Physics AS - Unit 2 - Mechanics, Materials and Waves - Revisin Ntes Mechanics Scalar and Vectr Quantities Definitin A Vectr Quantity has magnitude as well as directin while a scalar quantity nly had magnitude Scalar Distance, Speed, Mass Vectrs Displacement, Velcity, Acceleratin Resultant a b a a+b Pythagras - a = b 2 + c 2 Parallelgram Methd Nte: when yu knw the resultant and drawing the hrizntal and vertical cmpnent draw bth f the cmpnents frm the same pint that the resultant starts frm (resultant shuld be in- between 2 cmpnents) Reslutin Of Vectrs Reslves vectr int hrizntal and vertical cmpnents. y a θ x x = acsθ y = asinθ Balanced Frces If an bject is in equilibrium the tw frces acting upn a pint are equal and ppsite t each ther and the sum f the anticlckwise mments = the sum f the clckwise mments. N resultant frce acting. Equilibrium 3 frces 2 f the frces (resultant) are equal and ppsite t the third frce, there is n resultant frce acting. Can als be drawn in a clsed triangle. T calculate and unknwn frce r frces reslve each hrizntally and vertically and slve, if bject is in equilibrium there is n resultant frce!
Mments Definitin The mment f a frce abut a pint is equal t the frce multiplied by the perpendicular distance between the line f actin (f frce) and the pivt (place taking mment abut) The principle f mments states that fr an bject t be in equilibrium the sum f the anticlckwise mments must equal the sum f the clckwise mments at any pint. Tw Supprt Prblems If center f mass f beam is midway between bth supprts the frce each supprt exerts is equal. If nt: Take mments abut S 1 t calculate S 2 t wrk ut S 1 r vice versa Cuples Definitin A cuple is pair f equal and ppsite parallel frces acting n a bdy but nt at the same pint. The turning effect f a cuple f equal and ppsite frces is equal t One f the frces multiplied by the perpendicular distance between the tw frces. Mtin Alng a Straight Line Displacement Time Graphs Gradient = Velcity Velcity Time Graphs Gradient = Acceleratin Area Underneath = Displacement (if velcity is always +) Cnstant Acceleratin Frmulas S = ut + 1 2 at 2 S = 1 (u + v)t 2 v = u + at v 2 = u 2 + 2as Where V=final velcity, S=displacement, U=initial velcity, T=time Remember Acceleratin is ms - 2 When calculating distance cvered when smething is accelerating and starting velcity is nt 0 use frmulas!! IF EVER ASKS FOR DECELERATION NEVER USE A MINUS SIGN IS IT IS SLOWING DOWN! Prjectile Mtin Acceleratin due t gravity = 9.81ms - 2 Reslve hrizntally and vertically Gravity nly effects the vertical cmpnent f mtin Newtn s First Law Of Mtin Objects stay at rest r remain in unifrm mtin unless acted upn my an external frce S basically if an bject is mving and nthing is acting upn it (e.g air resistance) the bject will carry n ging frever, this happens in space as n air resistance
Newtn s Secnd Law f Mtin The frce applies t an bject is prprtinal t the mass and the acceleratin f the bject as a result f the being applied F = ma If 2 frces are acting upn the bject it is the resultant frce that is used in this equatin - F 1 F 2 if F 1 is bigger than F 2 With a rcket that is travelling directly upwards the bject has t vercme its wn weight ( mg ) in rder t accelerate s F mg = ma With a lift If mving at a cnstant speed upwards F = mg If mving upwards and accelerating F = mg + ma > mg If lift mving up and decelerating F = mg + ma < mg If mving dwnwards and accelerating (ging faster in dwnwards directin) F = mg + ma > mg If mving dwnwards and decelerating F = mg + ma < mg Terminal Speed Drag frce depends upn Velcity, viscsity f liquid mving thrugh and shape f an bject Drag perates in the ppsite directin f travel s F = mg d therefr ma = mg d s acceleratin equals a = g d m Remember when an bject is falling the frce acting dwnwards twards earth is its weight! ( mg ) Stpping Distances Thinking distance Braking Distance using v 2 = u 2 + 2as gives us a = v2 S if speed reduced frm 20mph t 10 mph the braking distance is reduced by mre than half Limiting frictin is the largest frictinal frce an bject can experience due t the rad befre skidding (measured in N) Impact Frces Impact frce can be in g which is the declaratin divides by g Impact Time time frm when 2 bjects cllide at 2 different velcities t the time there are bth travelling with the same velcity Cntact Time Time 2 bjects are in cntact with each ther 2s As F = ma the frce an bject experiences can be minimized by reducing the acceleratin r declaratin f an bject this is dne by increasing the time the bject is changing speed as a = Δv Δt s increasing t decreases a which subsequently reduces F This is why things like crumple znes and seat belts are used t increase time f declaratin t decrease average frce applied t a persn safer! Wrk Wrk = F d csθ F D
