GM r. v = For Newton s third law, the forces in the action/reaction pair always act on different objects

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1 SAT Physics Mechanics The dot poduct of two vectos: A B = AB cos θ The coss poduct of vectos: A B = AB sin θˆn. The magnitude of the coss poduct is equal to the aea of the paallelogam. We use the ight hand ule to find the diection.. Newton s laws F = m a F = G mm2 GM v = 2 R T 2 = 4π2 R 3 GM Fo Newton s thid law, the foces in the action/eaction pai always act on diffeent objects An object is in tanslational equilibium if F net = 0 (i.e a = 0 and v = constant). An object at est is in static equilibium F fiction = µf N, whee F N is the nomal foce. This is always in the opposite diection of intended motion. F N gives the maximum fictional foce, if the pushing foce is less than this then the object won t move. On an inclined plane, the foce paallel to the amp is F = mg sin θ, and the nomal foce is F N = mg cos θ..2 Enegy W = F d cos θ P = F v cos θ E P = GMm mg h E total = GMm 2 v escape = Wok is the scala poduct of foce and distance, i.e W = F d cos θ. Hence the wok done by the nomal foce is always zeo (since cos 90 = 0) The wok done by fiction is always negative (since it is antipaallel to the diection of motion). Fo a vaiable foce, W = F ds 2GM The Wok Enegy Theoem states that: W total = E K, so fo an acceleating body the wok done is 2 mv2 v 2 0 GP E = W by gavity. Fo small heights, E P = mg h. The path that the object takes is ielevant, hence gavity is a consevative foce E K = GMm 2, hence E total = GMm 2.

2 .3 Linea momentum p = mv F = p t Impulse = J = F t J = F KE = p2 2m The law of consevation of linea momentum states that in an isolated system, the total linea momentum will emain constant. Elastic collisions occu when momentum tansfes completely, KE is conseved. objects Nomally with had Inelastic collisions ae when the KE is not the same befoe and afte. Completely/pefectly/totally inelastic collisions ae when objects stick togethe. Fo two dimensional collisions, split up the x and y components..4 Cuved and Rotational Motion a c = v2 τ = F sin θ Angula momentum = L = mv = τ t = Iω v max = gµ I = m 2 Centipetal acceleation is the acceleation towads the cente of the cicle, its diection is always changing. Fo a satellite, the centipetal foce is equal to the gavitational foce. The centifugal foce is the outwad feeling due to inetia. If the centipetal foce wee emoved, the object would move at a tangent. When moving in a cicle but taking into account gavity, to wok out the nomal foce use F N +mg = mv2. To find the cente of mass of a system, select some point to be the oigin. Finding the x coodinate of the cente of mass: x cm = mx+m2+x2+ +mnxn m +m 2+ +m n, whee x n is the x coodinate of the nth object. The system woks as if its whole mass whee at x cm The same pocess can be done with the y-axis, and the final cente of mass is (x cm, y cm ) Centes of mass do not move, so if the coodinate has changed then the object has moved. All motion is a combination of tanslation and otation. Toque (τ) is a tuning foce that poduces angula acceleation The vetical component of the foce is F sin θ, and since τ = F, (whee is the distance fom the pivot), τ = F sin θ 2

