PHYS 2310 Engineering Physics I Formula Sheets Chapters 1-18 Chapter 1/Important Numbers Chapter 2 Units for SI Base Quantities Velocity Quantity Unit Name Unit Symbol 푑푖푠푝푙푎푐푒푚푒푛푡 ∆푥 Length Meter M Average Velocity 푉 = = 2.2 푎푣푔 푡푖푚푒 ∆푡 Time Second s Mass (not weight) Kilogram kg 푡표푡푎푙 푑푖푠푡푎푛푐푒 Average Speed 푠푎푣푔 = 2.3 푡푖푚푒 ∆푥̅ 푑푥 Common Conversions Instantaneous Velocity 푣 = lim = 2.4 1 kg or 1 m 1000 g or m 1 m 1 × 106 휇푚 ∆푡→0 ∆푡 푑푡 1 m 100 cm 1 inch 2.54 cm 1 m 1000 mm 1 day 86400 seconds Acceleration 1 second 1000 milliseconds 1 hour 3600 seconds 1 m 3.281 ft 360° 2휋 rad ∆푣 Average Acceleration 푎 = 2.7 푎푣푔 ∆푡 Important Constants/Measurements 푑푣 푑2푥 24 Instantaneous 2.8 Mass of Earth 5.98 × 10 kg 푎 = = 2 Acceleration 푑푡 푑푡 2.9 Radius of Earth 6.38 × 106 m 1 u (Atomic Mass Unit) 1.661 × 10−27 kg Density of water 1 푔/푐푚3 or 1000 푘푔/푚3 Motion of a particle with constant acceleration g (on earth) 9.8 m/s2 푣 = 푣0 + 푎푡 2.11 Density 1 ∆푥 = (푣 + 푣)푡 2.17 Common geometric Formulas 2 0 2 Circumference 퐶 = 2휋푟 Area circle 퐴 = 휋푟 1 2 ∆푥 = 푣0푡 + 푎푡 2.15 Surface area 4 3 2 푆퐴 = 4휋푟2 Volume (sphere) 푉 = 휋푟 (sphere) 3 2 2 푣 = 푣0 + 2푎∆푥 2.16 푉 = 푙 ∙ 푤 ∙ ℎ Volume (rectangular solid) 푉 = 푎푟푒푎 ∙ 푡ℎ푖푐푘푛푒푠푠 Chapter 3 Chapter 4 Adding Vectors ⃗⃗ ⃗⃗ 3.2 Position vector 푟⃗ = 푥푖̂ + 푦푗̂ + 푧푘̂ 4.4 Geometrically 푎⃗ + 푏 = 푏 + 푎⃗ Adding Vectors displacement ∆푟⃗ = ∆푥푖̂ + ∆푦푗̂ + ∆푧푘̂ 4.4 Geometrically (푎⃗ + 푏⃗⃗) + 푐⃗ = 푎⃗ + (푏⃗⃗ + 푐⃗) 3.3 (Associative Law) ∆푥 Average Velocity 푉⃗⃗푎푣푔 = 4.8 푎 = 푎푐표푠휃 ∆푡 Components of Vectors 푥 3.5 푎 = 푎푠푖푛휃 푑푟⃗ 4.10 푦 Instantaneous Velocity 푣⃗ = = 푣 푖̂ + 푣 푗̂ + 푣 푘̂ 푑푡 푥 푦 푧 4.11 Magnitude of vector |푎| = 푎 = √푎2 + 푎2 3.6 ∆푣⃗ 푥 푦 Average Acceleration 푎⃗ = 4.15 푎푣푔 ∆푡 Angle between x axis 푎푦 푡푎푛휃 = 3.6 푑푣⃗ and vector 푎 Instantaneous 푎⃗ = 4.16 푥 푑푡 Acceleration ̂ 4.17 Unit vector notation 푎⃗ = 푎푥푖̂ + 푎푦푗̂ + 푎푧푘̂ 3.7 푎⃗ = 푎푥푖̂ + 푎푦푗̂ + 푎푧푘 푟 = 푎 + 푏 3.10 Adding vectors in 푥 푥 푥 Projectile