Transcription
PHYS 2310 Engineering Physics I Formula SheetsChapters 1-18Chapter 1/Important NumbersChapter 2VelocityQuantityUnits for SI Base QuantitiesUnit NameUnit SymbolLengthMeterMTimeSecondsMass (not weight)Kilogramkg1 kg or 1 m1m1m1 second1mCommon Conversions1000 g or m1m100 cm1 inch1000 mm1 day1000 milliseconds 1 hour3.281 ft360 Average VelocityAverage Speed1 106 ππ2.54 cm86400 seconds3600 seconds2π radImportant Constants/MeasurementsMass of Earth5.98 1024 kgRadius of Earth6.38 106 m1 u (Atomic Mass Unit)1.661 10 27 kgDensity of water1 π/ππ3 or 1000 ππ/π3g (on earth)9.8 m/s 2CircumferenceSurface area(sphere)DensityCommon geometric FormulasArea circleπΆ 2ππππ΄ 4ππ 2Volume (rectangular solid)Instantaneous Velocityπππ£π πππ πππππππππ‘ π₯ π‘πππ π‘π‘ππ‘ππ πππ π‘πππππ‘πππΜ ππ₯ π₯π£ lim π‘ 0 π‘ππ‘π ππ£π Average Acceleration π£ π‘ππ£ π2 π₯π ππ‘ ππ‘ 2πππ£π Motion of a particle with constant accelerationπ΄ ππ4Volume (sphere)π ππ 33π π π€ βπ ππππ ionInstantaneousAcceleration22.2π£ π£0 ππ‘1 π₯ (π£0 π£)π‘21 π₯ π£0 π‘ ππ‘ 22π£ 2 π£02 2π π₯2.112.172.152.162.72.82.9
Chapter 3Adding VectorsGeometricallyAdding VectorsGeometrically(Associative Law)Chapter 4πβ πββ πββ πβ3.2(πβ πββ) πβ πβ (πββ πβ)3.33.5Magnitude of vector π π ππ₯2 ππ¦23.6Angle between x axisand vectorππ¦π‘πππ ππ₯3.6Unit vector notationπβ ππ₯ πΜ ππ¦ πΜ ππ§ πΜ3.7ππ₯ ππ₯ ππ₯ππ¦ ππ¦ ππ¦ππ§ ππ§ ππ§3.103.113.12πβ πββ πππππ π3.20Adding vectors inComponent FormScalar (dot product)Scalar (dot product)Projection of πβ ππ πββ orcomponent of πβ ππ πββVector (cross) productmagnitudeπβ πββ (ππ₯ πΜ ππ¦ πΜ ππ§ πΜ) (ππ₯ πΜ ππ¦ πΜ ππ§ πΜ )πβ πββ ππ₯ ππ₯ ππ¦ ππ¦ ππ§ ππ§displacementAverage AccelerationInstantaneousAcceleration4.4 π₯ π‘4.8ππβ π£π₯ πΜ π£π¦ πΜ π£π§ πΜππ‘ π£βπβππ£π π‘ππ£βπβ ππ‘πβ ππ₯ πΜ ππ¦ πΜ ππ§ πΜπ£β Projectile Motionπ£π¦ π£0 π πππ0 ππ‘3.241 π¦ π£0 π ππππ‘ ππ‘ 2222π£π¦ (π£0 π πππ0 ) 2π yπΜππ§ οΏ½οΏ½ππ¦ππ¦4.154.21π£π¦ π£0 π πππ0 ππ‘π πππ πππ4.104.114.231 π₯ π£0 πππ ππ‘ ππ₯ π‘ 22or π₯ π£0 πππ ππ‘ if ππ₯ 0πβ πββ π orπΜββπβπ₯π πππ‘ ππ₯ππ₯ πβ π₯πΜ π¦πΜ π§πΜββππ£π πInstantaneous Velocity3.22πβπ₯πββ (ππ₯ πΜ ππ¦ πΜ ππ§ πΜ)π₯(ππ₯ πΜ ππ¦ πΜ ππ§ πΜ ) (ππ¦ ππ§ ππ¦ ππ§ )πΜ (ππ§ ππ₯ ππ§ ππ₯ )πΜ (ππ₯ ππ¦ ππ₯ ππ¦ )πΜVector (cross product)4.4Average Velocityππ₯ ππππ πππ¦ ππ πππComponents of Vectorsπβ π₯πΜ π¦πΜ π§πΜPosition vectorRelative MotionUniform CircularMotionπ¦ (π‘πππ0 )π₯ π ππ₯2(π£0 πππ π0 )2π£02sin(2π0 )πβββββββπ£π΄πΆ βββββββπ£π΄π΅ βββββββπ£π΅πΆππ΄π΅ οΏ½οΏ½οΏ½π΄π π£2ππ 2πππ£4.254.264.444.454.344.35
