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00­Mathematics_for_this_course

Measured in radians, defines angle (in radians), where s is arclength and r is radius.The circumference of a circle is and the circle's area is is its area. The

surface area of a sphere is and sphere's volume is

A vector can be expressed as, , where , and are the x and y components. Alternative notation for the unit vectors

include and . An important vector is the displacement from theorigin, with components are typically written without subscripts: . Themagnitude (or absolute value or norm) of a vector is is ,

where the angle (or phase), , obeys , or (almost) equivalently, . As with anyfunction/inverse function pair, the tangent and arctangent are related by where . Thearctangent is not a true function because it is multivalued, with .

The geometric interpretations of and are shown in the figure.Vector addition and subtraction can also be defined through the components:

AND

01­Introduction

Text Symbol Factor Exponent

giga G 1 000 000 000 E9mega M 1 000 000 E6kilo k 1 000 E3(none) (none) 1 E0centi c 0.01 E−2milli m 0.001 E−3micro μ 0.000 001 E−6nano n 0.000 000 001 E−9pico p 0.000 000 000 001 E−12

1 kilometer = .621 miles and 1 MPH = 1 mi/hr ≈ .447 m/sTypically air density is 1.2kg/m3, with pressure 105Pa. The density of water is 1000kg/m3.

Earth's mean radius ≈ 6371km, mass ≈ 6 ×1024 kg, and gravitational acceleration = g ≈ 9.8m/s2

Universal gravitational constant = G ≈ 6.67 ×10−11

m3·kg−1·s−2

Speed of sound ≈ 340m/s and the speed of light = c ≈ 3×108m/s

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One light­year ≈ 9.5×1015m ≈ 63240AU (Astronomical unit)The electron has charge, e ≈ 1.6 × 10−19C and mass ≈ 9.11 × 10­31kg. 1eV = 1.602 × 10­19J is a unit of energy,defined as the work associated with moving one electron through a potential difference of one volt.1 amu = 1 u ≈ 1.66 × 10­27 kg is the approximate mass of a proton or neutron.Boltzmann's constant = kB≈ 1.38 × 10­23 J K−1 , and the gas constant is R = NAkB≈8.314 J  K−1  mol−1,where NA≈ 6.02 × 1023 is the Avogadro number.

≈ 8.987× 109 N·m²·C−2 is a fundamental constant of electricity; also ≈ 8.854 × 10−12

F·m−1 is the vacuum permittivity. = 4π × 10−7 NA ≈ 1.257 × 10−6 N A (magnetic permeability) is the fundamental constant of

magnetism: .

= h/(2π) ≈ 1.054×10−34 J·s the reduced Planck constant, and

≈ .526 × 10−10 m is the Bohr radius.

Two dimensional kinematics

Difference is denoted by , , or the Delta. or . Average, or mean, is denoted by , where is number and are probabilities. The average velocity is

, and the average acceleration is , where denotes position. In CALCULUS, instantaneous valuesare denoted by v(t)=dx/dt and a=dv/dt=d2x/dt2.

The equations of motion for uniform acceleration are: , and, . Also, , and, . Note that only if the acceleration is uniform.

03­Two­Dimensional_Kinematics

...in advanced notation this becomes .

In free fall we often set, ax=0 and ay= ­g. If angle is measured with respect to the x axis:

The figure shows a Man moving relative to Train withvelocity, , where the velocity of the train relative toEarth is, is the velocity of the Train relative to Earth.The velocity of the Man relative to Earth is,

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If the speeds are relativistic, define u=v/c where c is the speed of light, and this formula

must be modified to:

04­Dynamics:_Force_and_Newton's_Laws

Newton's laws of motion, can be expressed with two equations,

and . The second represents the fact that the force that the i­thobject exerts one object exerts on the j­th object is equal and opposite the forcethat the j­th exerts on the i­th object. Three non­fundamental fores are:

1. The normal force, , is a contact forces that is perpendicular to thesurface,

2. The force of friction, , is a contact force that is parallel to the surface.3. Tension, , is often associated with ropes and strings. If the rope hassufficiently low weight and of all external forces act at the two ends,then this tension is distributed uniformly along the rope.