Motion 푟푦 = 푎푦 + 푏푦 3.11 Component Form 푣푦 = 푣0푠푖푛휃0 − 푔푡 4.23 푟푧 = 푎푧 + 푏푧 3.12 1 ∆푥 = 푣 푐표푠휃푡 + 푎 푡2 4.21 Scalar (dot product) 푎⃗ ∙ 푏⃗⃗ = 푎푏푐표푠휃 3.20 0 2 푥 or ∆푥 = 푣0푐표푠휃푡 if 푎푥=0 푎⃗ ∙ 푏⃗⃗ = (푎푥푖̂ + 푎푦푗̂ + 푎푧푘̂) ∙ (푏푥푖̂ + 푏푦푗̂ + 푏푧푘̂) Scalar (dot product) 3.22 1 2 푎⃗ ∙ 푏⃗⃗ = 푎 푏 + 푎 푏 + 푎 푏 ∆푦 = 푣0푠푖푛휃푡 − 푔푡 4.22 푥 푥 푦 푦 푧 푧 2 2 2 Projection of 푎⃗ 표푛 푏⃗⃗ or 푎⃗ ∙ 푏⃗⃗ 푣푦 = (푣0푠푖푛휃0) − 2푔∆y 4.24 ⃗⃗ component of 푎⃗ 표푛 푏 |푏| 푣푦 = 푣0푠푖푛휃0 − 푔푡 4.23 Vector (cross) product 푔푥2 푐 = 푎푏푠푖푛휙 3.24 Trajectory 4.25 magnitude 푦 = (푡푎푛휃0)푥 − 2 2(푣0푐표푠휃0) 푎⃗푥푏⃗⃗ = (푎 푖̂ + 푎 푗̂ + 푎 푘̂)푥(푏 푖̂ + 푏 푗̂ + 푏 푘̂) 2 푥 푦 푧 푥 푦 푧 푣0 = (푎 푏 − 푏 푎 )푖̂ + (푎 푏 − 푏 푎 )푗̂ Range 푅 = sin(2휃 ) 4.26 푦 푧 푦 푧 푧 푥 푧 푥 푔 0 ̂ + (푎푥푏푦 − 푏푥푎푦)푘 Vector (cross product) or 3.26 ⃗푣⃗⃗⃗⃗⃗⃗ = ⃗푣⃗⃗⃗⃗⃗⃗ + 푣⃗⃗⃗⃗⃗⃗⃗ 푖̂ 푗 푘̂ Relative Motion 퐴퐶 퐴퐵 퐵퐶 4.44 푎⃗⃗⃗퐴퐵⃗⃗⃗⃗ = 푎⃗⃗⃗⃗퐵퐴⃗⃗⃗⃗ 푎⃗푥푏⃗⃗ = 푑푒푡 |푎푥 푎푦 푎푧| 4.45 2 푏푥 푏푦 푏푧 Uniform Circular 푣 2휋푟 4.34 푎 = 푇 = Motion 푟 푣 4.35 Chapter 5 Chapter 6 Newton’s Second Law Friction 퐹⃗푛푒푡 = 푚푎⃗ Static Friction General 5.1 푓⃗푠,푚푎푥 = 휇푠퐹푁 6.1 (maximum) 퐹푛푒푡,푥 = 푚푎푥 Kinetic Frictional 푓⃗푘 = 휇푘퐹푁 6.2 Component form 퐹푛푒푡,푦 = 푚푎푦 5.2 퐹푛푒푡,푧 = 푚푎푦 1 2 Drag Force 퐷 = 퐶휌퐴푣 6.14 2 Gravitational Force 2퐹푔 Terminal velocity 푣 = √ 6.16 푡 퐶휌퐴 Gravitational Force 5.8 퐹 = 푚푔 푔 Centripetal 푣2 Weight 푎 = 6.17 푊 = 푚푔 5.12 acceleration 푅 2 Centripetal Force 푚푣 퐹 = 6.18 푅 Chapter 7 Chapter 8 푥푓 Kinetic Energy 1 2 8.1 퐾 = 푚푣 7.1 Potential Energy ∆푈 = −푊 = − ∫ 퐹(푥)푑푥 2 푥 8.6 Work done by constant 7.7 Gravitational Potential Force ⃗ ⃗ ∆푈 = 푚푔∆푦 8.7 푊 = 퐹푑푐표푠휃 = 퐹 ∙ 푑 7.8 Energy 1 Work- Kinetic Energy Elastic Potential Energy 푈(푥) = 푘푥2 8.11 ∆퐾 = 퐾 − 퐾 = 푊 7.10 2 Theorem 푓 0 Work done by gravity Mechanical Energy 퐸푚푒푐 = 퐾 + 푈 8.12 푊 = 푚푔푑푐표푠휙 7.12 푔 Principle of Work done by ∆퐾 = 푊 + 푊 퐾1 + 푈1 = 퐾2 + 푈2 8.18 푎 푔 7.15 conservation of 퐸푚푒푐 = ∆퐾 + ∆푈 = 0 8.17 lifting/lowering object 푊푎 = 푎푝푝푙푖푒푑 퐹표푟푐푒 mechanical energy Spring