Transcript Physics 100
Physics 100 Definitions: 7/17/2015 Kinematics • • • • • x (or y): length / distance in meters, feet, etc. xi (subscripted): position (initial, final,1, 2, 3, etc.) Dx or Dy: change (D) in position x, xi, Dx: vector denotation, also with → t: time in seconds, hours, etc. ∆𝑥 𝑣= ∆𝑡 Average speed or velocity in m/s mph, etc. ∆𝑣 𝑎= ∆𝑡 Average acceleration in m/s2, ft/s2, etc. 𝑣𝑓 = 𝑣𝑖 + 𝑎𝑡 𝑥𝑓 = 1 2 𝑎𝑡 2 + 𝑣𝑖 𝑡 + 𝑥𝑖 For vertical motion a = -g, the acceleration of gravity, 9.8 m/s2 or 32 ft/s2 Two Dimensional Motion • Relative velocity between three inertial frames, a, b, c 𝑣𝑎𝑐 = 𝑣𝑎𝑏 + 𝑣𝑏𝑐 Triangle trig.: Courtesy Hyperphysics Ballistic (Projectile) Motion • vo: launch velocity • Qo: launch angle 𝑣𝑜 2 𝑅= sin 2𝜃 𝑔 𝑦𝑚𝑎𝑥 (𝑣𝑜 sin 𝜃)2 = 2𝑔 𝑡𝑓𝑙𝑖𝑔ℎ𝑡 = 2𝑣𝑜 sin 𝜃 𝑔 R: horizontal range Maximum height Time of flight Newton’s Laws • F: force in Newtons (1N = 1kg m/s2) or Dynes (g cm/s2) • m: coefficient of friction (dimensionless) o 0 < m < 1; mr < mk < ms (rolling, kinematic, static) • N: normal force; perpendicular to the plane of action 𝐹 = 𝑚𝑎 Newton’s Second Law 𝐹𝑓 = 𝜇𝑁 Contact friction force 𝐹=0 Static/equilibrium 𝐹 = 𝑚𝑎 Dynamical 𝑚𝑣 2 𝐹𝑐 = 𝑟 Centripetal force; 𝐹 = −𝑘𝑥 Hooke’s Law: k is the spring constant in N/m r = radius of curvature Energy ∆W=𝐹 ∙ ∆𝑥 = 𝐹𝑥 cos 𝜃 W: work in Joules (1J = 1Nm) 𝐾 = 1 2 m 𝑣𝑓 2 − 𝑣𝑖 2 K: kinetic energy 𝑈𝐺 = 𝑚𝑔𝑦 UG: gravitational potential energy 𝑈𝑒 = 1 2 𝑘 𝑥𝑓 2 − 𝑥𝑖 2 Ue: elastic potential energy ∆𝑊 = −∆𝐾 𝑃= 𝑊 =𝐹∙𝑣 𝑡 Work-energy theorem (conservation of energy) P: power in Watts; 1W = 1J/s Linear Momentum • Collisions o Elastic: no permanent deformation; kinetic energy is conserved o Inelastic: permanent deformation; K not conserved o Perfectly inelastic: objects stick together; K not conserved 𝑃 = 𝑚𝑣 P: momentum in kg m/s 𝑃𝑖 = 𝑃𝑓 Conservation of momentum 𝐽 = ∆𝑃 = 𝐹∆𝑡 J: impulse 𝑥= 𝑁 𝑖=1 𝑚𝑖 𝑥𝑖 𝑁 𝑖=1 𝑚𝑖 Center of mass (average x, y) Linear / Rotational Analogs Linear Rotational Equivalence x Q 𝑥 = 𝑟𝜃 v w 𝑣 = 𝑟𝜔 a a 𝑎 = 𝑟𝛼 m I F t 𝜏 = 𝐼𝛼 = 𝑟 × 𝐹 𝜏 = 𝑟𝐹 sin 𝜃 DWtrans DWrot ∆𝑊𝑟𝑜𝑡 = 𝜏 × 𝜃 Ktrans Krot 𝐾𝑟𝑜𝑡 = 1 2 𝐼𝜔2 Ptrans Prot 𝑃𝑟𝑜𝑡 = 𝜏 × 𝜔 = 𝜏𝜔 sin 𝜃 P L 𝐿 = 𝐼𝜔 𝐼= 𝑚𝑟 2 Gravity 𝐹𝐺 = 𝐺 𝑚1 𝑚2 𝑟 𝑟2 2 4𝜋 𝑇2 = 𝑟3 𝐺𝑀 𝑣𝑂𝑟𝑏 = 𝐺𝑀 𝑅 𝑣𝑒𝑠𝑐 = 2𝑣𝑂𝑟𝑏 G is the universal Gravitational Constant; r is the radial distance between