4. The fourth force is fundamental: Weight equals , and is the force ofgravity acting on an object of mass, . At Earth's surface,

.

The x and y components of the three forces of tension on the small greycircle where the three "massless" ropes meet are:

, ,

,

05­Friction,_Drag,_and_Elasticity

is an approximation for the force friction when an object is sliding on a surface, where μk ("mew­sub­k")is the kinetic coefficient of friction, and N is the normal force.

approximates the maximum possible friction (called static friction) that can occur before the objectbegins to slide. Usually μs > μk.

Also, air drag often depends on speed, an effect this model fails to capture.

06­Uniform_Circular_Motion_and_Gravitation

relates the radian, degree, and revolution.

is the number of revolutions per second, called frequency.

is the number of seconds per revolution, called period. Obviously .

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uniform circular motion (here theLatin d was used instead of theGreek Δ

is called angular frequency (ω is called omega, and θ is

measured in radians). Obviously

is the acceleration of uniform circular motion,

where v is speed, and r is radius., where T is period.

is the force of gravity between two objects, where the

universal constant of gravity is G ≈ 6.674 × 10­11 m3·kg−1·s−2.

07­Work_and_Energy

is kinetic energy, where m is mass and v is speed..

is gravitational potential energy,where y is height, and is the gravitational acceleration at

Earth's surface.

is the potential energy stored in a spring with spring constant .

relates the final energy to the initial energy. If energy is lost toheat or other nonconservative force, then Q>0.

(measured in Joules) is the work done by a force as it moves an object a distance . Theangle between the force and the displacement is θ.

describes the work if the force is not uniform. The steps, , taken by the particle are assumed smallenough that the force is approximately uniform over the small step. If force and displacement are parallel, then thework becomes the area under a curve of F(x) versus x.

is the power (measured in Watts) is the rate at which work is done. (v is velocity.)

08­Linear_Momentum_and_Collisions

is momentum, where m is mass and is velocity. The net momemtum is conserved if all forces between asystem of particles are internal (i.e., come equal and opposite pairs):

. is the impulse, or change in momentum associated with a brief force acting over a time interval .

(Strictly speaking, is a time­averaged force defined by integrating over the time interval.)

09­Statics_and_Torque

is the torque caused by a force, F, exerted at a distance ,r, from the axis. The angle between r and Fis θ.

The SI units for torque is the newton metre (N·m). It would be inadvisable to call this a Joule, even though a Joule is also a(N·m). The symbol for torque is typically τ, the Greek letter tau. When it is called moment, it is commonly denoted M.[1]The lever arm is defined as either r, or r⊥. Labeling r as the lever arm allows moment arm to be reserved for r⊥.

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10­Rotational_Motion_and_AngularMomentum

Linear motion Angular motion

The following table refers to rotation of a rigid body about a fixed axis: is arclength, is the distance from the axis to anypoint, and is the tangential acceleration, which is the component of the acceleration that is parallel to the motion. Incontrast, the centripetal acceleration, , is perpendicular to the motion. The component of the forceparallel to the motion, or equivalently, perpendicular, to the line connecting the point of application to the axis is . Thesum is over particles or points of application.

Analogy between Linear Motion and Rotational motion[2]

Linear motion Rotational motion Defining equationDisplacement = Angular displacement = Velocity = Angular velocity =

Acceleration = Angular acceleration =

Mass = Moment of Inertia =

Force = Torque =

Momentum= Angular momentum=

Kinetic energy = Kinetic energy =

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Pressure is theweight per unit areaof the fluid above apoint.

