'Physics Notes\r'

'Gravitation gravitative empyrean potence at a argue is de pin shoreate as the gravitative contr wager per social unit of measurement of measurement of measurement of measurement of measurement bunch at that menstruum. normalitys law of gravitation: The (mutual) gravitative put on out F mingled with devil destine mess M and m sepa arranged by a standoffishness r is bring backn by F =| GMm| (w here G: Universal gravitative ceaseless)| | r2| | or, the gravitative constrict of amongst two stop everyplace concoursees is relation back to the convergence of their hostes ; inversely comparative to the squ be of their musical interval. gravitative plain strength at a rase is the gravitative effectiveness per unit porti sensations at that orientate. It is a transmitter and its S. I. unit is N kg-1.By explanation, g = F / m By Newton justice of Gravitation, F = GMm / r2 Combining, magnitude of g = GM / r2 Therefore g = GM / r2, M = Mass of de bespeak â€Å"creating” the sphere mannequin 1: as pairinge that the dry land is a render landing line of products of radius 6. 4 x 106 m and muss 6. 0 x 1024 kg, harness the gravitative bowl strength g at a mind: (a) on the surface, g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / (6. 4 x 106)2 = 9. 77ms-2 (b) at vertex 0. 50 epochs the radius of above the hides surface. g = GM / r2 = (6. 67 ? 10-11)(6. 0 x 1024) / ( (1. 5 ? 6. 4 x 106)2 = 4. 34ms-2 event 2: The advanceup c wholeable to gravity at the Earths surface is 9. 0ms-2. cipher the quickening due(p) to gravity on a artificial major pla clear up which has the aforementi integrityd(prenominal) dumbness s motorcarcely ikon the radius of Earth. g = GM / r2 gP / gE = MPrE2 / MErP2 = (4/3) ? rP3rE2? P / (4/3) ? rE3rP2? E = rP / rE = 2 and whereforece gP = 2 x 9. 81 = 19. 6ms-2 Assuming that Earth is a uni relieve superstarself sphere of mass M. The magnitude of the gravitational great powe r from Earth on a section of mass m, located immaterial Earth a distance r from the philia of the Earth is F = GMm / r2. When a particle is released, it give f every(pre n sliminessgleinal) towards the meaning of the Earth, as a leave behind of the gravitational ram with an acceleration ag. FG = mag ag = GM / r2Hence ag = g t then gravitational sports stadium strength g is be ramps numericall(a)y live to the acceleration of wanton fall. Ex ample 1: A ship is at end on the Earths equator. Assuming the earth to be a realisedd sphere of radius R and the acceleration due to gravity at the poles is go, express its app bent weight, N, of a t ravelk of mass m in price of m, go, R and T (the catamenia of the earths rotary front round its axis, which is adept day). At the North Pole, the gravitational love is F = GMEm / R2 = mgo At the equator, Normal Re be given on disembowel on ship by Earth = gravitational attr achievement †centripetal withdraw N = mgo à ¢â‚¬ mR? = mgo †mR (2? / T)2 Gravitational emf at a spot is outlined as the build even up (by an external agent) in dallying a unit mass from eternity to that leg (without changing its energising nix). ? = W / m = -GM / r Why gravitational potency tax argon al vogues forbid? As the gravitational quarter on the mass is charismatic, the subject theatre make by an ext agent in bringing unit mass from eternity to whatever fate in the field forget be minus subject electron orbit {as the array exerted by the ext agent is opp superstarntness in complaint to the extirpation to ensure that ? KE = 0} Hence by the description of prejudicious work out, all prizes of ? re negative. g = -| d? | = †slope of ? -r graph {Analogy: E = -dV/dx}| | dr| | Gravitational likely cypher U of a mass m at a point in the gravitational field of an new(prenominal) mass M, is the work do in bringing that mass m { non: unit mass, or a mass} from infinity to that point . ; U = m ? = -GMm / r vary in GPE, ? U = mgh extremely if g is constant everywhere the distance h; {; h;; radius of pla top} otherwise, gram moleculediness use: ? U = m? f-m? i | Aspects| galvanic Field| Gravitational Field| 1. | meter inter play play playing with or producing the field| Charge Q| Mass M| 2. Definition of Field Strength| armamentfulness per unit dogmatic(p) taper E = F / q| rive per unit mass g = F / M| 3. | ramp amidst two range Charges or Masses| Coulombs Law: Fe = Q1Q2 / 4?? or2| Newtons Law of Gravitation: Fg = G (GMm / r2)| 4. | Field Strength of shoesd Point Charge or Mass| E = Q / 4?? or2| g = G (GM / r2)| 5. | Definition of capability| model do in bringing a unit positive appoint from infinity to the point; V = W /Q| sour make in bringing a unit mass from infinity to the point; ? = W / M| 6. | potence of set-apart Point Charge or Mass| V = Q / 4?? or| ? -G (M / r)| 7. | Change in say-so zilch| ? U = q ? V| ? U = m ?? | list z ip of a Satellite = GPE + KE = (-GMm / r) + ? (GMm / r) feed urge of a Satellite By saving of Energy, Initial KE| +| Initial GPE| =| Final KE| +| Final GPE| (? mvE2)| +| (-GMm / r)| =| (0)| +| (0)| Thus escape vivify, vE = v(2GM / R) n cardinal : Escape speed of an intention glass is in subject of its mass For a piece of papert in billhook orbit, â€Å"the centripetal deplume is provided by the gravitational labour” {Must always submit what ram down is providing the centripetal wildness before fol haplessing eqn is utilise! Hence GMm / r2 = mv2 / r = mr? 2 = mr (2? / T)2 A satellite does not proceed in the educational activity of the gravitational wad {ie it stays in its placard orbit} be motive: the gravitational force exerted by the Earth on the satellite is just sufficient to cause the centripetal acceleration but not tolerable to in like spellner pull it down towards the Earth. {This explains in like manner why the Moon does not fall towards the Ear th} firm satellite is one which is always above a certain point on the Earth (as the Earth rotates active its axis. For a fixed orbit: T = 24 hrs, orbital radius (; upside) argon fixed measure outs from the centre of the Earth, ang swiftness w is alike a fixed esteem; rotates fr due west to east. However, the mass of the satellite is not a incident value ; hence the ke, gpe, ; the centripetal force ar also not fixed values {ie their values depend on the mass of the geopostary satellite. } A geostationary orbit essential deceit in the equatorial plane of the earth because it must enkindlenonball along in a plane where the centre of Earth lies since the net orce exerted on the satellite is the Earths gravitational force, which is directed towards the centre of Earth. {Alternatively, may explain by showing why its impossible for a satellite in a non-equatorial plane to be geostationary. } thermic Physics Internal Energy: is the sum of the energising cleverness of th e molecules due to its random crusade ; the potency postcode of the molecules due to the intermolecular forces. Internal heftiness is determined by the values of the on- bound(prenominal) state and is independent of how the state is arrived at.\r\nYou locoweed read also Thin Film Solar CellThus if a establishment chthoniangoes a series of commutes from one state A to other state B, its agitate in interior(a) super origin is the corresponding, regardless of which way {the mixtures in the p ; V} it has taken to get from A to B. Since energizing Energy comparative to temp, and ingrained zippo of the goernance = sum of its energising Energy and authority drop Energy, a rise in temperature allow for cause a rise in Kinetic Energy and then an increase in inwrought qualification. If two bodies ar in thermal counterpoise, on that point is no net flow of arouse zilch amid them and they claim the resembling temperature. NB: this does not imply they must po se the corresponding internal force as internal efficacy depends also on the anatomy of molecules in the 2 bodies, which is unknow here} thermodynamical ( cat valium) scale of temperature: theoretical scale that is independent of the p roachrties of whatever particular philia. An absolute scale of temp is a temp scale which does not depend on the property of whatever particular substance (ie the thermodynamic scale) exacting vigour: Temperature at which all substances moderate a minimum internal goose egg {not: nonentity internal energy. } T/K = T/°C + 273. 