Kinetics: F=ma (Ch. 3 & 7) Review Lecture 32 ME 231: Dynamics Question of the Day What is the most important concept in mechanics? Free Body Diagram What is the most important B F ma concept in dynamics? B Fx max mx Equations of Motion B Fy may my B Fz maz mz ME 231: Dynamics 2 Outline for Today # Question ofof the dayday # Where are we in the course? # Inverse vs. forward dynamics # Kinetics: cause of motion # Possible solutions to kinetics problems # Direct application of Newton’s 2nd Law # Plane motion types for rigid bodies # Equations, equations, equations… # Exam 2a breakdown (kinetics: F=ma) ME 231: Dynamics 3 Where are we in the course? Concept: What is dynamics? Chapters 1, 2, 6 Chapters 3, 5, 7, 8 Kinematics Kinetics Dynamics Relationship Relationship among position, among forces velocity, and (and acceleration moments)and acceleration ME 231: Dynamics 4 Where are we in the course? Calculation: How do we use dynamics? Newton’s 2nd Law Force. A push or pull Mass. Measure of the resistance exerted on a body, of a body to linear acceleration. characterized by: m Acceleration. Velocity # magnitude F = a rate of change with # direction respect to # point of time application ME 231: Dynamics 5 Inverse vs. Forward Dynamics inverse Velocities d d Forces F ma dt dt Positions Velocities Forces F ma O O Positions forward MEME 231:231: DDynamicsyna 6 Kinetics: Cause of Motion? Concept: What is kinetics? ME 202 Chapters 1, 2, 6 Chapters 3, 5, 7, 8 Statics Kinematics Kinetics Relationship Relationship Relationship among forces among position, among forces (and moments) velocity, and (and and equilibrium acceleration moments)and acceleration ME 231: Dynamics 7 Possible Solutions to Kinetics Problems # Direct application of Newton’s 2nd Law – force-mass-acceleration method – Chapters 3 and 7 # UseUse of impulseimpulse andand momentummomentum methodsmethods – ChaptersChapters 5 and 8 # UseUse of workwork andand eenergynergy principlesprinciples – Chapter 4 ME 231: Dynamics 8 Step-by-Step Solution Process 1. Kinematics – Identify type of motion – Solve for linear and angular accelerations 2. Diagram – Assign inertial coordinate system – Draw complete free-body diagram –Draw kinetic diagram to clarify equations 3. Equations of motion – Apply 2 linear and 1 angular equations – Maintain consistent sense – Solve for no more than 5 scalar unknowns (3 scalar equations of motion and 2 scalar relations from the relative-acceleration equation) ME 231: Dynamics 9 Outline for Today # Question ofof the dayday # Where are we in the coursecourse?? # Inverse vs. forwardforward dynamicdynamicss # Kinetics: ccauseause ooff mmotionotion # Possible solutions to kinetics pproblemsroblems # Direct application of Newton’s 2nd Law # Plane motion types for rigid bodies # Equations, equations, equations… # Exam 2a breakdown (kinetics: F=ma) ME 231: Dynamics 10 Direct Application of Newton’s 2nd Law F1 F2 F3 f1 f2 f3 miri B F Bf B miri B F mr or B F ma B Fx max B Fy may B Fz maz ME 231: Dynamics 11 Rectangular (x-y) Coordinates: Exercise A particle with mass of 10 slugs moving in two-dimensions has a position vector (r) as a function of time (t) with coordinates given by x(t) = t2 –4t + 20 , y(t) = 3 sin(2t) where r is measured in feet and t is in seconds. Determine the magnitude of the net force (F) accelerating the particle at time t = 3 s. ME 231: Dynamics 12 Polar (r-) Coordinates: Exercise 2 a r r er r 2r e Tube A rotates about the vertical O-axis with constant angular velocity and contains a small cylinder B of mass m whose radial position is controlled by a cord passing through the tube and wound around a drum of radius b. Determine the tension T in the cord and - component of force F if the drum has a constant angular rate of rotation of as shown. ME 231: Dynamics 13 Normal and Tangential (n-t) Coordinates: Exercise v2 a e v e n t A 1500-kg car enters an s-curve and slows down from 100 km/h at A to a speed of 50 km/h as it passes C. Determine the total horizontal force exerted by the road on the tires at positions A, B, and C. ME 231: Dynamics 14 Outline for Today # Question ofof the