Description[3] Figure Moment(s) of inertia

Rod of length L and mass m(Axis of rotation at the end of the rod)

Solid cylinder of radius r, height h and mass m

Sphere (hollow) of radius r and mass m

Ball (solid) of radius r and mass m

11­Fluid_statics

Pressure versus Depth: A fluid's pressure is F/A where F is force and A is a (flat) area. Thepressure at depth, below the surface is the weight (per area) of the fluid above that point. Asshown in the figure, this implies:

where is the pressure at the top surface, is the depth, and is the mass density of the fluid.In many cases, only the difference between two pressures appears in the final answer to aquestion, and in such cases it is permissible to set the pressure at the top surface of the fluid equalto zero. In many applications, it is possible to artificially set equal to zero, for example atatmospheric pressure. The resulting pressure is called the gauge pressure, for below the surface of a bodyof water.

Buoyancy and Archimedes' principle Pascal's principle does not hold if two fluids are separated by a seal that prohibitsfluid flow (as in the case of the piston of an internal combustion engine). Suppose the upper and lower fluids shown in thefigure are not sealed, so that a fluid of mass density comes to equilibrium above and below an object. Let the objecthave a mass density of and a volume of , as shown in the figure. The net (bottom minus top) force on the objectdue to the fluid is called the buoyant force:

,

and is directed upward. The volume in this formula, AΔh, is called the volume of the displaced fluid, since placing thevolume into a fluid at that location requires the removal of that amount of fluid. Archimedes principle states:

A body wholly or partially submerged in a fluid is buoyed up by a force equal to the weight of thedisplaced fluid.

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A fluid element speeds up ifthe area is constricted.

Note that if , the buoyant force exactly cancels the force of gravity. A fluid element within a stationary fluid willremain stationary. But if the two densities are not equal, a third force (in addition to weight and the buoyant force) isrequired to hold the object at that depth. If an object is floating or partially submerged, the volume of the displaced fluidequals the volume of that portion of the object which is below the waterline.

12­Fluid_dynamics

the volume flow for incompressible fluid flow if viscosity

and turbulence are both neglected. The average velocity is and is the crosssectional area of the pipe. As shown in the figure, because isconstant along the developed flow. To see this, note that the volume of pipe is

along a distance . And, is the volume of fluid thatpasses a given point in the pipe during a time .

is Bernoulli's equation, where is

pressure, is density, and is height. This holds for inviscid flow.

13­Temperature,_Kinetic Theory,_and_Gas_Laws

converts from Celsius to Kelvins, and converts from Celsius to Fahrenheit.

is the ideal gas law, where P is pressure, V is volume, n is the number of moles and N isthe number of atoms or molecules. Temperature must be measured on an absolute scale (e.g. Kelvins).NAkB=R where NA= 6.02 × 1023 is the Avogadro number. Boltzmann's constant can also be written in eV andKelvins: kB ≈8.6 × 10­5 eV/deg.

is the average translational kinetic energy per "atom" of a 3­dimensional ideal gas.

is the root­mean­square speed of atoms in an ideal gas.

is the total energy of an ideal gas, where

14­Heat_and_Heat_Transfer

Here it is convenient to define heat as energy that passes between two objects of different temperature The SI unit is theJoule. The rate of heat trasfer, or is "power": 1 Watt = 1 W = 1J/s

is the heat required to change the temperature of a substance of mass, m. The change in temperatureis ΔT. The specific heat, cS, depends on the substance (and to some extent, its temperature and other factors suchas pressure). Heat is the transfer of energy, usually from a hotter object to a colder one. The units of specfic heat areenergy/mass/degree, or J/(kg­degree).

is the heat required to change the phase of a a mass, m, of a substance (with no change in temperature).The latent heat, L, depends not only on the substance, but on the nature of the phase change for any givensubstance. LF is called the latent heat of fusion, and refers to the melting or freezing of the substance. LV is called thelatent heat of vaporization, and refers to evaporation or condensation of a substance.

is rate of heat transfer for a material of area, A. The difference in temperature between two sides

separated by a distance, d, is . The thermal conductivity, kc, is a property of the substance used to insulate, orsubdue, the flow of heat.

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is the power radiated by a surface of area, A, at a temperature, T, measured on an absolute scalesuch as Kelvins. The emissivity, , is 1 for a black body, and 0 for a perfectly reflecting surface. TheStefan­Boltzmann constant is .