3 Anothe method is to extend the line of action of the foce, pependicula to the pivot point. The distance l is called the leve am o moment am, and the toque of F is then τ = F l An object is in otational equilibium if τ net = 0, o τ clockwise = τ anticlockwise. Angula momentum (L) is the analog of linea momentum. τ = dl = (mv) t Objects not moving in cicula paths can have angula momentums, which ae defined elative to a efeence point. The analog of p = mv is L = Iω, whee I is the moment of inetia (analogous to mass) and ω is the angula speed. Angula momentum is conseved, so if I deceases, ω inceases. Angula displacement = θ. If s is the ac length, then θ = s In a igid body, all points along a adial line have the same angula displacement. Angula acceleation (not centipetal acceleation): α = dω Keple s Fist law states that the obit of each planet is an ellipse, with the Sun being a focus. Keple s Second Law states that a lines dawn fom the sun to diffeent planets will sweep equal aeas in equal times. Keple s Thid law states that T 2 a 3 whee a is the semimajo axis..5 Oscillations F = kx U s = 2 kx2 f = T T = 2π m k ω = 2πf x = A cos(ωt + φ) = T = 2π KE = 2 (A2 x 2 ) E total = 2 ka2 v max = ka 2 m Thee is a minus sign fo Hooke s law F = kx because the estoing foce woks against extension. Spings that obey Hooke s law ae called ideal o linea. L g Since we know that F = ma and F = kx, we get m d2 x 2 x = A cos(ωt + φ) = kx. Solving this diffeential equation gives If x = A at t = 0, φ = 0 Fo a simple pendulum, the estoing foce comes fom gavity and is given by F estoing = mg sin θ. If θ max (the angle at the maximum height) is small, then by the small angle appoximation sin θ θ the estoing foce is mgθ which is popotional to θ, so the system can be teated as simple hamonic. 3

4 2 Electicity and Magnetism 2. Electic Foces and Fields F E = f qq2 2 k = Nm 2 C E = F q E = k Q 2 V = E d Chage is quantized, the smallest size is the elementay chage e = C Electic field and electic foce ae vectos, electic potential and potential electic enegy ae scala values. To find the effects of multiple point chages (o multiple field lines), use supeposition (vecto addition). Equal but opposite chages fom an electic dipole Thee can be no electic field inside the body of a conducto Fo electon obits, set F E = M Ev Electic Potential and Capacitance U E = qv = kqq2 W = U E = q V V = k Q Q = CV C = ɛ0a d E P = 2 Q V = 2 CV 2 = Q 2 2 C The change in a chage s potential enegy is the negative wok done: U E = W E. Equipotential sufaces ae pependicula to field lines, no wok done. Most capacitos ae paallel plate capacitos. Capacitance aea of plates and distance between plates : C = ɛ0a d. One Faad is one coulomb pe volt. When capacitos stoe up chage, they have moe potential enegy. Fo capacitos in paallel, capacitance adds up nomally C equivalent = C + C 2 + C Fo capacitos in seies, capacitance adds up like: C equivalent = C + C 2 + C 3... To maintain chage sepaation in a capacito, we can add an insulating mateial (a dielectic) between the plates. A dielectic always inceases the capacitance of a capacito. The dielectic mateial is polaised, so negative chage builds up nea the positive plate of the capacito. This ceates an induced electic field E i in the opposite diection, so E total = E E i We can say that the electic field has deceased by a facto of κ. Hence since V = Ed and C = Q/V, C dielectic = κc without dielectic. κ is the dielectic constant, and is always geate than. 2.3 Diect Cuent Cicuits I = Q T V = IR R = ρl A P = IV = V 2 R = I2 R 4

