PHYS 2310 Engineering Physics I Formula Sheets

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 𝜇𝑚2.54 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 𝑔/𝑐𝑚3 or 1000 𝑘𝑔/𝑚3g (on earth)9.8 m/s 2CircumferenceSurface area(sphere)DensityCommon geometric FormulasArea circle𝐶 2𝜋𝑟𝑆𝐴 4𝜋𝑟 2Volume (rectangular solid)Instantaneous Velocity𝑉𝑎𝑣𝑔 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑥 𝑡𝑖𝑚𝑒 𝑡𝑡𝑜𝑡𝑎𝑙 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑡𝑖𝑚𝑒̅ 𝑑𝑥 𝑥𝑣 lim 𝑡 0 𝑡𝑑𝑡𝑠𝑎𝑣𝑔 Average Acceleration 𝑣 𝑡𝑑𝑣 𝑑2 𝑥𝑎 𝑑𝑡 𝑑𝑡 2𝑎𝑎𝑣𝑔 Motion of a particle with constant acceleration𝐴 𝜋𝑟4Volume (sphere)𝑉 𝜋𝑟 33𝑉 𝑙 𝑤 ℎ𝑉 𝑎𝑟𝑒𝑎 ionInstantaneousAcceleration22.2𝑣 𝑣0 𝑎𝑡1 𝑥 (𝑣0 𝑣)𝑡21 𝑥 𝑣0 𝑡 𝑎𝑡 22𝑣 2 𝑣02 2𝑎 𝑥2.112.172.152.162.72.82.9

Chapter 3Adding VectorsGeometricallyAdding VectorsGeometrically(Associative Law)Chapter 4𝑎⃗ 𝑏⃗⃗ 𝑏⃗⃗ 𝑎⃗3.2(𝑎⃗ 𝑏⃗⃗) 𝑐⃗ 𝑎⃗ (𝑏⃗⃗ 𝑐⃗)3.33.5Magnitude of vector 𝑎 𝑎 𝑎𝑥2 𝑎𝑦23.6Angle between x axisand vector𝑎𝑦𝑡𝑎𝑛𝜃 𝑎𝑥3.6Unit vector notation𝑎⃗ 𝑎𝑥 𝑖̂ 𝑎𝑦 𝑗̂ 𝑎𝑧 𝑘̂3.7𝑟𝑥 𝑎𝑥 𝑏𝑥𝑟𝑦 𝑎𝑦 𝑏𝑦𝑟𝑧 𝑎𝑧 𝑏𝑧3.103.113.12𝑎⃗ 𝑏⃗⃗ 𝑎𝑏𝑐𝑜𝑠𝜃3.20Adding vectors inComponent FormScalar (dot product)Scalar (dot product)Projection of 𝑎⃗ 𝑜𝑛 𝑏⃗⃗ orcomponent of 𝑎⃗ 𝑜𝑛 𝑏⃗⃗Vector (cross) productmagnitude𝑎⃗ 𝑏⃗⃗ (𝑎𝑥 𝑖̂ 𝑎𝑦 𝑗̂ 𝑎𝑧 𝑘̂) (𝑏𝑥 𝑖̂ 𝑏𝑦 𝑗̂ 𝑏𝑧 𝑘̂ )𝑎⃗ 𝑏⃗⃗ 𝑎𝑥 𝑏𝑥 𝑎𝑦 𝑏𝑦 𝑎𝑧 𝑏𝑧displacementAverage AccelerationInstantaneousAcceleration4.4 𝑥 𝑡4.8𝑑𝑟⃗ 𝑣𝑥 𝑖̂ 𝑣𝑦 𝑗̂ 𝑣𝑧 𝑘̂𝑑𝑡 𝑣⃗𝑎⃗𝑎𝑣𝑔 𝑡𝑑𝑣⃗𝑎⃗ 𝑑𝑡𝑎⃗ 𝑎𝑥 𝑖̂ 𝑎𝑦 𝑗̂ 𝑎𝑧 𝑘̂𝑣⃗ Projectile Motion𝑣𝑦 𝑣0 𝑠𝑖𝑛𝜃0 𝑔𝑡3.241 𝑦 𝑣0 𝑠𝑖𝑛𝜃𝑡 𝑔𝑡 2222𝑣𝑦 (𝑣0 𝑠𝑖𝑛𝜃0 ) 2𝑔 y𝑘̂𝑎𝑧 ��𝑎𝑦𝑏𝑦4.154.21𝑣𝑦 𝑣0 𝑠𝑖𝑛𝜃0 𝑔𝑡𝑐 𝑎𝑏𝑠𝑖𝑛𝜙4.104.114.231 𝑥 𝑣0 𝑐𝑜𝑠𝜃𝑡 𝑎𝑥 𝑡 22or 𝑥 𝑣0 𝑐𝑜𝑠𝜃𝑡 if 𝑎𝑥 0𝑎⃗ 𝑏⃗⃗ 𝑏 or𝑖̂⃗⃗𝑎⃗𝑥𝑏 𝑑𝑒𝑡 𝑎𝑥𝑏𝑥 𝑟⃗ 𝑥𝑖̂ 𝑦𝑗̂ 𝑧𝑘̂⃗⃗𝑎𝑣𝑔 𝑉Instantaneous Velocity3.22𝑎⃗𝑥𝑏⃗⃗ (𝑎𝑥 𝑖̂ 𝑎𝑦 𝑗̂ 𝑎𝑧 𝑘̂)𝑥(𝑏𝑥 𝑖̂ 𝑏𝑦 𝑗̂ 𝑏𝑧 𝑘̂ ) (𝑎𝑦 𝑏𝑧 𝑏𝑦 𝑎𝑧 )𝑖̂ (𝑎𝑧 𝑏𝑥 𝑏𝑧 𝑎𝑥 )𝑗̂ (𝑎𝑥 𝑏𝑦 𝑏𝑥 𝑎𝑦 )𝑘̂Vector (cross product)4.4Average Velocity𝑎𝑥 𝑎𝑐𝑜𝑠𝜃𝑎𝑦 𝑎𝑠𝑖𝑛𝜃Components of Vectors𝑟⃗ 𝑥𝑖̂ 𝑦𝑗̂ 𝑧𝑘̂Position vectorRelative MotionUniform CircularMotion𝑦 (𝑡𝑎𝑛𝜃0 )𝑥 𝑅 𝑔𝑥2(𝑣0 𝑐𝑜𝑠𝜃0 )2𝑣02sin(2𝜃0 )𝑔⃗⃗⃗⃗⃗⃗⃗𝑣𝐴𝐶 ⃗⃗⃗⃗⃗⃗⃗𝑣𝐴𝐵 ⃗⃗⃗⃗⃗⃗⃗𝑣𝐵𝐶𝑎𝐴𝐵 ���𝐴𝑎 𝑣2𝑟𝑇 2𝜋𝑟𝑣4.254.264.444.454.344.35

