Conceptual Physics Summary of Formula Sheet

Summary Units (SI): Length: m = meter Time: s = second Mass: kg = kilogram; atomic mass unit u = 1.661.10-27 kg = m(12C)/12 Velocity: m/s Acceleration: m/s2 Momentum: kg m/s Force: N = Newton = kg m/s2 Energy: J = Joule = Nm = kg m2/s2 = Ws ; eV = electron-Volt = 1.602.10-19 J Power: W = Watt = J/s Charge: C = Coulomb Currents: A = Ampere = C/s Electric Field: N/C = V/m Electric Potential: V = Volt = J/C Electric Resistance: Ω = Ohm = V/A Magnetic Field: T = Tesla = Vs/m2 =10,000 Gauss Amount: mol (1 mol = NA molecules = A gram, where A is atomic mass) Density ρ: kg/m3 Pressure: Pa = Pascal = N/m2; 100,000 Pa ≈ 1 atmosphere Temperature: K = Kelvin (C = Celsius, F = Fahrenheit); o o o o 32 F = 0 C = 273.15 K; 212 F = 100 C = 373.15K Heat: J or 1 calorie = 4.187 J; 1 food calorie = 1000 calories = 4187 J Frequency: Hz = 1/s Prefixes: kilo = k = 103 = 1000 = Thousand Mega = M = 106 = 1,000,000 = Million Giga = G = 109 = 1,000,000,000 = Billion Tera = T = 1012 = Trillion, Peta = P = 1015 = Quadrillion centi = c = 10-2 = 0.01 = 1-hundreth milli = m = 10-3 = 0.001 = 1-thousandth micro = µ = 10-6 = 0.000,001 = 1-millionth nano = n = 10-9, pico = p = 10-12, femto = f = 10-15 Useful Constants: Gravitational constant: G = 6.67.10-11 Nm2/kg2 Gravitational acceleration at surface of Earth: g = 9.81 m/s2 Mass of Sun: 1.99.1030 kg Distance from Earth to Sun: 1.50.1011 m Mass of Moon: 7.35.1022 kg Distance from Earth to Moon: 3.84.108 m Mass of Earth: 5.97.1024 kg Radius of Earth: 6.38.106 m EarthÕs magnetic field: about 0.5 Gauss = 0.0005 T (5.10-5 T) (Magnetic north pole near Australia) Elementary charge: e = 1.602.10-19 C Permittivity constant: εo = 8.854.10-12 F/m Permeability constant: µo = 4π.10-7 H/m k (electrostatic force constant) = 1/4πεo = 8.99.109 Nm2/C2 Speed of Light: c = 2.998.108 m/s = 1/√ εo µo Avogadro's number: NA = 6.022.1023 molecules/mol = number of 12C atoms in 12 g of carbon Universal Gas Constant: R = 8.32 J/mol/K Density of water: 1000kg/m3, air (sea level): 1.25kg/m3, of iron: 7874kg/m3 Atmospheric pressure at sea level: 101,300 Pa Typical speed of sound: 330 m/s Ð 340 m/s in air Frequency of Òmiddle AÓ musical note: 440 Hz Electron mass: me = 9.109.10-31 kg; E = mc2 = 510,999 eV = 511 keV Proton mass: mp = 1.673.10-27 kg; E = mc2 = 938,272,030 eV = 938.3 MeV Neutron mass: mn = 1.675.10-27 kg; E = mc2 = 939,565,360 eV = 939.6 MeV PlanckÕs constant: h = 6.63.10-34 Js = 6.63.10-25 kg m/s x nm = 4.17 .10-15 eV / Hz => 1/h = 1.51.10-33 Hz/J = 2.4.1014Hz/eV; 2 eV corresponds to λ = 620 nm and f = 4.84.1014Hz (yellow light) Natural Science - The Scientific Method 1) Collect observations; categorize and develop detailed descriptions 2) Identify crucial parameters and conditions; develop systematic (quantitative, if possible) descriptions of these as well as measured outcomes (use precise definitions for all observables); abstract from ÒirrelevantÓ or ÒextraneousÓ perturbations and/or control for those. Avoid the Òcorrelation = causationÓ fallacy Ð there could 3) be Òhidden variablesÓ that influence both correlated observables If possible, do systematic experiments varying one of the relevant parameters at a time; look for systematic changes in outcome depending on each [in-class analog: Do labs with changing parameters]. 4) Develop a hypothesis (a Ð quantitative, if possible Ð relationship between parameters and outcomes); collect more data / do more experiments to test hypothesis 5) If your hypothesis describes your data approximately, call it a ÒModelÓ; if it agrees with all observations, call it a ÒLawÓ 6) A coherent set of ÒLawsÓ is called a ÒTheoryÓ; all parts of a Theory must be logically (and mathematically, if applicable) consistent with each other as well as 7) Apply ÒOccamÕs razorÓ: Your theory should have (only) the minimum number of 8) ingredients required to describe all relevant observations; do not invoke Òextranatural causesÓ Derive testable predictions from your theory or law; in particular, explore all new 9) and previously unforeseen consequences [in-class analog: Do problems] Test your predictions against observations and experiments; discard or modify with previously established theories (unless you can prove those wrong) theory (e.g., limit range of applicability) if test results disagree with predictions. 