GeneralComponent formChapter 5Chapter 6Newtonβs Second LawFrictionπΉβπππ‘ ππβπΉπππ‘,π₯ πππ₯πΉπππ‘,π¦ πππ¦πΉπππ‘,π§ πππ¦5.1Kinetic FrictionalWeightπΉπ πππ πππβπ ,πππ₯ ππ πΉπ6.1πβπ ππ πΉπ6.25.2Drag ForceGravitational ForceGravitational ForceStatic Friction(maximum)Terminal velocity1π· πΆππ΄π£ 222πΉππ£π‘ Centripetal Forceπ£2π π πΉ ππ£ 2π 6.176.18
Chapter 7Work- Kinetic EnergyTheorem7.1π πΉππππ π πΉβ πβ7.77.8Spring Force (Hookeβslaw)Work done by springWork done by VariableForceAverage Power(rate at which thatforce does work on anobject)Instantaneous οΏ½οΏ½οΏ½ Mechanical EnergyπΈπππ πΎ π8.127.15Principle ofconservation ofmechanical energyπΎ1 π1 πΎ2 π2πΈπππ πΎ π 08.188.177.207.21Force acting on particleπ§ππ 7.36π§ππ π‘ππ πΉππππ π πΉβ π£βππ‘8.78.117.25π πΉπ₯ ππ₯ πΉπ¦ ππ¦ πΉπ§ ππ§ π ππ π¦1 2ππ₯27.12 πΎ ππ ππππ πππππππ πΉπππππΉβπ ππβπΉπ₯ ππ₯ (along x-axis)Gravitational PotentialEnergy8.18.6π(π₯) ππ ππππππ π1 2 1 2ππ₯ ππ₯2 π 2 π π π πΉ(π₯)ππ₯Elastic Potential Energy7.10ππ Potential Energyπ₯π πΎ πΎπ πΎ0 πWork done by gravityWork done bylifting/lowering objectπ₯π1πΎ ππ£ 22Kinetic EnergyWork done by constantForceChapter 87.427.437.47Work on System byexternal forceWith no frictionWork on System byexternal forceWith frictionChange in thermalenergyConservation of Energy*if isolated W 0πΉ(π₯) ππ(π₯)ππ₯8.22π πΈπππ πΎ π8.258.26π πΈπππ πΈπ‘β8.33 πΈπ‘β ππ ππππ π8.31π πΈ πΈπππ πΈπ‘β πΈπππ‘8.35Average PowerInstantaneous Power**In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PEπππ£π π πΈ π‘ππΈππ‘8.408.41
Chapter 9Impulse and Momentumπ‘πImpulseπ½β πΉβ (π‘)ππ‘π‘π9.35πβ ππ£β9.22π½β Ξπβ πβπ πβπππβππ‘πΉβπππ‘ πβπβππππββ οΏ½ππ‘ Newtonβs 2nd lawSystem of Particles9.30π½ πΉπππ‘ π‘Linear MomentumImpulse-MomentumTheoremCollision continued πΉβπππ‘ 9.319.32Inelastic CollisionConservation of LinearMomentum (in 2D)Average forceElastic Collisionππ π π π£ π‘ π‘ ππΉππ£π π£ π‘2π1 ()π£π1 π2 1ππ9.149.259.27Center of mass locationπβπππ1 ππ πβππ9.67π 1ππ£βπππ 1 ππ π£βππRocket EquationsThrust (Rvrel)π π£πππ ππ9.68Change in velocity9.429.43πβ1π πβ2π πβ1π πβ2π9.509.519.78πΎ1π πΎ2π πΎ1π πΎ2π9.8π 1πββ ππππ π‘πππ‘πββπ πββππ1 π£π1 π2 π£12 π1 π£π1 π2 π£π29.379.40Center of Massππ‘π1 π2π£1π ()π£π1 π2 1π9.779.22Collisionπ£2ππββ1π πββ2π πββ1π πββ2ππΉππ£π Center of mass velocityFinal Velocity of 2objects in a head-oncollision where oneobject is initially at rest1: moving object2: object at restConservation of LinearMomentum (in 1D)π1 π£01 π2 π£02 (π1 π2 )π£πΞπ£ π£πππ ππππππ9.889.88