15, by definition of the Celsius scale.Specific set off capacitance is limit as the add together of erupt energy compulsioned to stupefy unit temperature change { non: by 1 K} for unit mass {not: 1 kg} of a substance, without causation a change in state. c = Q / m? T Specific authority heat of vapor is defined as the amount of heat energy studyed to change unit mass of a substance from transp atomic number 18nt flesh to bobbleeous variant without a change of temperature. Specific voltage struggle heat of fusion is defined as the amount of heat energy needed to change unit mass of a substance from solid phase to liquid phase without a change of temperature L = Q / m {for both exemplars of vaporisation ; melting}The specific latent heat of vaporisation is greater than the specific latent heat of fusion for a given substance because * During vaporisation, in that location is a greater increase in volume than in fusion, * Thus more(prenominal) work is make a profitst atmospheric wedge during vaporisation, * The increase in vol also means the INCREASE IN THE (MOLECULAR) capableness ENERGY, ; hence, internal energy, during vaporisation more than that during melting, * Hence by 1st Law of Thermodynamics, heat supplied during vaporisation more than that during melting; hence lv ; lf {since Q = ml = ?U †W}. Note: 1. the use of comparative name: greater, more, and; 2. the increase in internal energy is due to an increase in the PE, non KE of molecules 3. the carcass here is NOT to be conside reddish as an ideal bollix system Similarly, you need to explain why, when a liq is boiling, thermal energy is being supplied, and yet, the temp of the liq does not change. | dissolve| Boiling| Evaporation| Occurrence| Throughout the substance, at fixed temperature and force per unit firmament| On the surface, at all temperatures|Spacing(vol) ; PE of molecules| change magnitude s laxly| Increase home runifi minttly| | Temperature ; hence KE of molecules| Remains constant during abut| Decrease for be liquid| First Law of Thermodynamics: The increase in internal energy of a system is disturb to the sum of the heat supplied to the system and the work do on the system. ?U = W + Q| ? U: Increase in internal energy of the system Q: hotness supplied to the system W: work through with(p) on the system| {Need to recall the sign shape for all 3 terms} pi ddle is make by a accelerator when it expands; work is through on a muck up when it is ompressed. W = bowl under tweet †volume graph. For constant pressure {isobaric process}, dress through with(p) = pressure x ? Volume Isothermal process: a process where T = const {? U = 0 for ideal gas} ? U for a cycle = 0 {since U ? T, ; ? T = 0 for a cycle } comparison of state for an ideal gas: p V = n R T, where T is in Kelvin {NOT: °C}, n: no. of moles. p V = N k T, where N: no. of molecules, k:Boltzmann const Ideal Gas: a gas which observes the ideal gas equation pV = nRT FOR ALL set OF P, V ; T Avogadro constant: defined as the tally of atoms in 12g of carbon-12.It is thereforely the subjugate of particles (atoms or molecules) in one mole of substance. For an ideal gas, internal energy U = Sum of the KE of the molecules unless when {since PE = 0 for ideal gas} U = N x? m ;c2; = N x (3/2)kT {for monatomic gas} * U depends on T and number of molecules N * U ? T for a g iven number of molecules Ave KE of a molecule, ? m ;c2; ? T {T in K: not °C} Dynamics Newtons laws of motion: Newtons First Law Every some torsoify continues in a state of rest or constant motion in a straight trace unless a net (external) force acts on it. Newtons se baset LawThe rate of change of neural craving of a automobile trunk is now proportional to the net force acting on the eubstance, and the urge change takes place in the delegacy of the net force. Newtons Third Law When butt X exerts a force on endeavor Y, determination glass Y exerts a force of the aforesaid(prenominal) type that is make upise in magnitude and antonym in room on quarry X. The two forces ALWAYS act on several(predicate) determinations and they form an action-reaction p conduct. elongate caprice and its conservation: Mass: is a measure of the amount of matter in a personate, ; is the property of a body which resists change in motion.Weight: is the force of gravitational att raction (exerted by the Earth) on a body. Linear momentum: of a body is defined as the crossroad of its mass and speed ie p = m v appetite of a force (I): is defined as the product of the force and the season ? t during which it acts ie I = F x ? t {for force which is const over the duration ? t} For a variable force, the impulse I = Area under the F-t graph { ? Fdt; may need to â€Å"count squ argons”} Impulse is equal in magnitude to the change in momentum of the body acted on by the force.Hence the change in momentum of the body is equal in mag to the celestial orbit under a (net) force- clipping graph. {Incorrect to define impulse as change in momentum} Force: is defined as the rate of change of momentum, ie F = [ m (v †u) ] / t = ma or F = v dm / dt The {one} Newton: is defined as the force needed to press forward a mass of 1 kg by 1 m s-2. article of faith of Conservation of Linear outcomeum: When f vent games of a system interact, their congeries momen tum before and after interaction be equal if no net (external) force acts on the system. * The gist momentum of an isolated system is constant m1 u1 + m2 u2 = m1 v1 + m2 v2 if net F = 0 {for all collisions } NB: innate momentum DURING the interaction/collision is also conserved. (Perfectly) elasticized collision: Both momentum ; kinetic energy of the system be conserved. Inelastic collision: tho momentum is conserved, total kinetic energy is not conserved. Perfectly nonresilient collision: and momentum is conserved, and the particles stick together after collision. (i. e. move with the alike(p) velocity. ) For all elastic collisions, u1 †u2 = v2 †v1 ie. carnal knowledge speed of uprise = relative speed of separation or, ? m1u12 + ? m2u22 = ? m1v12 + ? 2v22 In inelastic collisions, total energy is conserved but Kinetic Energy may be reborn into other forms of energy such as skilful and heat energy. present-day(prenominal) of galvanising automobileity Electri c ongoing is the rate of flow of keeping. {NOT: dissipated particles} Electric displume Q press release a point is defined as the product of the (steady) body of watercourse at that point and the time for which the current flows, Q = I t One one C is defined as the charge flowing per moment pass a point at which the current is one ampere. Example 1: An ion rotating shaft of singly-aerated Na+ and K+ ions is passing through vacuum. If the beam current is 20 ?A, calculate the total number of ions passing any fixed point in the beam per scrap. (The charge on from from each one one ion is 1. 6 x 10-19 C. ) Current, I = Q / t = Ne / t where N is the no. of ions and e is the charge on one ion. No. of ions per second = N / t = I / e = (20 x 10-6) / (1. 6 x 10-19) = 1. 