10) All theories are ÒtentativeÓ at some level Ð there could always be contradictory observations in the future Ð but donÕt discard well-tested theories needlessly! Non-science ¥ ÒTheoriesÓ leading to untestable predictions (predictions that are too vague to be ÒfalsifiableÓ or that refer to unmeasurable phenomena) ¥ Ad-hoc hypotheses (a new one for every new observation), anecdotal evidence, testimonials value statements, matters of taste or opinion, exclusive reliance on authority or ¥ precedent, normative statements (Òyou shouldÉ!Ó) Éand now to PHYSICSÉ Motion in 1 dimension Average velocity for time interval Δt = t1Ét2 v av = x(t )# x(t1 ) "x change in position during the time "t = = 2 t 2 # t1 "t elapsed time "t Instantaneous velocity at time t = average velocity for a very short time interval around t ; dx/dt Relative velocity addition ! ! ! v (x relative to B )= v (x relative to A ) + v (A relative to B ) Acceleration for time interval Δ t = t1Ét2 a= change in velocity during time "t "v = elapsed time interval "t "t Motion in 1 dimension with constant acceleration a: v(t) = v(0) + a t = v 0 + a t vav, (0 É t 1) = 1/2 [v 0 + v(t 1)] x(t) = x0 + v 0 t + 1/2 a t 2 Forces Ð some Examples ! Weight (Gravity): F Weight = -mg (in negative vertical direction, up = +); g = 9.81 m/s2 near EarthÕs surface Force exerted by spring (Hooke's Law): Fel = -kx (opposite to the direction of the displacement x from relaxed state of spring; with k = spring constant) Normal force Fn: Equal and opposite to sum of perpendicular (to a surface) components of all other forces (makes net perpendicular force zero) Static Friction fstat (object at rest): equal and opposite to sum of parallel components of all other forces, canceling them if the sum is not too big: |fstat| ≤ µs |Fn|. Kinetic friction fkin (moving object): Force opposite to direction of motion along surface, |fkin| = µk |Fn|. Tension T: Force along direction of string, at the end it is the same as the force exerted by the string on the attachment point (towards the string). Newton's First Law When all forces applied to an object balance out to zero (cancel each other), then this object will not accelerate relative to an inertial system; it will be in equilibrium: If at rest, it will stay at rest, if in motion, it will continue to ! move in the same direction, with constant velocity " a = 0. Newton's Second Law a! = F/m; acceleration! equals net force divided by mass. More detailed: ! F i = m a (Sum over all forces, including direction!) F resultant = " all forces i=1...N Note: Forces add like vectors (parallelogram rule!) Newton's Third Law Forces always come in pairs (interaction between two objects A and B): F Action (A on B) = - F Reaction (B on A) Momentum ! ! For a single object: p = m v ! ! ! Change in momentum (impulse): J = Δ p = Σ F Δt Because of NewtonÕs 3rd Law, impulse transferred from object A to object B is equal in magnitude but opposite in sign (direction) to simultaneously transferred impulse from B to A. System of particle with no external force: Total momentum ! ! ! Σ p i = p 1 + p 2 + É is conserved (same before as after any collision between any particles in the system) Momentum conservation in two-particle collision: ! ! ! ! m1v 1i + m 2 v 2i = m1 v1 f + m 2 v 2 f (i = initial, f = final) If sum of all kinetic energies is also conserved (no work done on system as a whole, no energy converted to other forms; see below): ELASTIC collision. Otherwise (friction, conversion to heat/deformation,É): INELASTIC collision. If 2 objects stick together after collision: Completely inelastic ! ! ! If collision is completely inelastic: m1v 1i + m 2 v 2i = ( m1 + m 2 ) v f Work and Energy Work ΔW on some object (particle or system) due to a force F if that object is displaced by Δx in the direction of the force: ΔW = F Δx = F Δs cosφ (only the part of the total displacement Δs that is in the direction of the force counts; no displacement = no work). Work done on object can be positive (transferred TO object) or negative (transferred FROM object). Work-Energy Theorem: Net work done on an object changes the ENERGY of that object: ΔE (particle or system) = ΔW (done on that particle or system) Conservation of total Energy The total amount of energy within a system can only be changed by exchanging energy with another system. IF the energy of one object (particle, system,É) goes UP, then the energy of another object (particle, system,É) must go DOWN by the same amount, and vice versa! Power: Rate of work done on system per unit time: P = ΔW /Δt = F . v Types of Energy ! Kinetic energy of a moving particle: K.E. = m/2 v 2 (depends only on speed; is never negative; has no direction; depends on coordinate system used for v) If the only (relevant, changing) energy of a system is kinetic energy, then the work-energy theorem states: Δ K.E. = ΔW