the masses; r ‘hat’ merely defines the quantity as a vector and is not pertinent to 100 level Physics Kepler’s Third Law. T is orbital period in seconds, etc.; M is the central mass Orbital velocity; R is the distance to the center of the central mass Escape velocity Solids • Stress = Force/Area (AKA pressure) • Strain = Elongation/Original Length 𝐹𝐿𝑜 𝑌= ∆𝐿 𝐴 𝐹 𝑆= 𝐴∅ 𝐵 = −𝑉𝑜 Y: Young’s Modulus of Elasticity in N/m2 Tensile or Compressional S: shear Modulus in N/m2 ∆𝑝 ∆𝑉 B: bulk modulus in N/m2 Courtesy Tulane Un. Fluids 𝜌= 𝑚 𝑉 r: density in kg/m3. Constant in an ideal fluid 𝑝= 𝐹 𝐴 p: pressure in Pascals (1Pa = 1N/m2) 𝑝 = 𝜌𝑑𝑔 Pressure at depth in an ideal fluid 𝐹𝑏 = 𝜌𝑉𝑔 Buoyant force = weight of displace fluid 𝐴𝑊 = 𝑚𝑔 − 𝐹𝑏 𝐴1 𝑣1 = 𝐴2 𝑣2 Apparent weight Equation of continuity 𝑝 + 1 2 𝜌𝑣 2 + 𝜌𝑔𝑦 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Bernoulli’s equation Transverse Waves T: period in seconds A: amplitude (m, cm, etc.) l: wavelength (m, cm, etc. 𝑓= 1 𝑇 𝑐 = 𝑓l Frequency in Hertz (1Hz = 1 cycle/second) Speed of a wave in m/s 𝜔 = 2𝜋𝑓 = 2𝜋 𝑇 𝑦 = 𝐴 sin 𝜔𝑡 Vertical position of a point on a wave 𝑣 = 𝜔𝐴 cos 𝜔𝑡 Vertical speed of a point on a wave 𝑎 = −𝜔2 𝐴 sin 𝜔𝑡 Vertical acceleration of a point on a wave 𝐿 𝑇 = 2𝜋 𝑔 Period of a simple pendulum; L = length 𝑇 = 2𝜋 𝑚 𝑘 𝐼 𝑇 = 2𝜋 𝑚𝑔𝑑 Period of a mass on a spring Period of a physical pendulum; d = distance from the center of mass to the pivot Sound 𝐼= 𝑃 𝐴 I: sound intensity in W/m2 𝑑𝐵 = 10 log 𝑣= 𝑇 𝑚 𝐿 𝐼 𝐼𝑜 dB: sound level in decibels; Io is 10-12 W/m2 Speed of a wave on a string; T here is tension l 𝑑 sin 𝜃 = 𝑛 2 Path Length Difference (PLD): for destructive interference; n = 1, 3, 5… for constructive interference n = 0, 2, 4… l 𝐿=𝑛 4 Resonance condition for a tube of length L open at one end; n = 1, 3, 5… l 𝐿=𝑛 2 Resonance condition for a string of length L fixed at both ends; n = 1, 2, 3… Thermodynamics 𝐹= 9 𝐶 + 32 5 𝐶 = 𝐾 − 273 ∆𝐿 = 𝛼𝐿𝑜 ∆𝑇 ∆𝑉 = 𝛽𝑉𝑜 ∆𝑇 𝑄 = 𝑚𝑐∆𝑇 𝑄 = 𝑚𝐿𝑓,𝑣 Conversion factors among temperature scales Change of length/volume; a = coefficient of linear expansion, b = coefficient of volumetric expansion, both in /Co Q: heat required to change the temperature of a mass; c = specific heat capacity in J/kgCo Heat required to institute a phase change; L = latent heat of fusion or