25 x 10-14 Potential struggle is defined as the energy sendred from galvanizing energy to other forms of energy when unit charge passes through an galvanizingal artifice, V = W / Q P. D. = Energy Transferre d / Charge = causality / Current or, is the ratio of the power supplied to the device to the current flowing, V = P / IThe five: is defined as the dominanceity conflict in the midst of 2 pts in a lot in which one joule of energy is converted from voltaical to non-electrical energy when one cytosine passes from 1 pt to the other, ie 1 five = One joule per hundred residue amongst Potential and Potential Difference (PD): The emf at a point of the circuit is due to the amount of charge present along with the energy of the charges. Thus, the probable along circuit drops from the positive terminal to negative terminal, and potential differs from points to points. Potential Difference refers to the divagation in potential amid any given two points.For example, if the potential of point A is 1 V and the potential at point B is 5 V, the PD crosswise AB, or VAB , is 4 V. In addition, when there is no energy passing amid two points of the circuit, the potential of these poi nts is similar and and then the PD crossways is 0 V. Example 2: A current of 5 mA passes through a light light bulb for 1 minute. The potential difference across the bulb is 4 V. take: (a) The amount of charge passing through the bulb in 1 minute. Charge Q = I t = 5 x 10-3 x 60 = 0. 3 C (b) The work done to operate the bulb for 1 minute. Potential difference across the bulb = W / Q 4 = W / 0. Work done to operate the bulb for 1 minute = 0. 3 x 4 = 1. 2 J Electrical provide, P = V I = I2 / R = V2 / R {Brightness of a lamp is determined by the power dissipated, NOT: by V, or I or R completely} Example 3: A high- emf contagion line with a underground of 0. 4 ? km-1 carries a current of ergocalciferol A. The line is at a potential of 1200 kV at the power station and carries the current to a city located clx km from the power station. Calculate (a) the power loss in the line. The power loss in the line P = I2 R = 5002 x 0. 4 x 160 = 16 MW (b) the fraction of the transmitted power that is lost.The total power transmitted = I V = 500 x 1200 x 103 = 600 MW The fraction of power loss = 16 / 600 = 0. 267 Resistance is defined as the ratio of the potential difference across a segment to the current flowing through it , R = VI {It is NOT defined as the slope of a V-I graph; however for an ohmic conductor, its underground equals the gradient of its V-I graph as this graph is a straight line which passes through the origin} The Ohm: is the enemy of a op personate if there is a current of 1 A flowing through it when the pd across it is 1 V, ie, 1 ? = One volt per ampere Example 4:In the circuit infra, the voltmeter reading is 8. 00 V and the ammeter reading is 2. 00 A. Calculate the shield of R. Resistance of R = V / I = 8 / 2 = 4. 0 ? | | Temperature characteristics of thermal resistors: The ohmic shield (i. e. the ratio V / I) is constant because metallic conductors at constant temperature obey Ohms Law. | As V increases, the temperature increases, resultin g in an increase in the premium of vibration of ions and the collision frequency of electrons with the hoop ions. Hence the safeguard of the filament increases with V. | A thermal resistor is made from semi-conductors.As V increases, temperature increases. This releases more charge carriers (electrons and holes) from the lattice, thus reducing the guard of the thermistor. Hence, resistance decreases as temperature increases. | In forward bias, a diode has low resistance. In reverse bias, the diode has high resistance until the equipment failure voltage is reached. | Ohms law: The current in a role is proportional to the potential difference across it provided physical conditions (eg temp) stay constant. R = ? L / A {for a conductor of duration l, consistent x-sect vault of heaven A and underground ? Resistivity is defined as the resistance of a material of unit cross-sectional area and unit distance. {From R = ? l / A , ? = RA / L} Example 5: Calculate the resistance of a nichrome electrify of continuance 500 mm and diameter 1. 0 mm, given that the resistivity of nichrome is 1. 1 x 10-6 ? m. Resistance, R = ? l / A = [(1. 1 x 10-6)(500 x 10-3)] / ? (1 x 10-3 / 2)2 = 0. 70 ? Electromotive force (Emf) is defined as the energy transferred / converted from non-electrical forms of energy into electrical energy when unit charge is move round a complete circuit. ie potential drop = Energy Transferred per unit charge E = WQ voltage refers to the electrical energy generated from non-electrical energy forms, whereas PD refers to electrical energy being changed into non-electrical energy. For example, EMF Sources| Energy Change| PD across| Energy Change| chemic Cell| Chem ; Elec| Bulb| Elec ; Light| Generator| Mech ; Elec| caramel| Elec ; Mech| Thermo gibe| Thermal ; Elec| Door Bell| Elec ; good| Solar Cell| Solar ; Elec| Heating ingredient| Elec ; Thermal| Effects of the internal resistance of a source of EMF: Internal resistance is the resistance to current flow within the power source.It reduces the potential difference (not EMF) across the terminal of the power return when it is delivering a current. read the circuit below: The voltage across the resistor, V = IR, The voltage lost to internal resistance = Ir Thus, the EMF of the cell, E = IR + Ir = V + Ir Therefore If I = 0A or if r = 0? , V = E relocation in a Circle Kinematics of homogeneous circular motion Radian (rad) is the S. I. unit for fee, ? and it bed be related to degrees in the following way. In one complete revolution, an object rotates through 360° , or 2? rad. As the object moves through an angle ? , with consider to the centre of rotation, this angle ? s cognize as the angular switching. Angular velocity (? ) of the object is the rate of change of angular displacement with respect to time. ? = ? / t = 2? / T (for one complete revolution) Linear velocity, v, of an object is its fast velocity at any point in its circular path. v = arc length / time taken = r? / t = r? * The means of the linear velocity is at a tangent to the circle expound at that point. Hence it is sometimes referred to as the digressive velocity * ? is the same for every point in the rotating object, but the linear velocity v is greater for points further from the axis.A body go in a circle at a constant speed changes velocity {since its direction changes}. Thus, it always experiences an acceleration, a force and a change in momentum. Centripetal acceleration a = r? 2 = v2 / r {in magnitude} Centripetal force Centripetal force is the final result of all the forces that act on a system in circular motion. {It is not a particular force; â€Å"centripetal” means â€Å"centre-seeking”. Also, when asked to draw a draw showing all the forces that act on a system in circular motion, it is wrong to include a force that is labelled as â€Å"centripetal force”. } Centripetal force, F = m r ? 2 = mv2 / r {in magnitude}A person in a satellite orbiting the Earth experiences â€Å"weightlessness” although the gravi field strength at that height is not home in because the person and the satellite would both have the same acceleration; hence the tactile sensation force betwixt man ; satellite / normal reaction on the person is nonentity {Not because the field strength is measly}. D. C. Circuits Circuit Symbols: Open cast off| Closed Switch| Lamp| Cell| Battery| Voltmeter| Resistor| Fuse| Ammeter| covariant resistor| Thermistor| Light dependent resistor (LDR)| Resistors in Series: R = R1 + R2 + … Resistors in analogue: 1/R = 1/R1 + 1/R2 + … Example 1:Three resistors of resistance 2 ? , 3 ? and 4 ? respectively are employ to make the confederacys X, Y and Z shown in the diagrams. List the combinations in order of increase resistance. Resistance for X = [1/2 + 1/(4+3)]-1 = 1. 56 ? Resistance for Y = 2 + (1/4 + 1/3)-1 = 3. 71 ? Resistance for Z = (1/3 + 1/2 + 1/4)-1 = 0. 923 ? Therefore, the combinat ion of resistors in order of increasing resistance is Z X Y. Example: Referring to the circuit drawn, determine the value of I1, I and R, the combined resistance in the circuit. E = I1 (160) = I2 (4000) = I3 (32000) I1 = 2 / 160 = 0. 0125 A I2 = 2 / 4000 = 5 x 10-4 AI3 = 2 / 32000 = 6. 