Potential Energy Some forces can do work that depends only on initial and final position of the particle(s) that this work is being done on. Example: Gravity Ð only the difference in height between initial and final state matters: ΔW = mgΔh These forces are called ÒconservativeÓ because they can ÒstoreÓ work and return it without losses. Any work done against a conservative force increases stored ÒpotentialÓ for future positive work by the same force. Work ! done by conservative ! ! force stored in form of potential energy: U ( r ) = ΔWext ( r ref ->!r ) = -Fconservative Δs cos φ ! At reference point U ( r ref) = 0 ( r ref can be chosen for convenience). Examples: ¥ Particle attached to a spring, stretching the spring by moving a distance x from the springÕs relaxed state: U el = (k /2) x 2 (k = spring constant; x = 0 => U el = 0 for relaxed state of spring); more general: totally elastic distortion of objects ¥ Approximate Gravitational Potential Energy (near Earth surface): U grav = mgh (h is height above reference point) ¥ More general Gravitational Potential Energy: U grav = -GmM/r (see later in semester; here reference point is ∞ far away) ¥ Electrostatic Potential Energy (charged particle in static electric field) -> see even later Total mechanical energy Sum of kinetic and potential energy of a particle: Emech = K.E. + U IF no external (dissipative) forces present: Eparticle = Emech = const. (ΔE = 0) => sum of kinetic and potential energy stays constant; for example: going UP the ramp DEcreases kinetic energy (negative work done by gravity) and INcreases potential energy; going back DOWN INcreases kinetic energy (car speeds up) and DEcreases potential energy (mgh becomes less). Sum stays constant, so once you have Emech = m/2 v2 + mgh at one position, you can find velocity at any other position by solving m/2 v2 = Emech - mgh 2 m 2 If EXTERNAL forces DO work: Δ Emech = m/2 vf - /2 vi + ΔU = ΔWext Other forms of energy: (see later during PHYS101-102) Energy stored in objects (distortion, chemical energy, É, mass! 1) Kinetic energy due to random motion of particles in a system = internal energy (most disordered and least ÒusefulÓ form of energy; ÒheatÓ) Energy stored in fields (e.g. electromagnetic fields) Energy can be transmitted by fields and waves (sound, light, electromagnetic radiation,É) Energy can be positive (e.g., kinetic energy always is) or negative (e.g., general gravitational energy, binding energy of molecules in solid or liquid, binding energy of electrons in atoms, binding energy of nucleons in nuclei etc.) or both (depending on reference point) The total sum of ALL these forms of energy in a CLOSED system is always conserved; energy of an interacting system can only change if it is exchanged with other system (in form of work or heat). 1 Note: According to Einstein, mass is just one form of energy (E = mc2) and therefore only the sum of all energy types including mass energy is conserved! Motion in a circle Position described by angle θ [in radians], radius R Distance around the perimeter of a circle [in m]: D = θ [in radians] . R [in m] Period of 1 revolution: T; rotational speed in Òrounds per secondÓ rps = 1/T ! Angular velocity: ω = Δ θ [in radians] / Δt [in s] = 2π /T = 2π rps = | v | / R; linear (tangential) speed v = Rω = 2πR rps ! Centripetal acceleration: a centr = v2 /R = ω 2 R => required centripetal force F = m a centr Moment of Inertia I : - Single Particle of mass m at distance R from axis: I =mR2 - Extended Objects: I = Σp (mP rP 2); increases with total mass and with the square of the average distance rP from axis of that mass - Hollow cylindrical shell (total mass M, radius R): I =MR2 - Solid cylinder: I = MR2/2 - Solid sphere: I = 2/5 MR2 - Thin rod of length l (axis through center): 1/12 Ml2 Thin rod (axis through edge): 1/3 Ml2 Kinetic energy for rotational motion: K.E. = 1/2 Iω 2 Angular momentum: - Single Particle at distance R from axis: L = mRv = mR2ω = Ιω - Extended object: L = Iω (conserved if no external torque is present) Points along axis of rotation; positive if rotation is counter-clockwise, negative if rotation is clockwise (right hand rule: curling fingers = sense of rotation, thumb points in direction of L). Total angular momentum of closed system is conserved if no external force present -> if moment of inertia increases, angular velocity must decrease! A change in angular momentum (rotational status) requires external torque: tangential force F at distance l relative to axis: τ = l | F | sinθ = ΔL / Δt (lever arm l times force; only the part of F perpendicular to l