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Midterm this Friday March 13 Classical Mechanics Midterm 2 Review Force and Energy Newton’s Laws of Motion Forces and Free Body Diagrams Friction Work and Kinetic Energy Conservative Forces and Potential Energy Work and Potential Energy Knowledge of Units 1-3 will be useful “Playing with Blocks” Mechanics Lecture 9, Slide 1 Midterm Exam Multiple choice…but show your work and justification. Calculations Forces and Free-Body Diagrams Weight and Pulleys Springs Friction Gravitational Normal Work Calculations Conservation of Energy Conceptual questions…like checkpoint problems. Bring calculators and up to five sheets of notes Mechanics Lecture 8, Slide 2 Midterm Exam 2 Sample exam Folder under Files menu on PHYS1500 Canvas Website Phys 1500 Exams https://utah.instructure.com/courses/320947/files - Spring 2013: http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2.pdf - Solutions: http://www.physics.utah.edu/~springer/phys1500/exams/MidtermExam2Soln.pdf - Long Sample: https://utah.instructure.com/courses/320947/files/45779670/download?wrap=1 - Long Solutions:https://utah.instructure.com/courses/320947/files/45802143/download?wrap=1 Phys 2210 Exams - Practice : http://www.physics.utah.edu/~woolf/2210_Jui/rev2.pdf - Spring 2015: http://www.physics.utah.edu/~woolf/2210_Jui/ex2.pdf Mechanics Lecture 8, Slide 3 Force and Energy Summary Fnet ma Emechanical Wnonconservative Mechanics Lecture 8, Slide 4 Determining Motion Force Unbalanced Forces acceleration (otherwise objects velocity is constant) Fnet ma Energy Total Energy Motion, Location Fnet a m Determine Net Force acting on object Emechanical K U Emechanical Wnonconservative rf Work Fnet Fi i F12 F21 Use kinematic equations to determine resulting motion a a0 v(t ) a0t v0 x(t ) 1 2 a0t v0t x0 2 Wnet Fnet dl K r0 U Wnet K Emechanical U K 0 Motion from Energy conservation Emechanical, final K f U f Emechanical, final K i U i Emechanical K f K i U i Emechanical U f vf 2 K i U i Emechanical U f m Mechanics Lecture 8, Slide 5 Unit 4: Newton’s Laws Mechanics Lecture 4, Slide 6 Unit 5: Forces and Free-Body Diagrams FFloor ,box abox Box FMan,box FEarth ,box Mechanics Lecture 5, Slide 7 Inventory of Forces Weight Normal Force Tension Gravitational Springs …Friction Mechanics Lecture 5, Slide 8 Mechanics Lecture 5, Slide 9 Force Summary Fnet ma Fnet Fi i Mechanics Lecture 5, Slide 10 Unit 6: Friction Mechanics Lecture 6, Slide 11 Friction Mechanics Lecture 6, Slide 12 Unit 7: Work and Kinetic Energy Mechanics Lecture 7, Slide 13 Work-Kinetic Energy Theorem The work done by force F as it acts on an object that moves between positions r1 and r2 is equal to the change in the object’s kinetic energy: But again…!!! r2 W K W F dl r1 1 2 K mv 2 Mechanics Lecture 7, Slide 14 The Dot Product Mechanics Lecture 7, Slide 15 Vectors!!! Mechanics Lecture 8, Slide 16 Unit 8: Conservative Forces & Potential Energy Mechanics Lecture 7, Slide 17 Unit 8: Conservative Forces & Potential Energy Mechanics Lecture 8, Slide 18 Unit 9:Work and Potential Energy Mechanics Lecture 8, Slide 19 Energy Conservation Problems in general For systems with only conservative forces acting Emechanical 0 Emechanical is a constant Emechanical K i U i K f U f K (t ) U (t ) Mechanics Lecture 8, Slide 