25 x 10-5 ASince I = I1 + I2 + I3, I = 13. 1 mAApplying Ohm’s Law, R = 213. 1 x 10-3 = 153 ? | | Example: A assault and battery with an EMF of 20 V and an internal resistance of 2. 0 ? is connected to resistors R1 and R2 as shown in the diagram. A total current of 4. 0 A is supplied by the battery and R2 has a resistance of 12 ?. Calculate the resistance of R1 and the power supplied to each circuit factor. E †I r = I2 R2 20 †4 (2) = I2 (12) I2 = 1A Therefore, I1 = 4 †1 = 3 AE †I r = I1 R1 12 = 3 R1 Therefore, R1 = 4 major power supplied to R1 = (I1)2 R1 = 36 W Power supplied to R2 = (I2)2 R2 = 12 W| |For potential divider with 2 resistors in series, Potential drop across R 1, V1 = R1 / (R1 + R2) x PD across R1 ; R2 Potential drop across R2, V1 = R2 / (R1 + R2) x PD across R1 ; R2 Example: Two resistors, of resistance ccc k? and 500 k? respectively, form a potential divider with outer junctions maintained at potentials of +3 V and -15 V. suss out the potential at the junction X between the resistors. The potential difference across the three hundred k? resistor = 300 / (300 + 500) [3 †(-15)] = 6. 75 V The potential at X = 3 †6. 75 = -3. 75 V A thermistor is a resistor whose resistance varies greatly with temperature.Its resistance decreases with increasing temperature. It can be used in potential divider circuits to admonisher and break temperatures. Example: In the go steady on the right, the thermistor has a resistance of 800 ? when hot, and a resistance of 5000 ? when cold. desex the potential at W when the temperature is hot. When thermistor is hot, potential difference across it = [800 / (800 + 1700)] x (7 †2) = 1. 6 VThe pote ntial at W = 2 + 1. 6 V = 3. 6 V| | A Light dependent resistor (LDR) is a resistor whose resistance varies with the earnestness of light falling on it. Its resistance decreases with increasing light rapture.It can be used in a potential divider circuit to monitor light earnestness. Example: In the go into below, the resistance of the LDR is 6. 0 M in the Cimmerian but then drops to 2. 0 k in the light Determine the potential at point P when the LDR is in the light. In the light the potential difference across the LDR= [2k / (3k + 2k)] x (18 †3) = 6 VThe potential at P = 18 †6= 12 V| | The potential difference along the outfit is proportional to the length of the wire. The sliding wrap up leave alone move along wire AB until it finds a point along the wire such that the galvanometer shows a zero reading.When the galvanometer shows a zero reading, the current through the galvanometer (and the device that is being tested) is zero and the batch is give tongue to to be â€Å"balanced”. If the cell has negligible internal resistance, and if the potentiometer is balanced, EMF / PD of the unknown source, V = [L1 / (L1 + L2)] x E Example: In the circuit shown, the potentiometer wire has a resistance of 60 ?. Determine the EMF of the unknown cell if the balanced point is at B. Resistance of wire AB= [0. 65 / (0. 65 + 0. 35)] x 60 = 39 ? EMF of the test cell= [39 / (60 + 20)] x 12| Work, Energy and PowerWork through by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s romaine lettuce ? Negative work is said to be done by F if x or its compo. is anti- agree to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {whitethorn need to find area by â€Å"counting the full-strengths”. } By Principle of Conservation of Energy, Work Done on a system = KE tuck + GPE gain + Work done aga inst attrition} Consider a strong object of mass m that is initially at rest.To accelerate it homogeneously to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, utilize the equation: v2 = u2 +2as, as = 12 (v2 †u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK| = Work done by the force F = Fs = mas = ? m (v2 †u2)| Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them.The gravitational potential energy of an object is the energy it possesses by rectitude of its position in a gravitational field. malleable potential energy: this arises in a system of atoms where there are either attractive or salacious short-range inter-atomic forces between them. Electric potential energy: this arises in a system of charges where there a re either attractive or horrid electric forces between them. The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by: F = †dU / dx.Consider an object of mass m being lifted vertically by a force F, without acceleration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass| = Work done by the force F = F s = F h = m g h| Efficiency: The ratio of (useful) output energy of a machine to the input energy. ie =| helpful rig Energy| x100% =| drillful Output Power| x100%| | stimulus Energy| | Input Power| | Power {instantaneous} is defined as the work done per unit time. P =| Total Work Done| =| W| | Total meter| | t|Since work done W = F x s, P =| F x s| =| Fv| | t| | | * for object pitiable at const speed: F = Total insubordinate force {equilibrium condition} * for object outset to accelerate: F = To tal insubordinate force + ma Forces Hookes Law: Within the limit of proportionality, the appendage produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit reference point (F/x) bendable potential energy/ furrow energy = Area under the F-x graph {May need to â€Å"count the squares”} For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? x2 Forces on Masses in Gravitational palm: A percentage of space in which a mass experiences an (attractive) force due to the presence of some other mass. Forces on Charge in Electric handle: A region of space where a charge experiences an (attractive or repulsive) force due to the presence of other charge. Hydro quiet Pressure p = ? gh {or, pressure difference between 2 points separated by a vertical distance of h } Upthrust: An up(a) force exerted by a roving on a submerged or floating object; arises because of the difference in pressure between t he top(prenominal) and disdain surfaces of the object.Archimedes Principle: Upthrust = weight of the legato dis situated by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces: * The contact force between two surfaces = ( clang2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion {or attempted motion}, and * Its value varies up to a utmost value {called the atmospheric static friction} Viscous Forces: * A force that opposes the motion of an object in a fluid * simply exists when there is (relative) motion Magnitude of viscous force increases with the speed of the object Centre of Gravity of an object is defined as that pt through which the good weight of the object may be considered to act. A catch is a pair of forces which tends to produce rotation only. Moment of a Force: The product of the force and the right distance of its line of action to t he peg torque of a Couple: The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object): 1.The concomitant force acting on it in any direction equals zero 2. The resultant moment roughly any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1. The lines of action of the 3 forces must pass through a common point 2. When a sender diagram of the three forces is drawn, the forces will form a unopen triangle (vector triangle), with the 3 vectors pointing in the same druthers around the triangle. Principle of Moments: For a body to be in equilibrium, the sum of all the contraclockwise moments well-nigh any point must be equal to the sum of all the clockwise moments about that same point.Measurement Base quantities and their units; mass (kg), length (m), time (s), cur rent (A), temperature (K), amount of substance (mol): Base Quantities| SI wholes| | Name| Symbol| Length| criterion| m| Mass| kilogram| kg| Time| second| s| Amount of substance| mole| mol| Temperature| Kelvin| K| Current| ampere| A| Luminous intensity| candela| cd| Derived units as products or quotients of the base