5 2.4 Magnetic Foces and Fields F = q vb sin θ F = BIL sin θ B = µ0 I 2π The SI unit fo magnetic field stength is the tesla (T). One gauss is 0 4 T. F B is always pependicula to v and B. Magnetic fields cannot change the speed of an object, they do no wok. Only diection can be changed. The foce on a cuent caying wie is F = BIL sin θ, whee l is the length of the wie and θ is the angle between l and B. Magnetic field is popotional to the cuent and invesely popotional to the adius. The constant of popotionality is µ0 2π whee µ 0 is the pemeability of fee space (4π 0 7 T ma ). Fo a chage in a magnetic field, set mv2 = qvb. 2.5 Electomagnetic Induction ɛ = vbl Φ B = BA cos θ ɛ avg = dφ B When a wie moves pependicula to magnetic field lines, thee is a magnetic foce on the electons. This builds up a chage and induces a field. Hence the motion of a wie though a magnetic field ceates an emf, called the motional emf. The magnetic flux, Φ B though an aea A measues the density of the magnetic field lines though that aea. Φ B = BA cos θ The SI unit fo magnetic flux, T m 2, is a webe (Wb). Flux is scala, but we often descibe its diection. Faaday s law of electomagnetic induction states that the emf induced in a cicuit is equal to the ate of change of the magnetic flux: ɛ avg = dφ B The induced emf can poduce a cuent and its own magnetic field. Lenz s law states that the induced cuent will flow in the diection that opposes the change in magnetic flux that poduced it (consevation of enegy). ɛ = dφ B = d(blx) = Bl dx = Blv In a tansfome, V S V P = N S N P 3 Waves and Light 3. Waves c = fλ v = FT µ λ n = 2L n f n = nv 2L f n = nf v = f beat = f f 2 f d = v±v d v v s f s (highe sign fo motion towads) All waves of the same type in the same medium have the same speed. B ρ β = 0 log I I 0 Fo a tansvese wave in a sting of length L, its linea density µ = m/l. If the tension is F T, the speed of the wave is v = FT µ. We can then apply v = fλ. 5

6 When a wave passes into a new medium, its fequency is constant. Standing waves show intefeence, foming nodes (aeas of no displacement) and antinodes. The distance between two successive nodes o antinodes is 2 λ. Standing waves can only fom then the length of the sting is a whole multiple of 2 λ, i.e L = n( 2 λ). Solving fo λ gives λ n = 2L n ), which ae called the hamonic/esonant wavelengths (n is the hamonic numbe). We can convet this to hamonic fequencies if needed. The fist standing wave (n = ) is called the fundamental standing wave. In a tube closed at one end, fo a standing wave fomed by a sound wave, thee is an antinode at the open end. The distance between the node and the antinode is 4 λ. Standing waves can be established in a tube closed at one end if the tube s length is an odd multiple of 4 λ, so f n = n v 4L fo odd n. If the tube is open at both ends, the hamonics behave like those in stings. The speed of a sound wave is a function of the density and the bulk modulus (B), which measues a B medium s esponsiveness to compession. High B means had to compess (e.g solid). v = ρ. The loudness of a sound is measued by its intensity, and I 2. Loudness can also be measued though the decibel level (β) (elative intensity): β = 0 log I I 0. I 0 is the theshold of heaing, 0 2 W. The unit is db, but it is dimensionless. 0dB loude = 0 times moe intense. Two sound waves with simila fequencies who intefee often modulate in amplitude, the waves altenate between constuctive and destuctive intefeence. We call each constuctive intefeence a beat. f beat = f f 2. The Dopple effect esults fom elative motion between the souce and detecto. f d = v±v d v v s f s. We use the highe sign fo motion towads and the lowe fo motion away. 3.2 Optics y n = nλl d n = c v n sinθ = n 2 sinθ 2 f = R 2 Visible light is fom λ = 390nm 770nm f = d o + d i m = si s o = hi h o sin θ = mλ d Key powes of 0 fo the wavelength spectum ae: -, -3, -6, -7, -8, -2. Coheent waves have constant phase diffeence. Waves intefee constuctively if thei path diffeence is a whole numbe. l = nλ Waves intefee destuctively if thei path diffeence is a half numbe. l = (n + 2 )λ. Double slit intefeence causes finges to appea. To locate the position of the bight finges, we use: y m = mλl d. d is the length of the ba between the slits. L is the distance fom the slits that we obseve the patten. Hence the angle at which intefeence occus: sin θ = mλ d. 6

