial P}{\partial x}},{\frac {\partial P}{\partial y}},{\frac {\partial P}{\partial z}}\right),} where ∇ P

750-518: Is also derived for the case of deep water in which the fluid is not flowing in different speed or directions as a function of depth ("shear"). In cases where the water (or fluid) has shear, the results may be more complicated. Also, the deep water model neglects surface tension, which implies that the wave source is large compared to capillary length . "No wake zones" may prohibit wakes in marinas , near moorings and within some distance of shore in order to facilitate recreation by other boats and reduce

800-717: Is an infinitesimal volume element with respect to r' . Then E = − ∇ Φ = − 1 4 π ∇ ∫ R 3 div ⁡ E ( r ′ ) ‖ r − r ′ ‖ d V ( r ′ ) {\displaystyle \mathbf {E} =-\mathbf {\nabla } \Phi =-{\frac {1}{4\pi }}\mathbf {\nabla } \int _{\mathbb {R} ^{3}}{\frac {\operatorname {div} \mathbf {E} (\mathbf {r} ')}{\left\|\mathbf {r} -\mathbf {r} '\right\|}}\,dV(\mathbf {r} ')} This holds provided E

850-566: Is given by Φ ( r ) = 1 4 π ∫ R 3 div ⁡ E ( r ′ ) ‖ r − r ′ ‖ d V ( r ′ ) {\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\operatorname {div} \mathbf {E} (\mathbf {r} ')}{\left\|\mathbf {r} -\mathbf {r} '\right\|}}\,dV(\mathbf {r} ')} where dV ( r' )

900-811: Is moving toward the body. This phenomenon is often observed in wind tunnel testing of aircraft, and is especially important when parachute systems are involved, because unless the parachute lines extend the canopy beyond the reverse flow region, the chute can fail to inflate and thus collapse. Parachutes deployed into wakes suffer dynamic pressure deficits which reduce their expected drag forces. High-fidelity computational fluid dynamics simulations are often undertaken to model wake flows, although such modeling has uncertainties associated with turbulence modeling (for example RANS versus LES implementations), in addition to unsteady flow effects. Example applications include rocket stage separation and aircraft store separation. In incompressible fluids (liquids) such as water,

950-732: Is not even defined (the force is not a vector field, so one cannot evaluate its curl). Accordingly, some authors classify the magnetic force as conservative, while others do not. The magnetic force is an unusual case; most velocity-dependent forces, such as friction , do not satisfy any of the three conditions, and therefore are unambiguously nonconservative. Despite conservation of total energy, non-conservative forces can arise in classical physics due to neglected degrees of freedom or from time-dependent potentials. Many non-conservative forces may be perceived as macroscopic effects of small-scale conservative forces. For instance, friction may be treated without violating conservation of energy by considering

1000-470: Is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x, y, z . In some cases, mathematicians may use a positive sign in front of the gradient to define the potential. Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude

1050-479: Is the angle of inclination, and the component of F S perpendicular to gravity is F P = − m g   sin ⁡ θ   cos ⁡ θ = − 1 2 m g sin ⁡ 2 θ . {\displaystyle \mathbf {F} _{\mathrm {P} }=-mg\ \sin \theta \ \cos \theta =-{1 \over 2}mg\sin 2\theta .} This force F P , parallel to

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1100-436: Is the conservative force, U {\displaystyle U} is the potential energy, and s {\displaystyle s} is the position. Informally, a conservative force can be thought of as a force that conserves mechanical energy . Suppose a particle starts at point A, and there is a force F acting on it. Then the particle is moved around by other forces, and eventually ends up at A again. Though

1150-1183: Is the divergence of the convolution of E with Γ : Φ = div ⁡ ( E ∗ Γ ) . {\displaystyle \Phi =\operatorname {div} (\mathbf {E} *\Gamma ).} Indeed, convolution of an irrotational vector field with a rotationally invariant potential is also irrotational. For an irrotational vector field G , it can be shown that ∇ 2 G = ∇ ( ∇ ⋅ G ) . {\displaystyle \nabla ^{2}\mathbf {G} =\mathbf {\nabla } (\mathbf {\nabla } \cdot {}\mathbf {G} ).} Hence ∇ div ⁡ ( E ∗ Γ ) = ∇ 2 ( E ∗ Γ ) = E ∗ ∇ 2 Γ = − E ∗ δ = − E {\displaystyle \nabla \operatorname {div} (\mathbf {E} *\Gamma )=\nabla ^{2}(\mathbf {E} *\Gamma )=\mathbf {E} *\nabla ^{2}\Gamma =-\mathbf {E} *\delta =-\mathbf {E} } as required. More generally,