Measured in Jules Kinetic Energy E k = 1 2 mv2 Gravitatinal Ptential Energy ΔE g = mgδh As an bject f mass m is released abve the grund, it gravitatinal ptential energy is cnverted int kinetic energy as the height abve grund decreases as ΔE g = mgδh, in reality this is nt strictly true as sme gravitatinal ptential is cnverted int wrk against Air resistance and sme lst as heat and sund. Pendulum Bb if released frm height h i then at any pint where the height is h v the speed f the bb is such that Kinetic energy at that pint = lss in gravitatinal ptential (ignring all ver external frces) 1 2 mv2 = mg(h i h f ) Pwer Pwer Rate f transfer f energy measured in Watts p = ΔE Δt p = ΔW Δt p = fv Where E = Energy, W= Wrk and t = time Pwer = frce x velcity as it s the rate wrk is dne per secnd as its frce multiplied by distance mved each secnd Efficiency useful energy transfered by machine efficiency f a machine = energy supplied t machine Wrk Dne by machine efficiency f a machine = energy supplied t machine S basically efficiency is utput pwer ver input pwer
Materials Density Density A materials mass per unit vlume p = m v Mass f ally, m = p a v a + p b v b and the density f that ally p = m v = p a v a + p b v b m Springs Hke s Law Hke s Law The extensin f a spring is prprtinal t the frce applied t it as lng as the limit f prprtinality is nt exceeded F = kδl (where k is the spring cnstant) Elastic Limit Maximum stress that can be applied t a material withut plastic behavir ccurring defrmatin Nte : Elastic limit may be different t Limit f Prprtinality Yield Pint A pint is reached at which nticeably larger change in length due t frce Breaking Stress/Ultimate Tensile Stress mst stress a material can withhld withut breaking Brittleness Materials that cant extend withut breaking are said t be brittle hwever brittle materials are strng in cmpressin and used in building where cmpressin is cnstant. Springs in parallel - K = K 1 + K 2 where k is the effective spring cnstant Springs in series - 1 K = 1 K 1 + 1 K 2 In rder t cmpare 2 different materials elastic prperties use yung s mdulus Yung s Mdulus Yung s Mdulus Stiffness cnstant f material measured in Nm- 2 Tensile Stress = frce per unit area σ = f a Tensile Strain = rati f riginal length and extensin ε = ΔL L This is delta L as in change in L s final - initial E = σ ε = Stress Strain Yung s mdulus has units Nm- 2 r Pascal (Pa) E = FL AΔL NO UNITS
Experimental determinatin f Yung s Mdulus Of a Material Tw identical wires are fixed in parallel Bth wires initially laded t remve kinks/strengthen knts Micrmeter adjusted t make sprit level hrizntal Initial micrmeter reading taken Meter ruler measure riginal length f wire l (ne under test) Secnd micrmeter used t measure diameter f wire in several places t imprve accuracy Calculate area by using πr 2 (half d t give r) Test wire laded with mass and micrmeter adjusted t level sprit level Micrmeter reading taken and extensin calculated by final micrmeter initial Further lads are added and repeated until range btained Unlad secnd set Graph f frce against extensin Gradient calculated using big triangle y 2 y 1 x 2 x 1 As gradient = F ΔL y = Gradient L A FL and Y = Y can be calculated by gradient multiplied by AΔL Accuracy Imprved Lng thin wire gives large extensin per unit frce therefr percentage uncertainty decreases Cntrl wire used s temperature changes d nt impact results Measure diameter in several places t give average Large triangle/range used t calculate gradient Lading and Unlading If elastic limit is reached the unlading curve will nt have same value as lading curve at F=0 it has been misshaped metal extended Lading area Unlading area = Thermal Energy Lsses int Material As lading area = wrk dne and unlading area = energy given back Remember when cunting squares t calculate wrk dne that each square is wrth whatever the square represents vertically multiplied by whatever it represents hrizntally