7 The cental maximum (m=0) has the geatest intensity, followed by m = ±, then m = ±2 and so on. The moe slits thee ae, the shape the patten. These ae called diffaction gatings. Fo single-apetue diffaction, the cental maximum is vey ponounced. Fo a medium with efactive index n, n = c v Diffeent fequencies of light have diffeent wave speeds when in a medium. This means that dispesion occus. Spheical mios have ae cuved such that thei suface foms pat of a sphee. C is the cente of cuvatue. R is the adius of cuvatue. F is halfway between the mio and the cente of cuvatue, the focus o focal point. The vetex V is the intesection between the optic axis and the mio. The focal length is half the adius. Incident light ays nea the axis (paaxial ays) ae eflected to the focus. The mio equation is s o + s i = f, whee s o is the object s distance fom the mio, s i is the image distance fom the mio and f is the focal length. s o is positive, s i is positive fo eal images. The magnification equation is m = si s o Real images ae always inveted, vitual always upight. Concave mios poduce eal images, wheeas concave lenses poduce vitual images. Conveging lenses (convex) fom a eal focus. A diveging lens (e.g bi-concave) causes light to divege fom a vitual focus. Rays passing though optical cente ae unchanged When light passes though a polaising filte, the esulting light is polaised in a pependicula diection. 7

8 4 Themal Physics L = αl 0 T V = βv 0 T E K = 3 2 k bt v ms = e = Q H Q C Q H Q t = ka T L 3RT M Q = U + W W = P V U = 3 2 nr T Mateials expand when heated. α is the coefficient of linea expansion. Hence the change in length of a mateial is L = αl 0 T. β is the coefficient of volume expansion. Fo most solids, β 3α. Wate has a negative β value between 0 and 4 C. Fo a gas, the aveage tanslational KE of the molecules is popotional to the tempeatue. E K = 3 2 k bt, whee k b = R N A 3RT Fom this, we can deive v ms = M. The fist law of themodynamics states that Q = U + W The second law is about entopy Fo cyclic pocesses, U must be zeo. Fo an engine, Q net = W. The diffeence between heat in and heat out is Q H Q C. The efficiency of an engine is the atio of output to input, i.e W Q H. Hence e = Q H Q C Q H. Unless Q C is zeo, e <. Canot cycles ae the most efficient heat engines. Thei efficiency is given by: e = T C T H tempeatue of the esevoi in Kelvin. whee T is the The ate of heat tansfe though a od depends on the length, the amount of heat that must tavel, the coss sectional aea, the tempeatue diffeence, and the themal conductivity k. 5 Moden Physics 5. Nuclea and Quantum Physics Q t = ka T L E = hf E max = E φ = hf φ λ = h p A = A 0 e λt When light is shone on a metal, electons (called photoelectons) ae eleased if the fequency is above the theshold fequency (f 0 ). These electons ae ejected quickly, but they have a maximum kinetic enegy egadless of incident light intensity. Electons need enegy to be libeated - to exceed a metal s wok function (φ). E max = E φ = hf φ. φ is equal to the enegy of the gound state. If the photon enegy is less than φ (i.e f < φ h ), no electons will be emitted. The cuent is popotional to the intensity of the light. The enegy levels fo atoms with one electon ae given by E n = Z2 n 2 numbe. ( 3.6eV ), whee Z is the atomic Due to wave-paticle duality, paticles with linea momentum p = mv have a de Boglie wavelength λ = h p. The activity of a adioactive substance follows A = A 0 e λt. 8

9 5.2 Relativity v = u+v +uv/c T 2 2 = γ T γ = KE = (γ )mc 2 E ( v total = E est + KE = γmc 2 c )2 The laws of physics ae the same in all inetial efeence fames. The speed of light has the same value egadless of the motion of the souce o obseve. Relativistic addition of velocities: v = u+v +uv/c 2. If object 2 has elative velocity v to object, the time dilation is given by T 2 γ = ( v c )2 = γ T, whee If object 2 has elative velocity v to object then the length of the object L 2 = L γ. Since γ >, faste objects have contacted lengths. Fo elativistic kinetic enegy, KE = (γ )mc 2. The total enegy: E total = E est + KE = γmc 2 The Equivalence pinciple states that gavity causes spacetime to bend - light bends and time dilation occus. 9

OSCILLATIONS AND GRAVITATION

OSCILLATIONS AND GRAVITATION 1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,