1200-451: Is the rate of that decrease per unit length. In order for F to be described in terms of a scalar potential only, any of the following equivalent statements have to be true: The first of these conditions represents the fundamental theorem of the gradient and is true for any vector field that is a gradient of a differentiable single valued scalar field P . The second condition is a requirement of F so that it can be expressed as

1250-850: Is the volume of the unit n -ball. The proof is identical. Alternatively, integration by parts (or, more rigorously, the properties of convolution ) gives Φ ( r ) = − 1 n ω n ∫ R n E ( r ′ ) ⋅ ( r − r ′ ) ‖ r − r ′ ‖ n d V ( r ′ ) . {\displaystyle \Phi (\mathbf {r} )=-{\frac {1}{n\omega _{n}}}\int _{\mathbb {R} ^{n}}{\frac {\mathbf {E} (\mathbf {r} ')\cdot (\mathbf {r} -\mathbf {r} ')}{\|\mathbf {r} -\mathbf {r} '\|^{n}}}\,dV(\mathbf {r} ').} Wake (physics) In fluid dynamics ,

1300-606: The Newtonian potential . This is the fundamental solution of the Laplace equation , meaning that the Laplacian of Γ is equal to the negative of the Dirac delta function : ∇ 2 Γ ( r ) + δ ( r ) = 0. {\displaystyle \nabla ^{2}\Gamma (\mathbf {r} )+\delta (\mathbf {r} )=0.} Then the scalar potential

1350-450: The electric force (in a time-independent magnetic field, see Faraday's law ), and spring force . Many forces (particularly those that depend on velocity) are not force fields . In these cases, the above three conditions are not mathematically equivalent. For example, the magnetic force satisfies condition 2 (since the work done by a magnetic field on a charged particle is always zero), but does not satisfy condition 3, and condition 1

1400-480: The electromagnetic four-potential . If F is a conservative vector field (also called irrotational , curl -free , or potential ), and its components have continuous partial derivatives , the potential of F with respect to a reference point r 0 is defined in terms of the line integral : V ( r ) = − ∫ C F ( r ) ⋅ d r = − ∫

1450-411: The ambiguity of V reflects the freedom in the choice of the reference point r 0 . An example is the (nearly) uniform gravitational field near the Earth's surface. It has a potential energy U = m g h {\displaystyle U=mgh} where U is the gravitational potential energy and h is the height above the surface. This means that gravitational potential energy on

1500-425: The damage wakes cause. Powered narrowboats on British canals are not permitted to create a breaking wash (a wake large enough to create a breaking wave ) along the banks, as this erodes them. This rule normally restricts these vessels to 4 knots (4.6 mph; 7.4 km/h) or less. Wakes are occasionally used recreationally. Swimmers, people riding personal watercraft, and aquatic mammals such as dolphins can ride

1550-503: The direction of gravity; F . However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface: F S = − m g   sin ⁡ θ {\displaystyle \mathbf {F} _{\mathrm {S} }=-mg\ \sin \theta } where θ

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1600-401: The direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the surface, which can be characterized as the plane of zero pressure. If the liquid has a vertical vortex (whose axis of rotation is perpendicular to

1650-420: The energy from the large-scale motion of the bodies to small-scale movements in their interior, and therefore appear non-conservative on a large scale. General relativity is non-conservative, as seen in the anomalous precession of Mercury's orbit. However, general relativity does conserve a stress–energy–momentum pseudotensor . Scalar potential In mathematical physics , scalar potential describes

1700-561: The force is not conservative, then defining a scalar potential is not possible, because taking different paths would lead to conflicting potential differences between the start and end points. Gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force. Other examples of conservative forces are: force in elastic spring , electrostatic force between two electric charges, and magnetic force between two magnetic poles. The last two forces are called central forces as they act along

1750-642: The formula Φ = div ⁡ ( E ∗ Γ ) {\displaystyle \Phi =\operatorname {div} (\mathbf {E} *\Gamma )} holds in n -dimensional Euclidean space ( n > 2 ) with the Newtonian potential given then by Γ ( r ) = 1 n ( n − 2 ) ω n ‖ r ‖ n − 2 {\displaystyle \Gamma (\mathbf {r} )={\frac {1}{n(n-2)\omega _{n}\|\mathbf {r} \|^{n-2}}}} where ω n

1800-413: The gradient of a scalar function. The third condition re-expresses the second condition in terms of the curl of F using the fundamental theorem of the curl . A vector field F that satisfies these conditions is said to be irrotational (conservative). Scalar potentials play a prominent role in many areas of physics and engineering. The gravity potential is the scalar potential associated with