Chapter 10Angular displacement(in radiansAverage angularvelocityInstantaneous VelocityAverage angularaccelerationInstantaneous angularaccelerationπ πΞπ π2 π1 ππππ£π π‘πππ ππ‘ ππΌππ£π π‘πππΌ ππ‘10.110.4π Rotational Kinematicsπ π0 πΌπ‘1Ξπ π0 π‘ πΌπ‘ 2222π π0 2πΌΞπ1Ξπ (π π0 )π‘21Ξπ ππ‘ πΌπ‘ 2210.510.610.710.810.1310.1410.15πΌ πΌπππ πβ210.36π ππΉπ‘ π πΉ ππΉπ πππ10.3910.41Newtonβs Second Lawππππ‘ πΌπΌ10.45Rotational work doneby a toqueπ πππTorqueRotational KineticEnergyWork-kinetic energytheoremπ£ ππ10.18Tangential Accelerationππ‘ πΌπ10.192π£ π2 ππ10.232ππ 2π π£π10.1910.20π 10.35Power in rotationalmotionVelocityPeriodπΌ π 2 ππRotation inertia(discrete particlesystem)Parallel Axis Theoremh perpendiculardistance between twoaxesππRelationship Between Angular and Linear Variablesππ 10.3410.1210.16Radical component of πβπΌ ππ ππ2Rotation inertiaπππ π π (π constant)πππ ππππ‘1πΎ πΌπ2211 πΎ πΎπ πΎπ πΌππ2 πΌππ2 π2210.5310.5410.5510.3410.52
Moments of Inertia I for various rigid objects of Mass MThin walled hollow cylinder or hoopabout central axisAnnular cylinder (or ring) aboutcentral axisπΌ ππ 21πΌ π(π 12 π 22 )2Solid Sphere, axis through center2πΌ ππ 25Thin rod, axis perpendicular to rodand passing though endSolid Sphere, axis tangent to surface7πΌ ππ 25Thin Rectangular sheet (slab), axisparallel to sheet and passing thoughcenter of the other edgeSolid cylinder or disk about centralaxis1πΌ ππ 22Thin Walled spherical shell, axisthrough centerSolid cylinder or disk about centraldiameter11πΌ ππ 2 ππΏ2412Thin rod, axis perpendicular to rodand passing though center2πΌ ππ 23Thin Rectangular sheet (slab , axisalong one edgeπΌ 1ππΏ212Thin rectangular sheet (slab) aboutperpendicular axis through center1πΌ ππΏ23πΌ 1ππΏ2121πΌ ππΏ23πΌ 1π(π2 π 2 )12
Chapter 11Rolling Bodies (wheel)Speed of rolling wheelKinetic Energy of RollingWheelAcceleration of rollingwheelAcceleration along x-axisextending up the rampπ£πππ ππ 11.2Angular Momentum112πΎ πΌπππ π2 ππ£πππ2211.5Magnitude of AngularMomentumππππ οΏ½οΏ½οΏ½ πΌ1 πππ2ππ 11.10Magnitude of torqueNewtonβs 2nd Lawπβ πβ πΉβ11.14π ππΉ π πΉ οΏ½οΏ½οΏ½ ββπβππ‘β πππ£π πππβ ππ πππ£ 11.1811.1911.21πAngular momentum of asystem of particlesββ ββπΏβππ 1πβπππ‘ Torque as a vectorTorqueAngular Momentumββ βπ£β)π£ βββ βπβ βπβ π(πββππΏππ‘Angular Momentum continuedAngular Momentum of aπΏ πΌπrotating rigid bodyConservation of angularπΏββ π sion of a GyroscopePrecession rateΞ© ππππΌπ11.31