Force (Hooke’s 퐹⃗푠 = −푘푑⃗ 7.20 푑푈(푥) Force acting on particle 퐹(푥) = − 8.22 law) 퐹푥 = −푘푥 (along x-axis) 7.21 푑푥 1 1 Work on System by Work done by spring 2 2 7.25 8.25 푊푠 = 푘푥 − 푘푥푓 external force 푊 = ∆퐸 = ∆퐾 + ∆푈 2 2 푚푒푐 8.26 푥 푦 푧 With no friction Work done by Variable 푓 푓 푓 푊 = ∫ 퐹푥푑푥 + ∫ 퐹푦푑푦 + ∫ 퐹푧푑푧 7.36 Work on System by Force 푥푖 푦푖 푧푖 external force 푊 = ∆퐸푚푒푐 + ∆퐸푡ℎ 8.33 Average Power With friction (rate at which that 푊 푃 = 7.42 Change in thermal force does work on an 푎푣푔 ∆푡 ∆퐸 = 푓 푑푐표푠휃 8.31 energy 푡ℎ 푘 object) Conservation of Energy 푑푊 7.43 푊 = ∆퐸 = ∆퐸 + ∆퐸 + ∆퐸 8.35 Instantaneous Power 푃 = = 퐹푉푐표푠휃 = 퐹⃗ ∙ 푣⃗ *if isolated W=0 푚푒푐 푡ℎ 푛푡 푑푡 7.47 ∆퐸 Average Power 푃 = 8.40 푎푣푔 ∆푡 푑퐸 Instantaneous Power 푃 = 8.41 푑푡 **In General Physics, Kinetic Energy is abbreviated to KE and Potential Energy is PE Chapter 9 Impulse and Momentum Collision continued… 푡 Inelastic Collision 푚 푣 + 푚 푣 = (푚 + 푚 )푣 푓 9.30 1 01 2 02 1 2 푓 퐽⃗ = ∫ 퐹⃗(푡)푑푡 Impulse 푡푖 9.35 Conservation of Linear 푃⃗⃗ + 푃⃗⃗ = 푃⃗⃗ + 푃⃗⃗ 퐽 = 퐹푛푒푡∆푡 1 2 1푓 2푓 9.77 Momentum (in 2D) Linear Momentum 푝⃗ = 푚푣⃗ 9.22 푛 푛 퐹 = − ∆푝 = − 푚∆푣 푎푣푔 9.37 Impulse-Momentum 9.31 Average force ∆푡 ∆푡 퐽⃗ = Δ푝⃗ = 푝⃗ − 푝⃗ ∆푚 9.40 Theorem 푓 9.32 퐹 = − ∆푣 푎푣푔 ∆푡 푑푝⃗ nd Newton’s 2 law 퐹⃗푛푒푡 = 9.22 푑푡 Center of Mass 퐹⃗ = 푚푎⃗⃗⃗ 푛 푛푒푡 푐표푚 9.14 1 ⃗⃗ Center of mass location 푟⃗ = ∑ 푚 푟⃗ 9.8 System of Particles 푃 = 푀푣⃗푐표푚 9.25 푐표푚 ⃗⃗⃗ 푀 푑푃 9.27 =1 퐹⃗푛푒푡 = 푛 푑푡 1 Center of mass velocity 푣⃗ = ∑ 푚 푣⃗ 푐표푚 푀 Collision =1 Final Velocity of 2 푚1 − 푚2 objects in a head-on 푣 = ( ) 푣 Rocket Equations 1푓 푚 + 푚 1 9.67 collision where one 1 2 Thrust (Rvrel) 푅푣푟푒푙 = 푀푎 object is initially at rest 9.88 2푚1 9.68 1: moving object 푣2푓 = ( ) 푣1 푚1 + 푚2 Change in velocity 푀 2: object at rest Δ푣 = 푣푟푒푙푙푛 9.88 푀푓 Conservation of Linear 푃⃗⃗ = 푐표푛푠푡푎푛푡 9.42 Momentum (in 1D) 푃⃗⃗ = 푃⃗⃗ 9.43 푓 푝⃗1 + 푝⃗2 = 푝⃗1푓 + 푝⃗2푓 9.50 Elastic Collision 푚1푣1 + 푚2푣12 = 푚1푣푓1 + 푚2푣푓2 9.51 퐾1 + 퐾2 = 퐾1푓 + 