1850-486: The gravity per unit mass, i.e., the acceleration due to the field, as a function of position. The gravity potential is the gravitational potential energy per unit mass. In electrostatics the electric potential is the scalar potential associated with the electric field , i.e., with the electrostatic force per unit charge . The electric potential is in this case the electrostatic potential energy per unit charge. In fluid dynamics , irrotational lamellar fields have

1900-687: The ground, is greatest when θ is 45 degrees. Let Δ h be the uniform interval of altitude between contours on the contour map, and let Δ x be the distance between two contours. Then θ = tan − 1 ⁡ Δ h Δ x {\displaystyle \theta =\tan ^{-1}{\frac {\Delta h}{\Delta x}}} so that F P = − m g Δ x Δ h Δ x 2 + Δ h 2 . {\displaystyle F_{P}=-mg{\Delta x\,\Delta h \over \Delta x^{2}+\Delta h^{2}}.} However, on

1950-420: The leading edge of a wake. In the sport of wakeboarding the wake is used as a jump. The wake is also used to propel a surfer in the sport of wakesurfing . In the sport of water polo , the ball carrier can swim while advancing the ball, propelled ahead with the wake created by alternating armstrokes in crawl stroke , a technique known as dribbling . Furthermore, in the sport of canoe marathon, competitors use

2000-401: The line joining the centres of two charged/magnetized bodies. A central force is conservative if and only if it is spherically symmetric. For conservative forces, F c = − dU d s {\displaystyle \mathbf {F_{c}} =-{\frac {\textit {dU}}{d\mathbf {s} }}} where F c {\displaystyle F_{c}}

2050-410: The motion of individual molecules; however, that means every molecule's motion must be considered rather than handling it through statistical methods. For macroscopic systems the non-conservative approximation is far easier to deal with than millions of degrees of freedom. Examples of non-conservative forces are friction and non-elastic material stress . Friction has the effect of transferring some of

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2100-452: The negative pressure gradient along the surface of the object: F B = − ∮ S ∇ p ⋅ d S . {\displaystyle F_{B}=-\oint _{S}\nabla p\cdot \,d\mathbf {S} .} In 3-dimensional Euclidean space ⁠ R 3 {\displaystyle \mathbb {R} ^{3}} ⁠ , the scalar potential of an irrotational vector field E

2150-484: The notion of conservative force in physics. Examples of non-conservative forces include frictional forces, magnetic forces, and in fluid mechanics a solenoidal field velocity field. By the Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding vector potential . In electrodynamics, the electromagnetic scalar and vector potentials are known together as

2200-447: The particle may still be moving, at that instant when it passes point A again, it has traveled a closed path. If the net work done by F at this point is 0, then F passes the closed path test. Any force that passes the closed path test for all possible closed paths is classified as a conservative force. The gravitational force , spring force , magnetic force (according to some definitions, see below) and electric force (at least in

2250-426: The particle, taking path 1 from A to B and then path 2 backwards from B to A, is 0; thus, the work is the same in path 1 and 2, i.e., the work is independent of the path followed, as long as it goes from A to B. For example, if a child slides down a frictionless slide, the work done by the gravitational force on the child from the start of the slide to the end is independent of the shape of the slide; it only depends on

2300-423: The particle. This is illustrated in the figure to the right: The work done by the gravitational force on an object depends only on its change in height because the gravitational force is conservative. The work done by a conservative force is equal to the negative of change in potential energy during that process. For a proof, imagine two paths 1 and 2, both going from point A to point B. The variation of energy for

2350-408: The presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This buoyant force is the negative gradient of pressure : f B = − ∇ p . {\displaystyle \mathbf {f_{B}} =-\nabla p.} Since buoyant force points upwards, in

2400-408: The situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space : a directionless value ( scalar ) that depends only on its location. A familiar example is potential energy due to gravity . A scalar potential

2450-454: The surface), then the vortex causes a depression in the pressure field. The surface of the liquid inside the vortex is pulled downwards as are any surfaces of equal pressure, which still remain parallel to the liquids surface. The effect is strongest inside the vortex and decreases rapidly with the distance from the vortex axis. The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating

2500-438: The vertical displacement of the child. A force field F , defined everywhere in space (or within a simply-connected volume of space), is called a conservative force or conservative vector field if it meets any of these three equivalent conditions: The term conservative force comes from the fact that when a conservative force exists, it conserves mechanical energy. The most familiar conservative forces are gravity ,

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