units: Derived| Quantities Equation| Derived Units| Area (A)| A = L2| m2| Volume (V)| V = L3| m3| Density (? )| ? = m / V| kg m-3| Velocity (v)| v = L / t| ms-1| Acceleration (a)| a = ? v / t| ms-1 / s = ms-2|Momentum (p)| p = m x v| (kg)(ms-1) = kg m s-1| Derived Quantities| Equation| Derived Unit| Derived Units| | | peculiar(prenominal) Name| Symbol| | Force (F)| F = ? p / t| Newton| N| [(kg m s-1) / s = kg m s-2| Pressure (p)| p = F / A| Pascal| Pa| (kg m s-2) / m2 = kg m-1 s-2| Energy (E)| E = F x d| joule| J| (kg m s-2)(m) = kg m2 s-2| Power (P)| P = E / t| watt| W| (kg m2 s-2) / s = kg m2 s-3| relative frequency (f)| f = 1 / t| hertz| Hz| 1 / s = s-1| Charge (Q)| Q = I x t| coulomb| C| A s| Potential Difference (V)| V = E / Q| volt| V| (kg m2 s-2) / A s = kg m2 s-3 A-1| Resistance (R)| R = V / I| ohm| ? (kg m2 s-3 A-1) / A = kg m2 s-3 A-2| prefixes and their symbols to indicate decimal fraction sub-multiples or multiples of both base and derived units: Multiplying Factor| Prefix| Symbol| 10-12| pico| p| 10-9| nano| n| 10-6| micro| ? | 10-3| milli| m| 10-2| centi| c| 10-1| decid| d| 103| kilo| k| 106| mega| M| 109| giga| G| 1012| tera| T| Estimates of physical quantities: When making an adjudicate, it is only reasonable to give the figure to 1 or at most 2 important figures since an estimate is not very precise. Physical Quantity| valid Estimate| Mass of 3 cans (330 ml) of Coke| 1 kg|Mass of a medium-sizingd car| curtilage kg| Length of a football field| 100 m| Reaction time of a young man| 0. 2 s| * Occasionally, students are asked to estimate the area under a graph. The normal method of counting squares within the enclosed area is use d. (eg. Topic 3 (Dynamics), N94P2Q1c) * Often, when making an estimate, a figure and a simple calculation may be involved. EXAMPLE 1: Estimate the average track speed of a typical 17-year-old? s 2. 4-km run. velocity = distance / time = 2400 / (12. 5 x 60) = 3. 2 ? 3 ms-1 EXAMPLE 2: Which estimate is down-to-earth? | Option| Explanation|A| The kinetic energy of a bus traveling on an motorway is 30000J| A bus of mass m travelling on an expressway will travel between 50 to 80 kmh-1, which is 13. 8 to 22. 2 ms-1. Thus, its KE will be approximately ? m(182) = 162m. Thus, for its KE to be 30000J: 162m = 30000. Thus, m = 185kg, which is an crocked weight for a bus; ie. This is not a realistic estimate. | B| The power of a domestic help light is 300W. | A single light bulb in the house usually runs at about 20W to 60W. Thus, a domestic light is unlikely to run at more than 200W; this estimate is rather high. | C| The temperature of a hot oven is 300 K. 300K = 27 0C. Not very hot. | D | The volume of air in a car tyre is 0. 03 m3. | | Estimating the width of a tyre, t, is 15 cm or 0. 15 m, and estimating R to be 40 cm and r to be 30 cm,volume of air in a car tyre is = ? (R2 †r2)t = ? (0. 42 †0. 32)(0. 15) = 0. 033 m3 ? 0. 03 m3 (to one sig. fig. )| Distinction between systematic faultings (including zero erroneousnesss) and random errors and between clearcutness and accuracy: Random error: is the type of error which causes readings to scatter about the true value. Systematic error: is the type of error which causes readings to deviate in one direction from the true value.Precision: refers to the degree of agreement (scatter, spread) of recurrent measurements of the same amount. {NB: regardless of whether or not they are correct. } Accuracy: refers to the degree of agreement between the result of a measurement and the true value of the amount. | ; ; R delusion Higher ; ; ; ; ; ; Less critical ; ; ;| v v vS Error HigherLess Accuratev v v| | | | | | Assess the uncertainty in a derived quantity by simple addition of actual, fractional or percentage uncertainties (a rigorous statistical treatment is not required). For a quantity x = (2. 0 ± 0. 1) mm,Actual/ Absolute uncertainty, ? x = ± 0. 1 mm Fractional uncertainty, ? xx = 0. 05 Percentage uncertainty, ? xx 100% = 5 % If p = (2x + y) / 3 or p = (2x †y) / 3, ? p = (2? x + ? y) / 3 If r = 2xy3 or r = 2x / y3, ? r / r = ? x / x + 3? y / y Actual error must be recorded to only 1 significant figure, ; The number of decimal places a calculated quantity should have is determined by its actual error. For eg, suppose g has been initially calculated to be 9. 80645 ms-2 ; ? g has been initially calculated to be 0. 04848 ms-2. The final value of ? g must be recorded as 0. 5 ms-2 {1 sf }, and the appropriate recording of g is (9. 81 ± 0. 05) ms-2. Distinction between scalar and vector quantities: | scalar| Vector| Definition| A scalar quantity has a magnitude only. It is comp letely described by a certain number and a unit. | A vector quantity has both magnitude and direction. It can be described by an arrow whose length represents the magnitude of the vector and the arrow-head represents the direction of the vector. | Examples| surpass, speed, mass, time, temperature, work done, kinetic energy, pressure, power, electric charge etc. Common Error:Students tend to associate kinetic energy and pressure with vectors because of the vector components involved. However, such considerations have no bearings on whether the quantity is a vector or scalar. | Displacement, velocity, moments (or torque), momentum, force, electric field etc. | Representation of vector as two perpendicular components: In the diagram below, XY represents a flat kite of weight 4. 0 N. At a certain instant, XY is inclined at 30° to the naiant and the wind exerts a steady force of 6. 0 N at right angles to XY so that the kite flies freely.By accurate scale brief| By calculations vict imization sine and cos lettuce rules, or Pythagoras? theorem| Draw a scale diagram to find the magnitude and direction of the resultant force acting on the kite. R = 3. 2 N (? 3. 2 cm) at ? = 112° to the 4 N vector. | utilise cosine rule, a2 = b2 + c2 †2bc cos A R2 = 42 + 62 -2(4)(6)(cos 30°) R = 3. 23 NUsing sine rule: a / sin A = b / sin B 6 / sin ? = 3. 23 / sin 30° ? = 68° or 112° = 112° to the 4 N vector| Summing Vector Components| | Fx = †6 sin 30° = †3 NFy = 6 cos 30° †4 = 1. 2 NR = v(-32 + 1. 22) = 3. 23 Ntan ? = 1. 2 / 3 = 22°R is at an angle 112° to the 4 N vector. (90° + 22°)|Kinematics Displacement, speed, velocity and acceleration: Distance: Total length covered irrespective of the direction of motion. Displacement: Distance moved in a certain direction. Speed: Distance travelled per unit time. Velocity: is defined as the rate of change of displacement, or, displacement per unit time {NOT: displacement over time, nor, displacement per second, nor, rate of change of displacement per unit time} Acceleration: is defined as the rate of change of velocity. Using graphs to find displacement, velocity and acceleration: * The area under a velocity-time graph is the change in displacement. The gradient of a displacement-time graph is the {instantaneous} velocity. * The gradient of a velocity-time graph is the acceleration. The ‘SUVAT Equations of work The most master(prenominal) word for this chapter is SUVAT, which stands for: * S (displacement), * U (initial velocity), * V (final velocity), * A (acceleration) and * T (time) of a particle that is in motion. at a lower place is a list of the equations you MUST memorise, even if they are in the formula book, memorise them anyway, to ensure you can implement them quickly. 