20 Gravitational Potential Problems r rE rE r rM conservation of mechanical energy can be used to “easily” solve problems. Emechanical K U r rM Add potential energy from each source. GM E m U Earth (rE ) rE U Moon (rM ) 1 mv(h) 2 U (h) gravity 2 Define coordinates: where is U=0? U (r ) GM E m 0 as r r GM M m rM GM E m GM M m U total (r ) r rE r rM Mechanics Lecture 8, Slide 21 Example Problem : Block and spring A 2.5 kg box is held released from rest 1.5 m above the ground and slides down a frictionless ramp. It slides across a floor that is frictionless, except for a small section 0.5 m wide that has a coefficient of kinetic friction of 0.2. At the left end, is a spring with spring constant 250 N/m. The box compresses the spring, and is accelerated back to the right. What is the speed of the box at the bottom of the ramp? What is the maximum distance the spring is compressed by the box? Draw a free-body diagram for the box while at the top of the incline ? When the spring is maximally compressed? When the box is sliding on the rough spot to the right? What is the total work done by friction? Each way? What height does the box reach up the ramp after hitting the spring once? Where will the box come to rest? 2.5 kg k=250 N/m h=1.5 m mk = 0.4 d = 0.50 m Mechanics Review 2 , Slide 22 Example Problem: Free-Body Diagram 1) FBD m2 N f m2 T g T m2g m1 m1 m1g Mechanics Lecture 8, Slide 23 Example Problem: Free-Body Diagram 1) FBD 2) SF=ma m2 N T m2 f g T m2g N = m2g T – m m2g = m2a m1g – T = m1a m1 m1 m1g add m1g – m m2g = m1a + m2a a= m1g – m m2g m 1 + m2 Mechanics Lecture 8, Slide 24 Example Problem: Free-Body Diagram 1) FBD 2) SF=ma m2 N f m2 T g T m1 m1 m2g m1g a= m1g – m m2g m1 + m2 m1g – T = m1a T = m1g – m1a T is smaller when a is bigger Mechanics Lecture 8, Slide 25 Example Problems Mechanics Lecture 8, Slide 26 Example Problem Wtension T dl Tx Wnet Wtension W friction K W friction Wtension K 1 K m v 2f v02 2 1 W friction Wtension m v 2f v02 2 Mechanics Lecture 8, Slide 27 Block 1 2 2x at a 2 2 t F mg sin f k a net m m 2x f k mg sin ma m g sin 2 t x Mechanics Lecture 5, Slide 28 Pushing Blocks F23net a Fh1 F h1 (m1 m2 m3 m4 ) 4m1 F23netx (m3 m4 )a 2m1a 2m1 Fh1 Fh1 4m1 2 F23net y N (m3 m4 ) g 2m1 g 2 F23net F 2 23net x F 2 23net y F 2 h1 2m1 g 2 Mechanics Lecture 5, Slide 29 Example Problems Emechanical W friction W friction m k mgx Emechanical, final Emechanical,initial W friction mgh mghi m k mgx mg hi m k x h hi m k x Mechanics Lecture 8, Slide 30 Example: Pendulum v v 22ghgh hh Conserve Energy from initial to final position. 1 2 mgh mv 2 v 2 gh Mechanics Review 2 , Slide 31 Example Problems Mechanics Lecture 8, Slide 32 Example: Pulley and Two Masses A block of mass m1 = 1 kg sits atop an inclined plane of angle θ = 20o with coefficient of kinetic friction 0.2 and is connected to mass m2 = 3 kg through a string that goes over a massless frictionless pulley. The system starts at rest and mass m2 falls through a height H = 2 m. Use energy methods to find the velocity of mass m2 just before it hits the ground? θ m2 HH =2 m =2 2m kg Mechanics Lecture 19, Slide 33 Example Problems Mechanics Lecture 8, Slide 34 Example Problems Mechanics Lecture 8, Slide 35 Force and Energy Summary Fnet ma Emechanical Wnonconservative Mechanics Lecture 8, Slide 36