Waves Prgressive Waves Waves whse scillatins travel and d nt stay abut a fixed pint, this type f waves transfers energy Frequency Number f wave cycles that ccur in ne secnd measured in Hz Amplitude- Maximum displacement f a vibrating particle frm equilibrium Wavelength Distance between 2 adjacent pints/particles in phase in a wave Perid time fr ne cmplete wave t pass a fixed pint in space P = 1 f Transverse Waves Waves whse scillatins/vibratins are perpendicular t the directin f travel (light etc.) Lngitudinal Waves Waves whse scillatins/vibratin are in the same directin f travel (sund) Nte: Sund waves prpagate as a a series f cmpressins and rarefactins Wave Speed Speed f the waves is equal t distance traveled by wave in ne cycle divided by time taken fr ne cycle C = λ 1 f Therefr c = f λ Path/Phase Difference 2 pints are in phase if they are a whle wavelength apart (max displacement at same time)- there scillatins are in time with each ther 2 Pints are in Antiphase if they are half a wavelength apart (ne max displacement while ther experiences min displacement) Phase Difference " φ = 2π x x % " 1 2 $ ' (in radians) OR φ = 360 x x % 1 2 $ # λ & # λ & nte: x 1- x 2 is the distance between the tw pints Path Difference ' in degrees Between 2 different surces the phase difference between the 2 waves then they cnverge is: " φ = 360 S P S P % 1 2 $ ' # λ & nte: where S 1P is the distance frm surce 1 t the pint where the 2 waves cnverge
Plarizatin Plarizatin When scillatins f the electric field f an E.M wave are restricted t nly ne plane/directin the wave is said t be plarized Applicatin Plarid sunglasses reduce glare frm water as the light is partially plarized when reflected ff water thus the intensity f the light can be reduced This reduces glare Reflectin Angle f incidence is the same as angle f reflectin i = r Statinary Waves Superpsitin When 2 similar waves f similar frequency meet their resultant depends upn their amplitude and their relative phase difference If n phase difference angle then cnstructive interference ccurs If 180 degrees difference angle then destructive interference ccurs and waves cancel ut at that pint Principle Of Superpsitin States that the resultant displacement caused by 2 waves arriving at a pint is the vectr sum f the 2 displacements caused by each waves at that instant Statinary Waves Frmatin Frmed when 2 cntinuus waves travelling in ppsite directins f same frequency superimpse t frm n displacement (ndes - cmplete destructive) and pints f max displacement (antindes cmplete cnstructive interference) Statinary Wave Fixed pattern f vibratin where n energy is transferred alng the wave Nde particle/pint with zer displacement n amplitude Antinde Pint/particle with max displacement/amplitude The phase difference between 2 particles/pints n standing wave (that is the difference when ne is at max displacement and ther ne als is, in degrees) is zer if the pints/particles are between adjacent ndes r separated by even number f ndes r 180 degrees if they are separated by an dd number f ndes.
Fundamental Frequencies (as frm Distance between 2 Adjacent Ndes = λ 2 nde t nde = half a wavelength f ne f the cntinuus waves that frmed statinary wave seen in diagram) As seen in diagram the first fundamental that is when there is ne standing wave (1 anti- nde 1 nde) the length is half f the wavelength λ = 2L Using c = f λ We find: f 0 = c 2l (where c is speed f prpagatin f waves) Hence at the secnd harmnic r first vertne: λ = L Hence: f 1 = c l Furthermre at the third harmnic r secnd vertne: λ = 2 3 L Hence: f 2 = c 2 3 l
Refractin Refractin Change f directin f a wave when it prpagates thrugh a different medium Speed f light in air is 3.00 10 8 hwever when it enters mre ptically dense medium it reduces speed S when light enters a mre ptically dense medium its speed decreases and its directin changes mre twards the nrmal As c = f λ and the speed f light is reduced in mre ptically dense medium the wavelength f light changes (makes sense really the frequency cant suddenly change!) Abslute Refractive index Speed f light in Vacum n = Speed f Light in Medium nte: as the speed f different frequency f light is nt the same in a given medium (due t the fact the prpagatin directin changes t a different degree fr each frequency) t give abslute refractive index Yellw light frm a sdium bulb is used as a standard Sme typical R.Is Diamnd 2.4 Perspex 1.5 Air 1 Ice 1.31 Refractive Index between 2 materials n = Abslute f N! Abslute f N! If light frm ne medium 1 t medium 2 is N then light frm 2 t 1 is 1/N Snell s Law n 1 sinθ 1 = n 2 sinθ 2 OR sinθ 1 = n 2 = c 1 = λ 1 sinθ 2 n 1 