vaporization in J/kg 𝑄 𝑘𝐴∆𝑇 = 𝑡 𝑑 Heat transfer by conduction; k = coefficient of thermal conductivity 𝐿 = 𝐴𝜖𝜎 𝑇𝑖𝑛 4 − 𝑇𝑜𝑢𝑡 4 L: luminosity, heat transfer by radiation; є = emissivity, s = Stefan-Boltzmann constant 𝑝𝑉 = 𝑛𝑅𝑇 Ideal Gas Law; n = number of moles, R = Ideal Gas Constant in J/mol K 𝑝𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 Boyle’s Law 𝑝 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇 Gay-Lussac Law 𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑇 Charles’ Law ∆𝑈 = 𝑄 + 𝑝∆𝑉 Internal energy 𝑣𝑅𝑀𝑆 = 3 𝐾 = 𝑘𝑇 2 ∆𝑆 = ∆𝑄 𝑇 3𝑘𝑇 = 𝑚 3𝑅𝑇 𝑀 Average speed of a particle at a given temperature Kinetic energy of a particle at a given temperature Entropy Electric Fields / Energy 𝑄 𝐸= 𝑟 4𝜋𝜖𝑜 𝑟 2 1 E: electric field strength in N/C from a point charge; єo = permittivity of free space C2/Nm2, Q = electric charge in Coulombs (C) k = 4𝜋𝜖 Electric field constant 𝐹 = 𝑞𝐸 Coulomb’s Law; q is another charge 𝑜 𝑈𝐸 = 𝑘 𝑉=𝑘 𝑞𝑄 𝑟 𝑄 𝑟 Electric potential energy from a point V: electric potential from a point in Volts 𝐸∆𝐴 = 𝑄 𝜖𝑜 Gauss’ Law ∅𝐸 = 𝐸𝐴 cos 𝜃 Electric flux in Nm2/C 𝑉𝑎𝑏 = Generalized electric potential 𝐶= 𝐸∆𝑙 𝑄 𝑉 Capacitance in Farads 𝐴 𝐶 = 𝜖𝑜 𝑑 Capacitor construction; A = area, d = plate separation 𝑉 = 𝐸𝑑 Potential between capacitor plates 𝑈𝐶 = 1 2 𝐶𝑉 2 Energy stored in a capacitor 𝜖 = 𝐾𝐸 𝜖𝑜 Permittivity of a dielectric; KE is the dielectric constant (dimensionless) 1 𝐶𝑇 1 1 1 2 = 𝐶 + 𝐶 +… Capacitors added in series 𝑄𝑇 = 𝑄1 = 𝑄2 … Charge on capacitors in series 𝑉𝐵 = 𝑉1 + 𝑉2 + ⋯ Potential on capacitors in series 𝐶𝑇 = 𝐶1 + 𝐶2 + ⋯ Capacitors added in parallel 𝑄𝑇 = 𝑄1 + 𝑄2 +… Charge on capacitors in parallel 𝑉𝐵 = 𝑉1 = 𝑉1 = ⋯ Potential on capacitors in parallel Current and Resistance 𝐼= 𝑄 𝑡 I: current in Amperes (A) 𝑅= 𝑉 𝐼 R: resistance in Ohms (W), AKA Ohm’s Law 𝐿 𝑅=𝜌 𝐴 Resistor construction; r is resistivity 𝑅 = 𝑅𝑜 1 + 𝛼∆𝑇 Thermal dependence of resistance; a = coefficient of thermal resistivity 𝑅𝑇 = 𝑅1 + 𝑅2 + ⋯ Resistors added in series 𝐼𝑇 = 𝐼1 = 𝐼2 … Current in resistors in series 𝑉𝐵 = 𝑉1 + 𝑉2 + ⋯ Potential drop across resistors in series 1 𝑅𝑇 1 1 = 𝑅 + 𝑅 +… 1 2 Resistors added in parallel 𝐼𝑇 = 𝐼1 + 𝐼2 +… Current in resistors in parallel 𝑉𝐵 = 𝑉1 = 𝑉1 = ⋯ Potential on resistors in parallel Charging/Discharging 𝑡 𝑞 = 𝑄𝑜 1 − 𝑒 − 𝑡 𝑞 = 𝑄𝑜 𝑒 − 𝜏 = 𝑅𝐶 𝑅𝐶 𝑅𝐶 Capacitor charging Capacitor discharging t: RC time constant; 63% of maximum DC Circuits 𝑉 = 𝐼𝑅 Ohm’s Law 𝑃 = 𝐼𝑉 Power in a DC circuit 𝑃 = 𝐼2𝑅 Joule heating 𝑉𝑙𝑜𝑜𝑝 = 0 Kirchhoff’s Loop Law 𝐼𝑖𝑛 = Kirchhoff’s Junction Law 𝐼𝑜𝑢𝑡 Refraction and Reflection 𝑐 𝑛= ≈ 𝑣 𝑛= 𝐾𝐸 Index of refraction; 1 < n < 2.5 l𝑜 l 𝑛𝑖𝑛𝑐𝑖𝑑 sin 𝜃𝑖𝑛𝑐𝑖𝑑 = 𝑛𝑟𝑒𝑓𝑟𝑎𝑐 sin 𝜃𝑟𝑒𝑓𝑟𝑎𝑐 sin 𝜃𝑐𝑟𝑖𝑡 𝑛𝑖𝑛𝑐𝑖𝑑 = 𝑛𝑟𝑒𝑓𝑟𝑎𝑐 𝜃𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑐𝑒 = 𝜃𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 Snell’s Law Critical angle for total internal reflection Law of reflection Mirrors and Lenses 1 1 1 = + 𝑓 𝑝 𝑞 1 = 𝑛−1 𝑓 𝑅 = 2𝑓 Thin lens equation; f is focal length, p is object distance, q is image distance (sometimes so and si, respectively) 1 1 + 𝑅1 𝑅2 Lensmaker’s equation; R is radius of curvature Relationship between radius of curvature and focal length in a spherical mirror Sign Conventions Wave Optics 𝑃𝐿𝐷: 𝑠𝑒𝑒 𝑆𝑜𝑢𝑛𝑑 𝑚l = 𝑑 sin 𝜃 Constructive interference 𝐼 = 𝐼𝑜 cos 𝜃 2 𝑚 + 1 2 l = 𝑑 sin 𝜃 Destructive interference Intensity of light through two polarizing filters at relative angle Q Optical Instruments ℎ` 𝑞 𝑀= =− ℎ 𝑝 𝑀= 25 𝑓 𝑚𝑜 = − Magnification of a simple magnifier, f in cm. Image at infinity 𝐿 𝑓𝑜 𝑀𝑇 = 𝑚𝑜 𝑀 𝑀= 𝑓𝑜 𝑓𝑒 General magnification; h = height of object, h` = height of image Objective magnification of a microscope with length L Total magnification for a microscope Magnification of a telescope 𝐷= 1 𝑓 Power in diopters (D); f in meters 1 1 1 = + 𝑓 25 𝑞 Correction for farsightedness; q is < 0 1 1 = 𝑓 𝑞` Correction for nearsightedness; q` = q less distance from eye to glasses and < 0 𝑅= l ∆l sin 𝜃 = 1.220 R: resolving power of a grating (dimensionless) l 𝐷 Rayleigh criterion for resolving power of an aperture Modern Physics: Einstein 𝛾= 1 𝑣2 1− 2 𝑐 ∆𝑡` = ∆𝑡 𝛾 𝐿= 𝐿𝑜 𝛾 𝑚` = 𝛾𝑚 ℎ𝑐 𝐾= −∅ l Used in relativity equations Time dilation Length contraction Relativistic mass Photoelectric effect/kinetic energy of a photoelectron; f is the work function Modern Physics: Quantum Mechanics ℎl 𝐸= 𝑐 Energy of a photon; h is Planck’s Constant 𝑝= ℎ l Photon momentum l= ℎ 𝑚𝑣 deBroglie wavelength l= 0.0028977685 𝑚𝐾 𝑇 −𝑡 𝐴 = 𝐴𝑜 2 𝑡1/2 Wein Displacement Law Radioactive decay; Ao = original amount of the isotope, A = later amount, t1/2 = half-life