dayday # Where are we in the coursecourse?? # Inverse vs. forwardforward dynamicdynamicss # Kinetics: ccauseause ooff mmotionotion # Possible solutions to kinetics pproblemsroblems # Direct aapplicationpplication of Newton’s 2ndnd LawLaw # Plane motion types for rigid bodies # Equations, equations, equations… # Exam 2a breakdown (kinetics: F=ma) ME 231: Dynamics 15 Plane Motion Types A’A A’ for Rigid Bodies B’B B’ # Translation A’AA # Fixed-axis rotation B’BB’ # General plane motion A B ME 231: Dynamics 16 Rigid-Body Translation rectilinear curvilinear B M I 0 B M I 0 BF ma G G G G B M mad B M ma d 0 P A n A B M 0 B M ma d 0 A B t B ME 231: Dynamics 17 Rigid-Body Translation: Exercise A cleated conveyor belt transports solid cylinders up a 15º incline. The diameter of each cylinder is half its height. Determine the maximum acceleration for the belt without tipping the cylinders as it starts. ME 231: Dynamics 18 Fixed-Axis Rotation free-body kinetic diagram diagram # Mass center’s circular motion easily expressed in n-t coordinates # Plane-motion B F ma B M I BM I equations: G G O O ME 231: Dynamics 19 Fixed-Axis Rotation: Exercise Determine the angular acceleration and the force on the bearing at O for (a) the narrow ring of mass m and (b) the flat circular disk of mass m immediately after each is released from rest with OC horizontal. ME 231: Dynamics 20 General Plane Motoin: Combined Translation and Rotation B M P IG mad B M I BF ma G G B M P I P - maP ME 231: Dynamics 21 General Plane Motion: Exercise A truck has a mass of 2030 kg and carries a 1500-mm-diameter spool of cable with a mass of 0.75 kg per meter of length. There are 150 turns on the full spool. The empty spool has a mass of 140 kg with radius of gyration of 530 mm. Determine the tension T in the cable when the truck starts from rest with an acceleration of 0.2g. ME 231: Dynamics 22 Outline for Today # Question ofof the dayday # Where are we in the coursecourse?? # Inverse vs. forwardforward dynamicdynamicss # Kinetics: ccauseause ooff mmotionotion # Possible solutions to kinetics pproblemsroblems # Direct aapplicationpplication of Newton’s 2ndnd LawLaw # Plane motion types for ririgidgid bodiebodiess # Equations, equations, equations… # Exam 2a breakdown (kinetics: F=ma) ME 231: Dynamics 23 Equations, Equations, Equations… Particle Kinetics: F=ma Lecture Equations nd 18. Newton 2 Law B F ma B Fy may my 19. Eqs. of Motion B F ma mx B F ma mz 20. Rectilinear x x z z B F ma B Fr mar n n 21. Curvilinear B F ma B Ft mat t2 G mv G1 F dt G 2 Ot 27. Lin. ImImp.p. MoMom.m. F G 1 G 0 t2 HO r mv HO MO dt HO 1 Ot 2 28. Ang.Ang. Imp. MMom.om. 1 H 0 B MO HO O G mv HG B -i mi-i H P HG - mv 29. SSys.ys. ImImp.p. MomMom.. B M H B M H - ma HO B ri mi vi G G P G ME 231: Dynamics 24 Equations, Equations, Equations… Rigid Body Kinetics: F=ma Lecture Equations nd B M I mad 18. Newton 2 Law B MG HG P G 22. Gen. Eqs. Mot. I H B- F B M I - ma 23. Gen. Eqs. Mot. II G i i P P P B F ma B MG IG B MO IO 24. Fixed-Axis Rot. 2 2 IO I Gmr IO mkO B M I - ma B M I mad 25. Gen. Plane Mot. I P P P P G BF ma B M I 26. Gen. Plane Mot. II G G G mv HG IG2 H P IG2 mvmvdd HO IO2 B F G B M H B M H B M H 31. Body Imp. MomMom.. G G P P O O t t G 2 F dt G H 2 M dt H 1 O 2 G 1 O G G 2 t1 t1 ME 231: Dynamics 25 Exam 2a Breakdown (particle kinetics: F=ma) 40 35 30 25 20 40 40 34 34 30 15 26 10 number of points 5 0 0 18. Newton 19. Eqs. of 20. Rectilinear 21. 27. Lin. Imp.Imp. 28.28. AnAng.g. ImImp.p. 29.29. Sys.Sys. Imp.Imp. 2nd Law Motion Curvilinear Mom.Mom. Mom.Mom. Mom.Mom. lecture ME 231: Dynamics 26 Exam 2 Breakdown (rigid body kinetics: F=ma) 70 60 50 40 70 30 40 40 40 20 34 number of points 26 10 0 0 18. Newton 22. Gen. Eqs. 23. Gen. Eqs. 24. Fixed-Axis 25. Gen. 26. Gen. 31. BodyBody ImImp.p. 2nd Law of Motion I of Motion II Rotation Plane Mot. I Plane Mot. II Mom.Mom. lecture ME 231: Dynamics 27 For Next Time… # Review Chapters 3 & 7 # Review Lectures slides – http://rrg.utk.edu/resources/ME231/lectures.html # Review Examples from class – http://rrg.utk.edu/resources/ME231/examples.html # Exam #2a on Friday (11/9) ME 231: Dynamics 28.