Chapter 12Chapter 13Static Equilibrium12.3Gravitational Force(Newtonβs law ofgravitation)πβπππ‘ 012.5Principle ofSuperpositionπΉβπππ‘,π₯ 0, πΉβπππ‘,π¦ 012.712.8πΉβπππ‘ 0If forces lie on thexy-planeStress (force per unitarea)Strain (fractionalchange in length)Stress (pressure)Tension/CompressionE: Youngβs modulusShearing StressG: Shear modulusHydraulic StressB: Bulk modulusπβπππ‘,π§ 0π π‘πππ π ππππ’ππ’π π π‘πππππΉπ π΄πΉ πΏ πΈπ΄πΏπΉ π₯ πΊπ΄πΏ ππ π΅π12.912.22πΉ πΊπΉβ1,πππ‘ πΉβ1ππΉβ1 ππΉβEscape SpeedKeplerβs 3rd Law(law of periods)Energy for bject incircular orbit13.6πΊππ2πΊπππΉ ππ 3πΊπππ π13.11ππ Gravitational PotentialEnergy12.24π (13.1913.21πΊπ1 π2 πΊπ1 π3 πΊπ2 π3 )π12π13π232πΊππ£ π π2 (π 4π 2 3)ππΊππΊππππΎ πΈ οΏ½οΏ½πMechanical EnergyπΈ (elliptical orbit)2π 11*Note: πΊ 6.6704 10π π2 /ππ2Mechanical Energy(circular orbit)13.5π 2Gravitation within aspherical Shell12.2313.1πGravitational Forceacting on a particlefrom an extendedbodyGravitationalaccelerationPotential energy on asystem (3 particles)π1 π2π213.3413.2113.3813.4013.42
Chapter 14Density π πππ ππ Pressure and depth ina static FluidP1 is higher than P2Gauge PressureArchimedesβ principleMass Flow RateVolume flow rateBernoulliβs EquationEquation of continuityEquation of continuitywhen14.114.2 πΉ π΄πΉπ π΄14.314.4π2 π1 ππ(π¦1 π¦2 )π π0 ππβ14.714.8π PressureChapter 15Angular frequencyAccelerationKinetic and PotentialEnergyπΉπ ππ π14.16π π ππ π ππ΄π£14.25π π π΄π£14.24π π π΄π£ tyππβ1π ππ£ 2 πππ¦ ππππ π‘πππ‘2π π ππ π ππ΄π£ ππππ π‘πππ‘Frequencycycles per time14.29Angular frequencyπ 1π15.2π₯ π₯π cos(ππ‘ π)π 2π 2πππ15.315.5π£ ππ₯π sin(ππ‘ π)15.6π π2 π₯π cos(ππ‘ π)15.711πΎ ππ£ 2 π ππ₯ 222ππ π15.12Periodπ 2π ππ15.13Torsion pendulumπΌπ 2π π15.23Simple PendulumπΏπ 2π π15.28Physical PendulumπΌπ 2π πππΏ15.2914.2514.24Damping forceπΉβπ ππ£βdisplacementπ₯(π‘) π₯π π 2π cos(πβ² π‘ π)15.42Angular frequencyππ2πβ² π 4π215.43Mechanical Energy1 2 ππ‘πΈ(π‘) ππ₯ππ π215.44ππ‘
Chapter 16Sinusoidal WavesMathematical form(positive direction)Angular wave numberAngular frequencyWave speedAverage Powerπ¦(π₯, π‘) π¦π sin(ππ₯ ππ‘)2ππ π2ππ 2ππππ£ π π πππ π12πππ£π ππ£π2 π¦π2Traveling Wave Form16.216.5π¦(π₯, π‘) β(ππ₯ ππ‘)16.17ππ£ π16.2611π¦ β² (π₯, π‘) [2π¦π cos ( π)] sin (ππ₯ ππ‘ π)2216.51π¦ β² (π₯, π‘) [2π¦π sin(ππ₯)]cos(ππ‘)16.60Wave speed onstretched string16.9Resulting wave when 2waves only differ byphase constant16.13Standing wave16.33Resonant frequencyπ£π£π π π 2πΏ for n 1,2, 16.66