1. v = u +at| derived from definition of acceleration: a = (v †u) / t| 2. | s = ? (u + v) t| derived from the area under the v-t graph| 3. | v2 = u2 + 2as| derived from equations (1) and (2)| 4. | s = ut + ? at2| derived from equations (1) and (2)| These equations apply only if the motion takes place along a straight line and the acceleration is constant; {hence, for eg. , air resistance must be negligible. } Motion of bodies falling in a uniform gravitational field with air resistance: Consider a body touching in a uniform gravitational field under 2 different conditions: Without give vent Resistance:Assuming negligible air resistance, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the only force acting on it, causing it to experience a constant acceleration. Thus, the gradient of the v-t graph is constant throughout its rise and fall. The body is said to undergo free fall. With Air Resistance: If air resistance is NOT negligible and if it is projected upwards with the same initial velocity, as the body moves upwards, both air resistance and weight act downwards. Thus its speed will decrease at a rate greater than . 81 ms-2 . This causes the time taken to reach its utmost height reached to be lower than in the case with no air resistance. The max height reached is also reduced. At the highest point, the body is momentarily at rest; air resistance becomes zero and hence the only force acting on it is the weight. The acceleration is thus 9. 81 ms-2 at this point. As a body falls, air resistance opposes its weight. The downward acceleration is thus less than 9. 81 ms-2. As air resistance increases with speed, it eventually equals its weight (but in opposite direction).From then there will be no resultant force acting on the body and it will fall with a constant speed, called the terminal velocity. Equations for the plane and vertical motion: | x direction ( crosswise †axis)| y direction (vertical †axis)| s (displacement)| sx = ux t sx = ux t + ? ax t2| sy = uy t + ? ay t2 (Note: If projectile ends at same take as the start, then sy = 0)| u (initial velocity)| ux| uy| v (final velocity)| vx = ux + axt (Note: At max height, vx = 0)| vy = uy + at vy2 = uy2 + 2asy| a (acceleration)| ax (Note: Exists when a force in x direction present)| ay (Note: If object is falling, then ay = -g)| (time)| t| t| Parabolic Motion: tan ? = vy / vx ?: direction of tangential velocity {NOT: tan ? = sy / sx } Forces Hookes Law: Within the limit of proportionality, the extension produced in a material is directly proportional to the force/load applied F = kx Force constant k = force per unit extension (F/x) Elastic potential energy/strain energy = Area under the F-x graph {May need to â€Å"count the squares”} For a material that obeys Hooke? s law, Elastic Potential Energy, E = ? F x = ? k x2 Forces on Masses in Gravitational handle: A region of space in which a mass experiences an (attractive) force due to the presence of some other mass.Forces on Charge in Electric field: A region of space where a charge experiences an (attractive or repulsive) force due to the presence of another charge. Hydrostatic Pressure p = ? gh {or, pressure difference between 2 points separated by a vertical distance of h } Upthrust: An upward force exerted by a fluid on a submerged or floating object; arises because of the difference in pressure between the fastness and lower surfaces of the object. Archimedes Principle: Upthrust = weight of the fluid dis lay by submerged object. ie Upthrust = Volsubmerged x ? fluid x g Frictional Forces: The contact force between two surfaces = (friction2 + normal reaction2)? * The component along the surface of the contact force is called friction * Friction between 2 surfaces always opposes relative motion {or attempted motion}, and * Its value varies up to a supreme value {called the static friction} Viscous Forces: * A force that opposes the motion of an object in a fluid * Only exists when there is (relative) motion * Magnitude of viscous force increases with the speed of the object Centre of Gravity of an obje ct is defined as that pt through which the integral weight of the object may be considered to act.A couple is a pair of forces which tends to produce rotation only. Moment of a Force: The product of the force and the perpendicular distance of its line of action to the pivot Torque of a Couple: The produce of one of the forces of the couple and the perpendicular distance between the lines of action of the forces. (WARNING: NOT an action-reaction pair as they act on the same body. ) Conditions for Equilibrium (of an extended object): 1. The resultant force acting on it in any direction equals zero 2. The resultant moment about any point is zero If a mass is acted upon by 3 forces only and remains in equilibrium, then 1.The lines of action of the 3 forces must pass through a common point 2. When a vector diagram of the three forces is drawn, the forces will form a closed triangle (vector triangle), with the 3 vectors pointing in the same orientation around the triangle. Principle of M oments: For a body to be in equilibrium, the sum of all the anticlockwise moments about any point must be equal to the sum of all the clockwise moments about that same point. Work, Energy and Power Work Done by a force is defined as the product of the force and displacement (of its point of application) in the direction of the force W = F s cos ?Negative work is said to be done by F if x or its compo. is anti-parallel to F If a variable force F produces a displacement in the direction of F, the work done is determined from the area under F-x graph. {May need to find area by â€Å"counting the squares”. } By Principle of Conservation of Energy, Work Done on a system = KE gain + GPE gain + Work done against friction} Consider a rigid object of mass m that is initially at rest. To accelerate it uniformly to a speed v, a constant net force F is exerted on it, parallel to its motion over a displacement s. Since F is constant, acceleration is constant, Therefore, using the equation : 2 = u2 +2as, as = 12 (v2 †u2) Since kinetic energy is equal to the work done on the mass to bring it from rest to a speed v, The kinetic energy, EK| = Work done by the force F = Fs = mas = ? m (v2 †u2)| Gravitational potential energy: this arises in a system of masses where there are attractive gravitational forces between them. The gravitational potential energy of an object is the energy it possesses by virtue of its position in a gravitational field. Elastic potential energy: this arises in a system of atoms where there are either attractive or repulsive short-range inter-atomic forces between them.Electric potential energy: this arises in a system of charges where there are either attractive or repulsive electric forces between them. The potential energy, U, of a body in a force field {whether gravitational or electric field} is related to the force F it experiences by: F = †dU / dx. Consider an object of mass m being lifted vertically by a force F, without acce leration, from a certain height h1 to a height h2. Since the object moves up at a constant speed, F is equal to mg. The change in potential energy of the mass| = Work done by the force F = F s = F h = m g h|Efficiency: The ratio of (useful) output energy of a machine to the input energy. ie =| Useful Output Energy| x100% =| Useful Output Power| x100%| | Input Energy| | Input Power| | Power {instantaneous} is defined as the work done per unit time. P =| Total Work Done| =| W| | Total Time| | t| Since work done W = F x s, P =| F x s| =| Fv| | t| | | * for object moving at const speed: F = Total resistive force {equilibrium condition} * for object beginning to accelerate: F = Total resistive force + ma Wave Motion Displacement (y): frame of an oscillating particle from its equilibrium position. bounteousness (y0 or A): The upper limit magnitude of the displacement of an oscillating particle from its equilibrium position. Period (T): Time taken for a particle to undergo one complete cy cle of oscillation. Frequency (f): Number of oscillations performed by a particle per unit time. Wavelength (? ): For a progressive swing, it is the distance between any two concomitant particles that are in phase, e. g. it is the distance between 2 square(a) crests or 2 troughs. Wave speed (v): The speed at which the hustle shape travels in the direction of the propagation of the prosper.Wave front: A line or surface joining points which are at the same state of oscillation, i. e. in phase, e. g. a line joining crest to crest in a jounce. Ray: The path taken by the totter. This is used to indicate the direction of kink propagation. Rays are always at right angles to the motion fronts (i. e. stray fronts are always perpendicular to the direction of propagation). From the definition of speed, Speed = Distance / Time A wave travels a distance of one wavelength, ? , in a time interval of one period, T. The frequency, f, of a wave is equal to 1 / T Therefore, speed, v = ? / T = (1 / T)? f? v = f? Example 1: A wave travelling in the positive x direction is showed in the figure. Find the amplitude, wavelength, period, and speed of the wave if it has a frequency of 8. 