퐾2푓 9.78 Chapter 10 푠 Angular displacement 휃 = 10.1 푟 Rotation inertia 퐼 = ∑ 푚 푟2 10.34 (in radians 10.4 Δ휃 = 휃2 − 휃1 Average angular ∆휃 Rotation inertia 휔 = 10.5 2 velocity 푎푣푔 ∆푡 (discrete particle 퐼 = ∫ 푟 푑푚 10.35 system) 푑휃 Instantaneous Velocity 휔 = 10.6 Parallel Axis Theorem 푑푡 h=perpendicular 퐼 = 퐼 + 푀ℎ2 10.36 Average angular ∆휔 distance between two 푐표푚 훼 = 10.7 acceleration 푎푣푔 ∆푡 axes Instantaneous angular 푑휔 10.39- 훼 = 10.8 Torque 휏 = 푟퐹푡 = 푟⊥퐹 = 푟퐹푠푖푛휃 acceleration 푑푡 10.41 Rotational Kinematics Newton’s Second Law 휏푛푒푡 = 퐼훼 10.45 휔 = 휔0 + 훼푡 10.12 휃푓 Rotational work done 푊 = ∫ 휏푑휃 10.53 1 2 Δ휃 = 휔0푡 + 훼푡 10.13 by a toque 휃푖 10.54 2 푊 = 휏∆휃 (휏 constant) 2 2 휔 = 휔0 + 2훼Δ휃 10.14 Power in rotational 푑푊 푃 = = 휏휔 10.55 1 motion 푑푡 Δ휃 = (휔 + 휔0)푡 10.15 2 Rotational Kinetic 1 퐾 = 퐼휔2 10.34 1 Energy Δ휃 = 휔푡 − 훼푡2 10.16 2 2 Work-kinetic energy 1 1 ∆퐾 = 퐾 − 퐾 = 퐼휔2 − 퐼휔2 = 푊 10.52 theorem 푓 2 푓 2 Relationship Between Angular and Linear Variables Velocity 푣 = 휔푟 10.18 Tangential Acceleration 푎푡 = 훼푟 10.19 푣2 Radical component of 푎⃗ 푎 = = 휔2푟 10.23 푟 푟 2휋푟 2휋 10.19 Period 푇 = = 푣 휔 10.20 Moments of Inertia I for various rigid objects of Mass M Thin walled hollow cylinder or hoop Annular cylinder (or ring) about Solid cylinder or disk about central Solid cylinder or disk about central about central axis central axis axis diameter 2 1 퐼 = 푀푅 퐼 = 푀(푅2 + 푅2) 1 2 1 2 퐼 = 푀푅2 2 1 1 퐼 = 푀푅2 + 푀퐿2 4 12 Solid Sphere, axis through center Solid Sphere, axis tangent to surface Thin Walled spherical shell, axis Thin rod, axis perpendicular to rod through center and passing though center 2 퐼 = 푀푅2 7 5 퐼 = 푀푅2 2 1 5 퐼 = 푀푅2 퐼 = 푀퐿2 3 12 Thin rod, axis perpendicular to rod Thin Rectangular sheet (slab), axis Thin Rectangular sheet (slab_, axis Thin rectangular sheet (slab) about and passing though end parallel to sheet and passing though along one edge perpendicular axis through center center of the