counts) ! If net external torque (sum of all torques) = 0 -> L is conserved Requirement for equilibrium: Sum of all forces AND sum of all torques must both be zero! Center of gravity must be above support area. Gravitation Gravitational Force due to mass M on mass m at distance r : F = -GMm/r2 r is distance from center of mass M to center of mass m Gravitational Force F points back to its source (the center of the ÒattractorÓ) Gravitational acceleration of m towards M: a = GM/r2 (independent of m) Tidal acceleration difference across an object of size D due to a mass M at distance r: Δa = 2 GM D/r 3 Gravitational Potential Energy between two masses at distance r (center to center): Ugrav = - GmM /r (reference point assumed at infinite separation) Escape speed from planet with mass M and radius R: v esc = 2GM / R KeplerÕs Laws 1) Orbits of satellites (moons, planets,É) are elliptical 2) A line from the gravitating body (sun, planet,É) to the satellite sweeps out equal areas in equal times (conservation of angular momentum) => speed is larger if distance is smaller 3) The square of the period T of an orbit around a mass M is proportional to the cube of a: T2 ∝ a 3 (a = major half axis of ellipse) => T ∝ √a3 . # & Complete formula T = 2" %a GM ( $ ' => on a circular orbit of radius R the velocity is v orbit = GM / R 3 Projectile motion near EarthÕs surface (neglect air resistance) Horizontal (x) and vertical (y) motion are independent of each other Horizontal components: vx = vx0 = const. ; x(t) = x0 + vx0 t Vertical components: vy (t) = vy0 -gt ; y(t) = y0 + vy0t - 1/2 gt 2 Total motion is simply combination of horizontal and vertical one. Vertical motion usually determines total time for path (until impact). Initial launch at angle θ above horizontal: v x0 = v 0 cos(" 0 ) ; v y0 = v0 sin(" 0 ) 0 #v & Total speed = v = v x2 + v 2y ; final angle " = arctan% y v (. $ x' Maximum height of trajectory is ymax = y0 + 1/2vy02/g (assuming vy0 > 0); Total time elapsed for trajectory from ground back to ground: Δt = 2 vy0 /g Maximum distance traveled (from ground to ground) is Δxmax = 2 vx0 vy0 /g Electrostatics: Charge Positive or negative, measured in Coulomb [C], conserved, quantized (in units of e). Equal sign charges repel, opposite sign charges attract. Electron: Charge q = -e . Proton: Charge q = +e. Most atoms have q = 0. Force between charges: ! Qq , k = 8.99.109 Nm2/C2. Coulomb Force F on q due to Q at distance r = k r2 Alternative Formulation: Charge Q (or any distribution of such charges) create an electric field E. Charge q experiences a force F = qE in this field (include sign of q !). Field lines begin at positive charges and end at negative ones or go on forever; density of field lines indicates strength of field. Examples: - Electric field at distance r from a single spherical charge: E = kQ/r2 - Electric field between two large, conducting parallel plates: E = const. Conductors - Contain huge amounts of ÒfreeÓ charges - In presence of electric field, charges will flow to counteract field - Unless new charges are constantly supplied (current), field will be canceled -> no field inside conductors - All charges sit on the outside surface of the conductor - External field will rearrange free charges such that conductor experiences net force towards external charge even in absence of net charge (ÒinductionÓ) - Conductor connected to ground will have net charge if one type of its charges is pushed away by nearby other charges of same sign Insulators - Contain no free charges - Can pick up a few Òextra chargesÓ Ð a little net charge - External field can polarize individual molecules such that material experiences net force towards external charge - Polarization can weaken but never cancel external field Potential Moving charge q along electric field E does work: W = F.d = qE.d Electrostatic forces are conservative -> work stored in electrostatic potential energy Upot. Electrostatic potential energy Upot of a charged object divided by its charge q equals potential V [measured in Volt = V]. A charge q changes its energy by q.