15­Thermodynamics

Pressure (P), Energy (E), Volume (V), and Temperature (T) are statevariables (state functionscalled state functions). The number of particles(N) can also be viewed as a state variable.Work (W), Heat (Q) are not state variables.

, is the entropy of an

ideal , monatomic gas. The constant is arbitrary only in classical (non­quantum) thermodynamics. Since it is afunction of state variables, entropy is also a state function.

A point on a PV diagram define's the system's pressure (P) and volume (V).Energy (E) and pressure (P) can be deduced from equations of state:E=E(V,P) and T=T(V,P). If the piston moves, or if heat is added or taken from thesubstance, energy (in the form of work and/or heat) is added or subtracted. If thepath returns to its original point on the PV­diagram (e.g., 12341 along therectantular path shown), and if the process is quasistatic, all state variables(P, V, E, T) return to their original values, and the final system is indistinguishablefrom its original state.

The net work done per cycle is area enclosed by the loop. This work equals thenet heat flow into the system, (valid only for closed loops).

Remember: Area "under" is the work associated with a path; Area "inside" is the total workper cycle.

is the work done on a system of

pressure P by a piston of voulume V. If ΔV>0 the substance is expanding as it exerts an outward force, so thatΔW<0 and the substance is doing work on the universe; ΔW>0 whenever the universe is doing work on the system.

is the amount of heat (energy) that flows into a system. It is positive if the system is placed in a heat bath ofhigher temperature. If this process is reversible, then the heat bath is at an infinitesimally higher temperature and afinite ΔQ takes an infinite amount of time.

is the change in energy (First Law of Thermodynamics).

CALCULUS: .

In an isothermal expansion (contraction), temperature, T, is constant. Hence P=nRT/V and substitution yields,

16­Oscillatory_Motion_and_Waves

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describes oscillatory motion with period (here we use the zero­subscript to denote constants

that do not vary with time).. For example, .

for a mass­spring system with mass, m, and spring constant, ks.

for a low amplitude pendulum of length, L, in a gravitational field, g.

is the potential energy of a mass spring system.

Let describe position:

, where is maximum velocity., where , is maximum

acceleration., relates maximum force to maximum acceleration.

is the total energy.

CALCULUS: x(t) obeys the linear homogeneous differential equation (ODE),

relates the frequency, f, wavelength, λ,and the the phase speed, vp of thewave (also written as vw) This phase speed is the speed of individual crests, whichfor sound and light waves also equals the speed at which a wave packet travels.

describes the n­th normal mode vibrating wave on a string that is fixed

at both ends (i.e. has a node at both ends). The mode number, n = 1, 2, 3,..., asshown in the figure.Beat frequency: The frequency of beats heard if two closely space frequencies, and , are played is

.Musical acoustics: Frequency ratios of 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 8/5 are called the (just) "octave", "fifth", "fourth","major­sixth", "major­third", "minor­third", and "minor­sixth", respectively.

17­Physics_of_Hearing

is the the approximate speed near Earth's surface, where the temperature, T, is measured in

Kelvins. A theoretical calculation is where for a semi­classical gas with degrees of

freedom. For a diatomic gas such as Nitrogen, γ = 1.4.

is the speed of a wave in a stretched string if is the tension and is the linear mass density (kilograms

per meter).

18­Electric_charge_and_field

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is Coulomb's law for the force between two charged particles separated by a distance r:

ke≈8.987×109N·m²·C−2, and ε0≈8.854×10−12 F·m−1.

is the electric force on a "test charge", q, where is the magnitude of the electric field situated

a distance r from a charge, Q.

Consider a collection of particles of charge , located at points (called source points), the electric field at (calledthe field point) is:

is the electric field at the field point, , due to point charges at the

source points, , and points from source points to the field point.

CALCULUS supplement:

is the electric field due to distributed charge, where

, and denote linear, surface, andvolume density (or charge density), respectively.

Cartesian coordinates (x, y, z). Volume element: . Lineelement: . Three basic area elements: ,or, , or, .