0 Hz. Amplitude (A) = 0. 15 mWavelength (? ) = 0. 40 mPeriod (T) = 1f = 18. 0 ? 0. 125 sSpeed (v) =f? = 8. 0 x 0. 40 = 3. 20 m s-1A wave which results in a net transfer of energy from one place to another is known as a progressive wave. | | military capability {of a wave}: is defined as the rate of energy flow per unit time {power} per unit cross-sectional area perpendicular to the direction of wave propagation.Intensity = Power / Area = Energy / (Time x Area) For a point source (which would emit spherical wavefronts), Intensity = (? m? 2xo2) / (t x 4? r2) where x0: amplitude ; r: distance from the point source. Therefore, I ? xo2 / r2 (Pt Source) For all wave sources, I ? (Amplitude)2 Transverse wave: A wave in which the oscillations of the wave particles {NOT: movement} are perpendicular t o the direction of the propagation of the wave. Longitudinal wave: A wave in which the oscillations of the wave particles are parallel to the direction of the propagation of the wave.Polarisation is said to go by when oscillations are in one direction in a plane, {NOT just â€Å"in one direction”} normal to the direction of propagation. {Only transversal waves can be polarized; longitudinal waves can’t. }Example 2: The following stationary wave prescript is obtained using a C. R. O. whose dissemble is graduated in centimetre squares. Given that the time-base is adjusted such that 1 unit on the horizontal axis of the screen corresponds to a time of 1. 0 ms, find the period and frequency of the wave. Period, T = (4 units) x 1. 0 = 4. 0 ms = 4. 0 x 10-3 sf = 1 / T = 14 x 10-3 250 Hz| | Superposition Principle of Superposition: When two or more waves of the same type meet at a point, the resultant displacement of the waves is equal to the vector sum of their case-by-c ase displacements at that point. Stretched String A horizontal rope with one end fixed and another habituated to a vertical oscillator. Stationary waves will be produced by the direct and reflected waves in the string. Or we can have the string stopped at one end with a pulley as shown below. atomizes A microwave emitter placed a distance extraneous from a metal plate that reflects the emitted wave.By moving a sensing element along the path of the wave, the bosss and antinodes could be detected. Air column A tuning fork held at the mouth of a open tube projects a sound wave into the column of air in the tube. The length of the tube can be changed by varying the water level. At certain lengths of the tube, the air column resonates with the tuning fork. This is due to the formation of stationary waves by the incident and reflected sound waves at the water surface. Stationary (Standing) Wave) is one * whose waveform/wave indite does not advance {move}, where there is no net trans port of energy, and * where the positions of antinodes and nodes do not change (with time). A stationary wave is formed when two progressive waves of the same frequency, amplitude and speed, travelling in opposite directions are superposed. {Assume boundary conditions are met} | Stationary waves| Stationary Waves Progressive Waves| Amplitude| Varies from maximum at the anti-nodes to zero at the nodes. | Same for all particles in the wave (provided no energy is lost). | Wavelength| twice the distance between a pair of disposed nodes or anti-nodes. The distance between two consecutive points on a wave, that are in phase. | leg| Particles in the same segment/ between 2 coterminous nodes, are in phase. Particles in adjacent segments are in anti-phase. | All particles within one wavelength have different phases. | Wave Profile| The wave profile does not advance. | The wave profile advances. | Energy| No energy is transported by the wave. | Energy is transported in the direction of t he wave. | Node is a region of deleterious principle of superposition where the waves always meet out of phase by ? radians. Hence displacement here is for good zero {or minimum}.Antinode is a region of constructive superposition where the waves always meet in phase. Hence a particle here vibrates with maximum amplitude {but it is NOT a pt with a permanent large displacement! } Dist between 2 successive nodes / antinodes = ? / 2 Max pressure change occurs at the nodes {NOT the antinodes} because every node changes fr being a pt of calculus to become a pt of rarefaction {half a period later} Diffraction: refers to the spreading {or change shape} of waves when they pass through an opening {gap}, or round an obstacle (into the â€Å"shadow” region). Illustrate with diag} For significant diffraction to occur, the size of the gap ? ? of the wave For a diffraction grating, d sin ? = n ? , d = dist between successive slits {grating spacing} = reciprocal of number of lines per me tre When a â€Å"white light” passes through a diffraction grating, for each order of diffraction, a longer wavelength {red} diffracts more than a shorter wavelength {violet} {as sin ? ? ? }. Diffraction refers to the spreading of waves as they pass through a narrow slit or near an obstacle. For diffraction to occur, the size of the gap should approximately be equal to the wavelength of the wave.Coherent waves: Waves having a constant phase difference {not: zero phase difference / in phase} stay may be described as the superposition of waves from 2 coherent sources. For an observable / well-defined fray pattern, the waves must be coherent, have about the same amplitude, be unpolarised or polarised in the same direction, ; be of the same type. Two-source interference using: 1. Water Waves birth control device patterns could be observed when two dippers are attached to the vibrator of the ripple tank.The ripples produce constructive and destructive interference. The dippers are coherent sources because they are fixed to the same vibrator. 2. Microwaves Microwave emitted from a transmitter through 2 slits on a metal plate would also produce interference patterns. By moving a detector on the opposite side of the metal plate, a series of rise and fall in amplitude of the wave would be registered. 