ΔV when moving from a point with potential V1 to a point with potential V2 = V1 + ΔV. 1 V potential difference between 2 points means there is a potential to do 1 J of work per Coulomb moved from the higher potential point to the other. Current Net flow of charge in a given direction per unit time, measured in Ampere [A = C/s] Count positive charges +, negative charges Ð and reverse sign for charges moving in opposite direction. Net current = sum of all currents Current in conductor with resistance R requires electric field -> potential difference ΔV = -RI. (OhmÕs Law). Resistance measured in Ohm [Ω], is proportional to length of conductor, is inversely proportional to cross section, and is usually greater if conductor is hot. Currents require complete (closed circuit) path and non-electrostatic ÒpumpÓ (EMF = ElectroMotive Force = energy per charge) to keep running. Current is identically the same everywhere around a simple closed loop circuit. Currents heat up conductors; power (energy transfer per unit time) is P [Watt] = ΔV [Volt] x I [Ampere] = RI2 = ΔV2/R Typical speed of electrons in wire: 106 m/s random (internal energy), but only 0.1 mm/s on average in direction of electric force. Electric fields (that Òtell the electrons to moveÓ) travel with (nearly) speed of light, c. Power can be delivered both by currents going in one direction only (DC, ex.: battery driven) and by currents going back and forth (AC, currents driven by generators, household and HV circuits). Series Circuit Same current I has to go through all elements in series (doesnÕt matter which one comes ÒfirstÓ). Voltage supplied by battery etc. gets Òdivided upÓ among various elements. Total resistance is sum of individual resistances. The largest voltage drop occurs across the largest resistor. Current (and therefore power) in each element smaller than if connected directly (without the others) to battery. Any break in circuit and all current ceases. Parallel Circuit Current that must be supplied by battery is the sum of the currents to all of the various branches. Each branch sees the full battery voltage, independent from the other branches. Smallest resistor draws the largest current and therefore the largest power. Any one element (branch) can have a break without affecting the others. Magnetic fields Due to charges in motion (electric currents, spinning electrons,É) Field lines can form closed loops, but never end or begin. Example: Permanent dipole, solenoid (or short coil) : Field lines can be described as if emanating from ÒNorth poleÓ and converging towards ÒSouth poleÓ (but connect through the interior of the dipole!) Cannot separate Ònorth poleÓ from Òsouth poleÓ Ð chopping in half will simply yield 2 new dipoles. Earth has a magnetic dipole field with magnetic north pole close to geographic south pole (Australia). Field of straight wire: circles around it, falls off like 1/r. Magnetic materials Most materials react only slightly to magnetic fields. All materials have a slight diamagnetic response (see below) to strong magnetic fields. In some special materials, there are additional responses which can totally overpower the usual diamagnetic response. Examples: Ferromagnets: Iron, Cobalt, Nickel and some Rare Earth compounds Magnetic properties due to electron spins (aligned within domains). Normally random orientation of domains Ð no net magnetism External field can orient or grow/shrink domains Ð strong magnetic response (magnetization) Ð leading to attraction towards other magnets For some alloys, domains can keep orientation indefinitely Ð> permanent magnets. Permanent dipoles can attract (unlike poles) or repel (like poles); net force on dipole is zero in homogeneous magnetic field but torque tends to align dipole with field direction Paramagnets: A weak magnetization in the direction of external field; attraction to strong magnets. Some materials like (liquid) oxygen, gadolinium,É Diamagnets: A (usually extremely weak) magnetization opposite to the direction of external field; always leads to (usually feeble) repulsion from strong magnets. Property of nearly all materials (including water; -> floating frog) usually too weak to detect and/or masked by ferro- or paramagnetism. Exception: Superconductors are perfect diamagnets (strong expulsion of all external magnetic fields -> floating permanent magnet above cold superconducting disk). Magnetic forces Acts only on moving charges (including permanent or induced magnets: equal poles repel, opposite poles attract) Magnitude proportional to q, v and Bperp (the part of B perpendicular to v): F = qv |Bperp| Acts perpendicular to both v and B (right hand rule). In vector notation: F = qvxB Typical motion in homogeneous field: Circle of radius R = mv/qB with angular velocity ω = qB/m, neither speeding up nor slowing down (no change in kinetic energy -> magnetic forces donÕt do work on moving charges). Force on wire: ILB (L=Length), sideways (perpendicular to field and wire). Two parallel wires with current in the same direction attract: F = 2.10-7 N per 1 m of length if wires are 1 m apart and both carry I = 1 A.