c 2 λ 2 Nte where n is the abslute refractive index f a certain medium and θ 1 is the angle f incidence and θ 2 is the angle f refractin Ttal Internal Reflectin If angle f incidence is s large and the light is mving frm a mre dense t less dense medium (hence prpagatin directin changing away frm nrmal) the angle f incidence will be larger than 90 and subsequently the light des nt exit but is reflected back int medium The angle that this happens at that is the angle at which light is reflected at 90 t the nrmal (θ 2 = 90) is called the critical angle and usually dented by i sini sin90 = n 1 n 2 and as sin 90 = 1: sini = n 2 n 1 This is hw fiber ptic cables wrk: Cladding In Fiber Optic Cables Imprve tensile strength f the cable as the cre has t be very thin. Increase the critical angle needed fr ttal internal reflectin: hence reducing the multipath dispersin (merging f signals f light dwn a cable)
Stps cre being scratched as scratches cause light t be dispersed if it hits the scratch Imprves security as withut cladding light wuld be able t travel between fibers, as with cladding ttal internal reflectin ccurs and light cannt travel between fibers hwever withut cladding light wuld be free t travel between medians f similar ptical density (cre t cre) Used in Internet bradband t deliver fast internet cnnectin t husehlds. Interference When 2 similar waves meet at a pint, by the principle f superpsitin the cmbined displacement is fund by the vectr sum f the 2 displacements f each wave at that pint, if they arrive exactly in phase the waves will cnstructively interfere t frm a duble height wave, if exactly ut f phase (180 degrees) they will add tgether destructively and cancel In rder t view a steady interference pattern the wave surces have t be cherent Cherent Cnstant phase relatinship and same frequency The pattern we see will depend upn the phase difference f the 2 surces when they hit that pint A bright fringe When the waves cnstructively interfere, ccurs when path difference = nλ A dark fringe When the waves destructively interfere, ccurs when path difference = (n + 1 )λ as the waves have t be 180 degrees ut f phase 2 Tw- Slit interference patterns (Yung s Slits) Single slit diffractin spreads ut ne wave and causes it t act as 2 cherent surces fr 2 further slits. Shwed evidence fr wave thery f light Light bands ccur whenever the path difference between the light waves are a whle number f wavelengths (and dark when half number) The distance between tw successive maxima w depends upn: The distance between the 2 slits; increasing S makes fringes clser tgether (r w smaller) The wavelength f light, W is smaller at smaller wavelengths Distance D between the slits and the screen, if D increase W increases. w = λd s Als can use laser Laser light mnchrmatic Laser light Is cherent (cnstant phase relatinship same Hz) Highly directinal very little divergence Pattern Maxima Similar intensity and same width as central fringe Diffractin When waves pass thrugh a gap r arund an bstacle the waves spread ut this is called Diffractin When the slit width is the same as the wavelength perfect diffractin ccurs When slit is large than wavelength less diffractin ccurs Single Slit Diffractin Pattern Central maximum is twice as wide as the thers
Rest f the fringes decrease slightly in intensity as way frm the middle (same width) Maxima ccurs with cnstructive interference and zer intensity with destructive interference If gap gets smaller f distance between screen and slit increases s des width f maxima Central maxima gets larger if wavelength is lnger r the gap is smaller Diffractin Grating Series f unifrm narrw slits in parallel d sinθ = nλ Where d= the distance frm center t center f adjacent slits (N=1/d t wrk ut d if yu nly have number f slits in say a meter) θ is the angle f the rder yu are trying t calculate, n is the rder number trying t calculate 1 2 r 3 etc. Fractins f a degree usually expressed in minutes T find max number f rder substitute θ fr 90 (therefr sinθ = 1) and use equatin Applicatin: Spectrmeter Uses cllimatr t prduce parallel light and then uses diffractin grating t prduce spectrum pattern Use sample light frm exciting an atm t prduce pattern and analysis can be dne n this12 Always a white line at the Zer rder as all wavelengths arrive in phase at this pint The shrter the wavelength the shrter the angle Hence patterns can be cmplicated first rder red light line can be clse t secnd rder blue