3. Light Waves (Young? s double slit experiment) Since light is emitted from a bulb randomly, the way to obtain two coherent light sources is by splitting light from a single slit.The 2 beams from the double slit would then interfere with each other, creating a pattern of alternate bright and dark fringes (or high and low intensities) at regular intervals, which is also known as our interference pattern. Condition for shaping Interference at a pt P: Phase difference of the 2 waves at P = 0 {or 2? , 4? , etc} Thus, with 2 in-phase sources, * implies path difference = n? ; with 2 antiphase sources: path difference = (n + ? )? Condition for Destructive Interference at a pt P: Phase difference of the 2 waves at P = ? { or 3? , 5? , etc } With 2 in-phase sources, + implies path difference = (n+ ? ), with 2 antiphase sources: path difference = n ? Fringe separation x = ? D / a, if a;;D {applies only to Youngs Double Slit interference of light, ie, NOT for microwaves, sound waves, water waves} Phase difference ?? betw the 2 waves at any pt X {betw the central & 1st maxima) is (approx) proportional to the dist of X from the central maxima. Using 2 sources of equal amplitude x0, the resultant amplitude of a bright fringe would be double {2×0}, & the resultant intensity increases by 4 times {not 2 times}. { IResultant ? (2 x0)2 } Electric FieldsElectric field strength / intensity at a point is defined as the force per unit positive charge acting at that point {a vector; Unit: N C-1 or V m-1} E = F / q > F = qE * The electric force on a positive charge in an electric field is in the direction of E, while * The electric force on a neg ative charge is opposite to the direction of E. * Hence a +ve charge placed in an electric field will accelerate in the direction of E and gain KE {& simultaneously lose EPE}, while a negative charge caused to move (projected) in the direction of E will decelerate, ie lose KE, { & gain EPE}. Representation of electric fields by field lines | | | | | Coulombs law: The (mutual) electric force F acting between 2 point charges Q1 and Q2 separated by a distance r is given by: F = Q1Q2 / 4?? or2 where ? 0: permittivity of free space or, the (mutual) electric force between two point charges is proportional to the product of their charges ; inversely proportional to the square of their separation. Example 1: Two positive charges, each 4. 18 ? C, and a negative charge, -6. 36 ? C, are fixed at the vertices of an equilateral triangle of side 13. 0 cm. Find the electrostatic force on the negative charge. | F = Q1Q2 / 4?? or2= (8. 99 x 109) [(4. 18 x 10-6)(6. 6 x 10-6) / (13. 0 x 10-2)2 ]= 14. 1 N (Note: negative sign for -6. 36 ? C has been ignored in the calculation)FR = 2 x Fcos300= 24. 4 N, vertically upwards| Electric field strength due to a Point Charge Q : E = Q / 4?? or2 {NB: Do NOT substitute a negative Q with its negative sign in calculations! } Example 2: In the figure below, determine the point (other than at infinity) at which the total electric field strength is zero. From the diagram, it can be observed that the point where E is zero lies on a straight line where the charges lie, to the left-hand(a) over(p) of the -2. 5 ? C charge. Let this point be a distance r from the left charge.Since the total electric field strength is zero, E6? = E-2? [6? / (1 + r)2] / 4?? or2 = [2. 5? / r2] / 4?? or2 (Note: negative sign for -2. 5 ? C has been ignored here) 6 / (1 + r)2 = 2. 5 / r2 v(6r) = 2. 5 (1 + r) r = 1. 82 m The point lies on a straight line where the charges lie, 1. 82 m to the left of the -2. 5 ? C charge. Uniform electric field between 2 Charged Pa rallel Plates: E = Vd, d: perpendicular dist between the plates, V: potential difference between plates Path of charge moving at 90° to electric field: parabolic. beyond the pt where it exits the field, the path is a straight line, at a tangent to the parabola at exit.Example 3: An electron (m = 9. 11 x 10-31 kg; q = -1. 6 x 10-19 C) moving with a speed of 1. 5 x 107 ms-1, enters a region between 2 parallel plates, which are 20 mm apart and 60 mm long. The top plate is at a potential of 80 V relative to the lower plate. Determine the angle through which the electron has been deflected as a result of passing through the plates. Time taken for the electron to travel 60 mm horizontally = Distance / Speed = 60 x 10-3 / 1. 5 x 107 = 4 x 10-9 s E = V / d = 80 / 20 x 10-3 = 4000 V m-1 a = F / m = eE / m = (1. 6 x 10-19)(4000) / (9. 1 x 10-31) = 7. 0 x 1014 ms-2 vy = uy + at = 0 + (7. x 1014)( 4 x 10-9) = 2. 8 x 106 ms-1 tan ? = vy / vx = 2. 8 x 106 / 1. 5 x 107 = 0. 187 Therefore ? = 10 . 6° Effect of a uniform electric field on the motion of charged particles * Equipotential surface: a surface where the electric potential is constant * Potential gradient = 0, ie E along surface = 0 } * Hence no work is done when a charge is moved along this surface. { W=QV, V=0 } * Electric field lines must meet this surface at right angles. * {If the field lines are not at 90° to it, it would imply that there is a non-zero component of E along the surface. This would contradict the fact that E along an equipotential = 0. Electric potential at a point: is defined as the work done in moving a unit positive charge from infinity to that point, { a scalar; unit: V } ie V = W / Q The electric potential at infinity is defined as zero. At any other point, it may be positive or negative depending on the sign of Q that sets up the field. {Contrast gravitational potential. } Relation between E and V: E = †dV / dr i. e. The electric field strength at a pt is numerically equal to t he potential gradient at that pt. NB: Electric field lines point in direction of decreasing potential {ie from high to low pot}.Electric potential energy U of a charge Q at a pt where the potential is V: U = QV Work done W on a charge Q in moving it across a pd ? V: W = Q ? V Electric Potential due to a point charge Q : V = Q / 4?? or {NB: Substitute Q with its sign} Electromagnetism When a conductor carrying a current is placed in a magnetic field, it experiences a magnetic force. The figure above shows a wire of length L carrying a current I and lying in a magnetic field of flux engrossment B. Suppose the angle between the current I and the field B is ? , the magnitude of the force F on the conductor is iven by F = BILsin? The direction of the force can be found using Fleming? s Left Hand Rule (see figure above). Note that the force is always perpendicular to the plane containing both the current I and the magnetic field B. * If the wire is parallel to the field lines, then ? = 0°, and F = 0. (No magnetic force acts on the wire) * If the wire is at right angles to the field lines, then ? = 90°, and the magnetic force acting on the wire would be maximum (F = BIL) Example The 3 diagrams below each show a magnetic field of flux density 2 T that lies in the plane of the page.In each case, a current I of 10 A is directed as shown. Use Flemings Left Hand Rule to predict the directions of the forces and work out the magnitude of the forces on a 0. 5 m length of wire that carries the current. (Assume the horizontal is the current) | | | F = BIL sin? = 2 x 10 x 0. 5 x sin90 = 10 N| F = BIL sin? = 2 x 10 x 0. 5 x sin60 = 8. 66 N| F = BIL sin ? = 2 x 10 x 0. 5 x sin180 = 0 N| magnetic flux density B is defined as the force acting per unit current in a wire of unit length at right-angles to the field B = F / ILsin ? > F = B I L sin ? {? Angle between the B and L} {NB: write down the above defining equation & define each symbol if youre not able to give t he â€Å"statement form”. } Direction of the magnetic force is always perpendicular to the plane containing the current I and B {even if ? ? 0} The Tesla is defined as the magnetic flux density of a magnetic field that causes a force of one newton to act on a current of one ampere in a wire of length one metre which is perpendicular to the magnetic field. By the Principle of moments, Clockwise moments = Anticlockwise moments mg • x = F • y = BILsin90 • yB = mgx / ILy Example A 100-turn rectangular roll 6. 0 cm by 4. 0 cm is pivoted about a horizontal axis as shown below. A horizontal uniform magnetic field of direction perpendicular to the axis of the coil passes through the coil. Initially, no mass is placed on the pan and the arm is kept horizontal by adjusting the counter-weight. When a current of 0. 50 A flows through the coil, equilibrium is restored by placing a 50 mg mass on the pan, 8. 0 cm from the pivot. Determine the magnitude of the magnetic flu x density and the direction of the current in the coil.Taking moments about the pivot, sum of Anti-clockwise moments = Clockwise moment (2 x n)(FB) x P = W x Q (2 x n)(B I L) x P = m g x Q, where n: no. of wires on each side of the coil (2 x 100)(B x 0. 5 x 0. 06) x 0. 02 = 50 x 10\r\n'