GeneralComponent formChapter 5Chapter 6Newton’s Second LawFriction𝐹⃗𝑛𝑒𝑡 𝑚𝑎⃗𝐹𝑛𝑒𝑡,𝑥 𝑚𝑎𝑥𝐹𝑛𝑒𝑡,𝑦 𝑚𝑎𝑦𝐹𝑛𝑒𝑡,𝑧 𝑚𝑎𝑦5.1Kinetic FrictionalWeight𝐹𝑔 𝑚𝑔𝑊 𝑚𝑔𝑓⃗𝑠,𝑚𝑎𝑥 𝜇𝑠 𝐹𝑁6.1𝑓⃗𝑘 𝜇𝑘 𝐹𝑁6.25.2Drag ForceGravitational ForceGravitational ForceStatic Friction(maximum)Terminal velocity1𝐷 𝐶𝜌𝐴𝑣 222𝐹𝑔𝑣𝑡 Centripetal Force𝑣2𝑎 𝑅𝐹 𝑚𝑣 2𝑅6.176.18

Chapter 7Work- Kinetic EnergyTheorem7.1𝑊 𝐹𝑑𝑐𝑜𝑠𝜃 𝐹⃗ 𝑑⃗7.77.8Spring Force (Hooke’slaw)Work done by springWork done by VariableForceAverage Power(rate at which thatforce does work on anobject)Instantaneous ��� Mechanical Energy𝐸𝑚𝑒𝑐 𝐾 𝑈8.127.15Principle ofconservation ofmechanical energy𝐾1 𝑈1 𝐾2 𝑈2𝐸𝑚𝑒𝑐 𝐾 𝑈 08.188.177.207.21Force acting on particle𝑧𝑓𝑃 7.36𝑧𝑖𝑊 𝑡𝑑𝑊 𝐹𝑉𝑐𝑜𝑠𝜃 𝐹⃗ 𝑣⃗𝑑𝑡8.78.117.25𝑊 𝐹𝑥 𝑑𝑥 𝐹𝑦 𝑑𝑦 𝐹𝑧 𝑑𝑧 𝑈 𝑚𝑔 𝑦1 2𝑘𝑥27.12 𝐾 𝑊𝑎 𝑊𝑔𝑊𝑎 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝐹𝑜𝑟𝑐𝑒𝐹⃗𝑠 𝑘𝑑⃗𝐹𝑥 𝑘𝑥 (along x-axis)Gravitational PotentialEnergy8.18.6𝑈(𝑥) 𝑊𝑔 𝑚𝑔𝑑𝑐𝑜𝑠𝜙1 2 1 2𝑘𝑥 𝑘𝑥2 𝑖 2 𝑓 𝑈 𝑊 𝐹(𝑥)𝑑𝑥Elastic Potential Energy7.10𝑊𝑠 Potential Energy𝑥𝑖 𝐾 𝐾𝑓 𝐾0 𝑊Work done by gravityWork done bylifting/lowering object𝑥𝑓1𝐾 𝑚𝑣 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 0𝐹(𝑥) 𝑑𝑈(𝑥)𝑑𝑥8.22𝑊 𝐸𝑚𝑒𝑐 𝐾 𝑈8.258.26𝑊 𝐸𝑚𝑒𝑐 𝐸𝑡ℎ8.33 𝐸𝑡ℎ 𝑓𝑘 𝑑𝑐𝑜𝑠𝜃8.31𝑊 𝐸 𝐸𝑚𝑒𝑐 𝐸𝑡ℎ 𝐸𝑖𝑛𝑡8.35Average PowerInstantaneous Power**In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PE𝑃𝑎𝑣𝑔 𝑃 𝐸 𝑡𝑑𝐸𝑑𝑡8.408.41