Chapter 17Sound WavesStanding Waves Patterns in Pipesπ΅π£ π17.3π π π cos(ππ₯ ππ‘)17.12Change in pressureΞπ Ξππ sin(ππ₯ ππ‘)17.13Standing wavefrequency (open atboth ends)Standing wavefrequency (open atone end)Pressure amplitudeΞππ (π£ππ)π π17.14beatsSpeed of sound wavedisplacementπ£π π π£π π ππ£2πΏππ£4πΏfor n 1,2,317.39for n 1,3,517.41πππππ‘ π1 π217.46InterferencePhase differenceFully ConstructiveInterferenceFull DestructiveinterferenceMechanical EnergyΞπΏ2πππ π(2π) for m 0,1,2 ΞπΏ 0,1,2ππ (2π 1)π for m 0,12ΞπΏ .5,1.5,2.5 ππ 1 2 ππ‘πΈ(π‘) ππ₯ππ π217.2117.2217.2317.2417.2515.44Doppler EffectSource Moving towardstationary observerSource Moving awayfrom stationaryobserverObserver movingtoward stationarysourceObserver moving awayfrom stationary sourceSound IntensityπΌ IntensityIntensity -uniform inall directionsππ΄12πΌ ππ£π2 π π2πΌ ππ 4ππ 2πβ² ππ£π£ π£π 17.53πβ² ππ£π£ π£π 17.54π£ π£π·π£π£ π£π·πβ² ππ£πβ² πShockwave17.2617.2717.29Intensity level indecibelsπΌπ½ (10ππ΅) log ( )πΌπ17.29Mechanical Energy1 2 ππ‘πΈ(π‘) ππ₯ππ π215.44Half-angle π of Machconeπ πππ π£π£π 17.4917.5117.57
Chapter 18Temperature ScalesFahrenheit to CelsiusCelsius to Fahrenheit5ππΆ (ππΉ 32)99ππΉ ππΆ 325π ππΆ 273.15Celsius to KelvinFirst Law of Thermodynamics18.818.818.7 πΈπππ‘ πΈπππ‘,π πΈπππ‘,π π πFirst Law of18.26Thermodynamics18.27ππΈπππ‘ ππ ππNote: πΈπππ‘ Change in Internal EnergyQ (heat) is positive when the system absorbs heat and negative when itloses heat. W (work) is work done by system. W is positive when expandingand negative contracts because of an external forceThermal ExpansionApplications of First LawLinear Thermal Expansion πΏ πΏπΌ π18.9Volume Thermal Expansion π ππ½ π18.10Q 0 πΈπππ‘ πAdiabatic(no heat flow)W 0 πΈπππ‘ π πΈπππ‘ 0Q Wπ π πΈπππ‘ 0(constant volume)HeatHeat and temperaturechangeπ πΆ(ππ ππ )π ππ(ππ ππ )18.1318.14Heat and phase changeπ πΏπ18.16Power (Conducted)ππππππππ» ππΆ ππ΄π‘πΏFree expansionsMisc.ππP Q/tPowerCyclical process18.32Rate objects absorbsenergy4ππππ πππ΄ππππ£18.39Power from radiationππππ πππ΄π 418.38Work Associated withVolume Changeπ ππ πππππ18.25π π π£π 5.6704 10 8 π/π2 πΎ 4Revised 7/20/17
Jul 20, 2017Β Β· PHYS 2310 Engineering Physics I Formula Sheets Chapters 1-18 Chapter 1/Important Numbers Chapter 2 Units for SI Base Quantities Quantity Unit Name Unit Symbol Length Meter M Time Second s Mass (not weight) Kilogram kg Common Conversions 1 k