Cylindrical coordinates (ρ, φ, z): Volume element: .Line element: . Basic area elements:

(side), and, (top end).

  Spherical coordinates (r, θ, φ): Volume element: (if symmetry holds). Lineelement: . Basic area element of a sphere: , where dΩ is a solid angle.

19­Electric_Potential_and_Electric_Field

is the potential energy of a particle of charge, q, in the presence of an electric potential V. (measured in Volts) is the variation in electric potential as one moves through an electric

field . The angle between the field and the displacement is θ. The electric potential, V, decreases as one movesparallel to the electric field.

describes the electric potential if the field is not uniform.

due to a set of charges at where .

is the (equal and opposite) charge on the two terminals of a capacitor of capicitance, C, that has a voltagedrop, V, across the two terminals.

is the capacitance of a parallel plate capacitor with surface area, A, and plate separation, d. Thisformula is valid only in the limit that d2/A vanishes. If a dielectric is between the plates, then ε>ε0≈ 8.85 × 10−12 dueto shielding of the applied electric field by dielectric polarization effects.

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closed surfaces | Ω & ∂ΩTo the left are closed surfaces. Tothe right are open surfaces, Ω,that possess closed boundaries,∂Ω.

is the energy stored in a capacitor.

is the energy density (energy per unit volume, or Joules per cubic meter) of an electric field.

CALCULUS supplement

is the gradient theorem.

is Stokes' theorem

is the divergence theorem

Here, Ω is a (3­dimensional) volume and ∂Ω is the boundary of the volume, which isa (two­dimensional) surface. Also a surface is Σ, which, if open, has the boundary∂Σ, which is a (one­dimensional) curve.

in the limit that the Riemann sum becomes an

integral. where is the del operator.

is Gauss's law for the surface integral of the electric field over any closed surface, and

is the total charge inside that surface.

is a useful variant if the medium is dielectric. D=εE is the electric displacement field. The

permittivity, ε = (1+χ)ε0, where ε0≈ 8.85 × 10−12, and the electric susceptibility, χ, represents the degree to whichthe medium can be polarized by an electric field. The free charge, Qfree, represents all charges except thoserepresented by the susceptibility, χ.

20­Electric_Current,_Resistance,_and_Ohm's_Law

defines the electric current as the rate at which charge flows past a given point on a wire. The direction of

the current matches the flow of positive charge (which is opposite the flow of electrons if electrons are the carriers.) is Ohm's Law relating current, I, and resistance, R, to the difference in voltage, V, between the terminals.

The resistance, R, is positive in virtually all cases, and if R > 0, the current flows from larger to smaller voltage. Anydevice or substance that obeys this linear relation between I and V is called ohmic.

relates the density (n), the charge(q), and the average drift velocity (vdrift) of the carriers. The area(A) is measured by imagining a cut across the wire oriented such that the drift velocity is perpendicular to the surfaceof the (imaginary) cut.

expresses the resistance of a sample of ohmic material with a length (L) and area (A). The 'resistivity', ρ

("row"), is an intensive property of matter.

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The current enteringany junction is equalto the current leavingthat junction. i2 + i3 =i1 + i4

The sum of all the voltagesaround the loop is equal tozero. v1 + v2 + v3 ­ v4 = 0

Resistors in parallel

Resistors in series

voltage divider

In this example, we assumethat the rectangular element isa resistor, R, and that theinternal resistance of thevoltage source (not shown) isalso R. The ammeter andvoltmeter shown are ideal.

thumb

Power is energy/time, measured in joules/second or J/s. Often called P (never p). It is measured in watts (W)Current is charge/time, measured in coulombs/second or C/s. Often called I or i. It is measured in amps or ampheres(A)Electric potential (or voltage) is energy/charge, measured in joules/coulomb or J/C. Often called V (sometimes E,emf, ). It is measured in volts (V)Resistance is voltage/current , measured in volts/amp or V/A. Often called R (sometimes r, Z) It is measured inOhms (Ω).

is the power dissipated as current flows through a resistor

21­Circuits,_Bioelectricity,_and_DC_Instruments

and are Kirchoff's Laws[4]

for the voltage divider shown.