Chapter 9Impulse and Momentum𝑡𝑓Impulse𝐽⃗ 𝐹⃗ (𝑡)𝑑𝑡𝑡𝑖9.35𝑝⃗ 𝑚𝑣⃗9.22𝐽⃗ Δ𝑝⃗ 𝑝⃗𝑓 𝑝⃗𝑖𝑑𝑝⃗𝑑𝑡𝐹⃗𝑛𝑒𝑡 𝑚⃗𝑎⃗𝑐𝑜𝑚𝑃⃗⃗ �𝑒𝑡 Newton’s 2nd lawSystem of Particles9.30𝐽 𝐹𝑛𝑒𝑡 𝑡Linear MomentumImpulse-MomentumTheoremCollision continued 𝐹⃗𝑛𝑒𝑡 9.319.32Inelastic CollisionConservation of LinearMomentum (in 2D)Average forceElastic Collision𝑛𝑛 𝑝 𝑚 𝑣 𝑡 𝑡 𝑚𝐹𝑎𝑣𝑔 𝑣 𝑡2𝑚1 ()𝑣𝑚1 𝑚2 1𝑖𝑛9.149.259.27Center of mass location𝑟⃗𝑐𝑜𝑚1 𝑚𝑖 𝑟⃗𝑖𝑀9.67𝑖 1𝑛𝑣⃗𝑐𝑜𝑚 1 𝑚𝑖 𝑣⃗𝑖𝑀Rocket EquationsThrust (Rvrel)𝑅𝑣𝑟𝑒𝑙 𝑀𝑎9.68Change in velocity9.429.43𝑝⃗1𝑖 𝑝⃗2𝑖 𝑝⃗1𝑓 𝑝⃗2𝑓9.509.519.78𝐾1𝑖 𝐾2𝑖 𝐾1𝑓 𝐾2𝑓9.8𝑖 1𝑃⃗⃗ 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑃⃗⃗𝑖 𝑃⃗⃗𝑓𝑚1 𝑣𝑖1 𝑚2 𝑣12 𝑚1 𝑣𝑓1 𝑚2 𝑣𝑓29.379.40Center of Mass𝑑𝑡𝑚1 𝑚2𝑣1𝑓 ()𝑣𝑚1 𝑚2 1𝑖9.779.22Collision𝑣2𝑓𝑃⃗⃗1𝑖 𝑃⃗⃗2𝑖 𝑃⃗⃗1𝑓 𝑃⃗⃗2𝑓𝐹𝑎𝑣𝑔 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)𝑚1 𝑣01 𝑚2 𝑣02 (𝑚1 𝑚2 )𝑣𝑓Δ𝑣 𝑣𝑟𝑒𝑙 𝑙𝑛𝑀𝑖𝑀𝑓9.889.88