other edge 1 퐼 = 푀퐿2 3 1 1 퐼 = 푀퐿2 퐼 = 푀(푎2 + 푏2) 1 3 12 퐼 = 푀퐿2 12 Chapter 11 Rolling Bodies (wheel) Angular Momentum Speed of rolling wheel 푣푐표푚 = 휔푅 11.2 Angular Momentum 푣ℓ⃗⃗ = 푟⃗⃗ × 푝⃗⃗⃗ = 푚(푟⃗⃗ × 푣⃗⃗⃗) 11.18 Kinetic Energy of Rolling 1 1 2 2 11.5 Magnitude of Angular ℓ = 푟푚푣푠푖푛휙 11.19- Wheel 퐾 = 퐼푐표푚휔 + 푀푣푐표푚 2 2 Momentum ℓ = 푟푝⊥ = 푟푚푣⊥ 11.21 Acceleration of rolling 푛 푎푐표푚 = 훼푅 11.6 wheel ⃗⃗ 퐿⃗⃗ = ∑ ℓ 11.26 푔푠푖푛휃 Angular momentum of a Acceleration along x-axis =1 11.29 푎푐표푚,푥 = − 11.10 system of particles extending up the ramp 퐼푐표푚 푑퐿⃗⃗ 1 + 2 휏⃗ = 푀푅 푛푒푡 푑푡 Torque as a vector Angular Momentum continued Angular Momentum of a Torque 휏⃗ = 푟⃗ × 퐹⃗ 11.14 퐿 = 퐼휔 11.31 rotating rigid body 11.15- ⃗⃗ Magnitude of torque 휏 = 푟퐹 = 푟 퐹 = 푟퐹푠푖푛휙 Conservation of angular 퐿 = 푐표푛푠푡푎푛푡 11.32 ⊥ ⊥ 11.17 momentum 퐿⃗⃗ = 퐿⃗⃗푓 11.33 nd 푑ℓ⃗⃗ Newton’s 2 Law 휏⃗ = 11.23 푛푒푡 푑푡 Precession of a Gyroscope 푀푔푟 Precession rate Ω = 11.31 퐼휔 Chapter 12 Chapter 13 Static Equilibrium Gravitational Force 푚1푚2 (Newton’s law of 퐹 = 퐺 13.1 푟2 퐹⃗푛푒푡 = 0 12.3 gravitation) 푛 Principle of 퐹⃗ = ∑ 퐹⃗ 13.5 휏⃗푛푒푡 = 0 12.5 Superposition 1,푛푒푡 1 =2 12.7 Gravitational Force 퐹⃗ = 0, 퐹⃗ = 0 If forces lie on the 푛푒푡,푥 푛푒푡,푦 12.8 acting on a particle 퐹⃗1 = ∫ 푑퐹⃗ 13.6 xy-plane from an extended 휏⃗푛푒푡,푧 = 0 12.9 body Gravitational 퐺푀 13.11 푎푔 = 2 Stress (force per unit acceleration 푟 area) Gravitation within a 퐺푚푀 푠푡푟푒푠푠 = 푚표푑푢푙푢푠 × 푠푡푟푎푖푛 12.22 퐹 = 푟 13.19 Strain (fractional spherical Shell 푅3 change in length) Gravitational Potential 퐺푀푚 푈 = − 13.21 퐹 Energy Stress (pressure) 푃 = 푟 퐴 Potential energy on a 퐺푚1푚2 퐺푚1푚3 퐺푚2푚3 Tension/Compression 퐹 ∆퐿 푈 = − ( + + ) 13.22 = 퐸 12.23 system (3 particles) 푟12 푟13 푟23 E: Young’s modulus 퐴 퐿 Shearing Stress 퐹 ∆푥 2퐺푀 = 퐺 12.24 Escape Speed 푣 = √ 13.28 G: Shear modulus 퐴 퐿 푅 Hydraulic Stress ∆푉 2 rd 푝 = 퐵 Kepler’s 3 Law 2 4휋 3 B: Bulk modulus 푉 푇 = ( ) 푟 13.34 (law of periods) 퐺푀 Energy for bject in