Simple RC circuit[5] The figure to the right depicts acapacitor being charged by an ideal voltage source. If, at t=0,the switch is thrown to the other side, the capacitor willdischarge, with thevoltage, V ,undergoingexponential decay:

where V0 is thecapacitor voltage attime t = 0 (when theswitch was closed).

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The time required for the voltage to fall to is called the RC time constant and is given by

22­Magnetism

is the force on a particle with charge q moving at velocity v with in the presence of a magnetic fieldB. The angle between velocity and magnetic field is θ and the force is perpeduclar to both velocity and magneticfield by the right hand rule.

expresses this result as a cross product. is the force a straight wire segment of length carrying a current, I. expresses thus sum over many segments to model a wire.

CALCULUS: In the limit that we have the integral, .

Defining magnetic force and field without calculus:

1. is the magnetic field at a distance r from an infinitely long wire carrying a

current, where μ0 = 4π × 10−7 N A. This field points azimuthally around the wire in adirection defined by the right hand rule. Application of the force law on a current element,we have

2. is the force between two long wires of length separated by a short distance . The

currents are I1 and I2, with the force being attractive if the currents are flowing in the same direction.

Cyclotron motion: For a particle moving perpendicular to B, we have cyclotron motion. Recall that for uniform circularmotion, the acceleration is a=v2/r, where r is the radius. Since sin θ =1, Newton's second law of motion (F=ma) yields,

Since, sin θ =0, for motion parallel to a magnetic field, particles in a uniform magnetic field move in spirals at a radius whichis determined by the perpendicular component of the velocity:

Hall effect: The Hall effect occurs when the magnetic field, velocity, and electric field are mutually perpendicular. In thiscase, the electric and magnetic forces are aligned, and can cancel if qE=qvB (since sinθ = 1). Since both terms areporportional to charge, q, the appropriate ratio of electric to magnetic field for null net force depends only on velocity:

,

where we have used the fact that voltage (i.e. emf or potential) is related to the electric field and a displacement parallel tothat field: ΔV = ­E Δs cosθ

CALCULUS supplement:

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In rod's frame the force oncarriers is electric, not magnetic.(See railgun)

and the volume integral , where is current density.

is Ampere's law relating a closed integral involving magnetic field to the total current enclosed

by that path.

23­Electromagnetic_Induction,_AC_Circuits,_and_Electrical_Technologies

is a consequence of the magnetic force law as seen in thereference frame of a moving charged object, where E is the electric fieldperceived by an observer moving at velocity v in the presence of a magneticvield, B. Also written as, E = vBsinθ, this can be used to derive Faraday'slaw of induction. (Here, θ is the angle between the velocity and the magneticfield.)

is the magnetic flux, where θ is the angle betweenthe magnetic field and the normal to a surface of area, A.

is Faraday's law where t is time and N is the number of

turns. The minus sign reminds us that the emf, or electromotive force, actsas a "voltage" that opposes the change in the magnetic field or flux.

24­Electromagnetic_Waves

is called the displacement current because it replaces thecurrent density when using Ampère's circuital law to calculate the lineintegral of the magnetic field around a closed loop.

Maxwell's equations hold for all volumes and closed surfaces. In vacuum,

electromagnetic waves travel at the speed, .

25­Geometric_Optics

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relates the focal

length f of the lens, the imagedistance S1, and the object distanceS2. The figure depicts the situationfor which (S1, S2, f) are all positive:(1)The lens is converging (convex);(2) The real image is to the right ofthe lens; and (3) the object is to theleft of the lens. If the lens isdiverging (concave), then f < 0. Ifthe image is to the left of the lens(virtual image), then S2 < 0 .

27­Wave_Optics

where describes the constructive interference associatedwith two slits in the Fraunhoffer (far field) approximation.

where is the high frequency carrier and

is the slowly varying envelope. Here,

and . Consequently, the beat frequency

heard when two tones of frequency and is .

models the addition of two waves of equal amplitude but different pathlength, .

References

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