Oscillations about Equilibrium: Equation: Variables: Units:

Transcription

1 Physics 111 Fall 2017 Exam 3 cheat sheet Oscillations about Equilibriu Equation: Variables: Units: ω = 2π T = 2πf F = kx x = Acos(ωt) 4 = ωasin(ωt) a 4 = ω 8 Acos(ωt) ω = k m E = KE + PE E = 1 2 m kx8 E = 1 2 ka8 Angular frequency T: Period f: frequency F: Force exerted on a spring Spring constant x: displacement(distance a spring is stretched or compressed) x: displacement or position of an object in oscillation amplitude angular frequency time elocity of an object in oscillation a: acceleration of an object in oscillation angular frequency of the oscillations of a mass attached to a SPRING spring constant mass Total mechanical energy K kinetic energy P Spring potential energy mass of the oscillator elocity of the oscillator x: displacement of the oscillator amplitude rad/s T: seconds f: Hz (1/seconds) F: N N/m x: m x: m m rad/s seconds m/s a: m/s^2 w For all: Joules(Nm) x:

2 Tips: Practice using the position, elocity, and acceleration equations Don t forget that = 8 m8 + = 8 k = = 8 ka8, this will be useful, and something that i m seeing lots of people forget While it s not on the equation sheet outright, remember that T = 1 f this can be found using the first equation listed by cancelling out the 2p from both sides. This will be ery useful, so try to remember this Waes and Sound: Equation: Variables: Units: = λ T = λf speed of the wae propogation (if it is a sound wae, is the speed of sound, 343 m/s l: waelength (distance between l: meters peaks of the wae) T: period f: frequency T: f: = F F: Tension in the string/wire that F: N μ the wae propagates along µ: linear mass density µ: kg/m μ = m L I = P A = P 4πr 8 β = 10dB log ( I I M ) µ: linear mass density mass of the string/wire L: length of the string/wire I: Intensity of the sound P: power produced by the sound source The surface area of the wae propagation at the distance the obserer is listening to the sound b: Sound intensity in decibels I: Sound intensity in W/m^2 I M : Base sound intensity (constant) 2 Doppler effect equations: f M : frequency heard by the obserer f P : frequency produced by the source M : elocity of the obserer P : elocity of the source µ: kg/m L: I: Watts/m^2 P: Watts (J/s) m^2 b: decibles I: Watts/m^2 I M = 10 N=8 w m 8 f M : f P : M : P :

3 2 Interference equations (Constructie and Destructie) λ = 8R : On a string, open column λ = SR : column closed at one end d = : distance between the obserer and the first source d 8 : distance between the obserer and the second source l: waelength of the waes n: just a multiplier, meaning the difference between the two distances should be a multiple of the waelength, or a multiple of half the waelength λ : waelength L: length of pipe or string n: just a multiplier, meaning the waelength should be a multiple of the length. d = : d 8 : l: n: unitless λ : L: n: unitless f = λ f = n T : On a string, open 8R column f = n T : column closed at SR one end f UVWX = f = f 8 f : frequency of the nth harmonic speed of sound L: length of column or string n: just a multiplier, meaning the frequency should be a multiple of the length. f UVWX : Beat frequency f = : frequency of the first sound wae f 8 : frequency of the second sound wae f : L: n: f: Tips: For problems inoling linear mass density, you can find it using the total mass of the string/wire and its total length, or you can find it by using the known mass of the string/wire per unit length of it such as 1 meter. (for example, you can find µ from knowing a 20 meter rope has mass 10kg, or that eery meter of the rope weighs 0.5 kg). The second line of the equation sheet is just some properties of the log function, look them oer and make sure you re familiar with them. Remember that the log function is actually log base 10 The fundamental frequency is when n = 0 for the waelength and frequency of a standing wae on a string, or in an open or closed column You should always get a positie number for the beat frequency. The ertical bars mean absolute alue, so if you get a negatie number, just take the negatie sign off and you ll still get the right answer.

4 Fluids: ρ = m V P = F A P \W]\V = P P WX^ P 8 = P = + ρgh F a]bcwx = ρ de]fg Vg Conseration of mass: Bernoulli s Equation ρ: density mass V: olume P: pressure F: force area The gauge pressure is the pressure inside the body that you re measuring (like a tire) minus the atmospheric pressure P 8 : Pressure at the bottom of a reseroir containing a fluid P = : Pressure at the top of a reseroir containing a fluid (atmospheric pressure) r: density of the fluid g: acceleration due to graity h: height/depth of the reseroir F: the buoyant force of an object in a fluid ρ de]fg : density of the fluid V: olume of the object submerged in the fluid change in mass change in time r: density cross sectional area of the pipe or essel through which the fluid is flowing elocity of the fluid flowing P: Pressure r: density elocity of the fluid flowing g: acceleration due to graity y: the height of the fluid(for example, if the pipe water flows through turns upwards) ρ: kg/m^3 V: P: N/m^2 F: P: P: r: g: h: F: ρ: V: r: P: r: g: y:

5 Tips: In the equation P 8 = P = + ρgh, it is not necessarily a reseroir of water. It could be a gas, like Carbon Dioxide, or it could be a liquid, like oil. If there are multiple layers of fluid (like oil on top of water), You would find the pressure at the bottom if the oil, and that would become the new P =, before you find the pressure at the bottom of the water. In the buoyant force equation, the olume is the olume of the object that is under the surface of the fluid. So if the object is only partially submerged, only the olume that is under the fluid should be considered in this equation. In the conseration of mass equation, A is the cross sectional area of the pipe or other essel through which the fluid is flowing. Remember the area of a circle is A = πr 8. This is worth memorizing for the exam. If you use the radius of the pipe instead of the area, you will get the wrong answer. Temperature and Hea T i = (T j 32) 5 9 T n = T i L = αl M T V = βl M T = mc T d]pfb = ml d T i : Temperature in Celsius T j : Temperature in Fahrenheit T n : Temperature in Kelin T i : L: change in length α: coefficient of linear expansion L M : Original length Change in temperature V: change in olume β: coefficient of olumetric expansion (= 3α) L M : Original length Change in temperature : heat mass c: specific heat capacity (the heat required to change the temperature of the unit mass of a gien substance by one degree) change in teperature d]pfb : Heat of fusion: the heat required to melt/freeze a substance mass of substance T i : degrees Celsius T j : degrees Fahrenheit T n : Kelin T i : L: α: 1/degree (of temperature) L M : L: β: 1/degree (of temperature) L M : : Joules grams c: Joule/gram*degree celsius d]pfb : Joules grams

6 TWsbtfuWXfb = ml T t = ka T L t = eσats L d : specific heat of fusion (different constants for different substances) TWsbtfuWXfb : Heat of fusion: the heat required to aporize/condense a substance mass of substance L T : specific heat of aporization (different constants for different substances) X : The rate of heat transfer by conduction thermal conductiity of the material per unit thickness area of the material change in temperature L: thickness of the material X : The rate of heat transfer by radiation e: emissiity σ: Stefan-Boltzmann constant L d : J/g TWsbtfuWXfb : L T : X : J/m/s/degree celsius L: meters X : The rate of heat transfer by radiation e: unitless σ = Nz W(m 8 K S ) Tips: ALWAYS conert temperatures to Kelin. Some calculations will work if you leae temperatures as Fahrenheit or Celsius, but they ALL will work in Kelin, so it s easiest to always conert to kelin to begin with to aoid mistakes. Don t forget to account for the heat of aporization/fusion when you use a change in heat calculation that causes the substance to change state (melt, freeze, aporize, condense). GOOD LUCK!!