Chapter 10Angular displacement(in radiansAverage angularvelocityInstantaneous VelocityAverage angularaccelerationInstantaneous angularacceleration𝑠𝑟Δ𝜃 𝜃2 𝜃1 𝜃𝜔𝑎𝑣𝑔 𝑡𝑑𝜃𝜔 𝑑𝑡 𝜔𝛼𝑎𝑣𝑔 𝑡𝑑𝜔𝛼 𝑑𝑡10.110.4𝜃 Rotational Kinematics𝜔 𝜔0 𝛼𝑡1Δ𝜃 𝜔0 𝑡 𝛼𝑡 2222𝜔 𝜔0 2𝛼Δ𝜃1Δ𝜃 (𝜔 𝜔0 )𝑡21Δ𝜃 𝜔𝑡 𝛼𝑡 2210.510.610.710.810.1310.1410.15𝐼 𝐼𝑐𝑜𝑚 𝑀ℎ210.36𝜏 𝑟𝐹𝑡 𝑟 𝐹 𝑟𝐹𝑠𝑖𝑛𝜃10.3910.41Newton’s Second Law𝜏𝑛𝑒𝑡 𝐼𝛼10.45Rotational work doneby a toque𝑊 𝜏𝑑𝜃TorqueRotational KineticEnergyWork-kinetic energytheorem𝑣 𝜔𝑟10.18Tangential Acceleration𝑎𝑡 𝛼𝑟10.192𝑣 𝜔2 𝑟𝑟10.232𝜋𝑟 2𝜋 𝑣𝜔10.1910.20𝑇 10.35Power in rotationalmotionVelocityPeriod𝐼 𝑟 2 𝑑𝑚Rotation inertia(discrete particlesystem)Parallel Axis Theoremh perpendiculardistance between twoaxes𝜃𝑓Relationship Between Angular and Linear Variables𝑎𝑟 10.3410.1210.16Radical component of 𝑎⃗𝐼 𝑚𝑖 𝑟𝑖2Rotation inertia𝜃𝑖𝑊 𝜏 𝜃 (𝜏 constant)𝑑𝑊𝑃 𝜏𝜔𝑑𝑡1𝐾 𝐼𝜔2211 𝐾 𝐾𝑓 𝐾𝑖 𝐼𝜔𝑓2 𝐼𝜔𝑖2 𝑊2210.5310.5410.5510.3410.52

Moments of Inertia I for various rigid objects of Mass MThin walled hollow cylinder or hoopabout central axisAnnular cylinder (or ring) aboutcentral axis𝐼 𝑀𝑅 21𝐼 𝑀(𝑅12 𝑅22 )2Solid Sphere, axis through center2𝐼 𝑀𝑅 25Thin rod, axis perpendicular to rodand passing though endSolid Sphere, axis tangent to surface7𝐼 𝑀𝑅 25Thin Rectangular sheet (slab), axisparallel to sheet and passing thoughcenter of the other edgeSolid cylinder or disk about centralaxis1𝐼 𝑀𝑅 22Thin Walled spherical shell, axisthrough centerSolid cylinder or disk about centraldiameter11𝐼 𝑀𝑅 2 𝑀𝐿2412Thin rod, axis perpendicular to rodand passing though center2𝐼 𝑀𝑅 23Thin Rectangular sheet (slab , axisalong one edge𝐼 1𝑀𝐿212Thin rectangular sheet (slab) aboutperpendicular axis through center1𝐼 𝑀𝐿23𝐼 1𝑀𝐿2121𝐼 𝑀𝐿23𝐼 1𝑀(𝑎2 𝑏 2 )12

Chapter 11Rolling Bodies (wheel)Speed of rolling wheelKinetic Energy of RollingWheelAcceleration of rollingwheelAcceleration along x-axisextending up the ramp𝑣𝑐𝑜𝑚 𝜔𝑅11.2Angular Momentum112𝐾 𝐼𝑐𝑜𝑚 𝜔2 𝑀𝑣𝑐𝑜𝑚2211.5Magnitude of AngularMomentum𝑎𝑐𝑜𝑚 ��� 𝐼1 𝑐𝑜𝑚2𝑀𝑅11.10Magnitude of torqueNewton’s 2nd Law𝜏⃗ 𝑟⃗ 𝐹⃗11.14𝜏 𝑟𝐹 𝑟 𝐹 ��� ⃗⃗𝑑ℓ𝑑𝑡ℓ 𝑟𝑚𝑣𝑠𝑖𝑛𝜙ℓ 𝑟𝑝 𝑟𝑚𝑣 11.1811.1911.21𝑛Angular momentum of asystem of particles⃗⃗ ⃗⃗𝐿ℓ𝑖𝑖 1𝜏⃗𝑛𝑒𝑡 Torque as a vectorTorqueAngular Momentum⃗⃗ ⃗𝑣⃗)𝑣 ⃗ℓ⃗ ⃗𝑟⃗ ⃗𝑝⃗ 𝑚(𝑟⃗⃗𝑑𝐿𝑑𝑡Angular Momentum continuedAngular Momentum of a𝐿 𝐼𝜔rotating rigid bodyConservation of angular𝐿⃗⃗ 𝑖 sion of a GyroscopePrecession rateΩ 𝑀𝑔𝑟𝐼𝜔11.31