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Midterm this Friday November 8 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 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 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 3 Determining Motion Force Unbalanced Forces acceleration (otherwise objects velocity is constant) Energy Total Energy Motion, Location F a F a m F12 F21 Emechanical K U Emechanical Wnonconservative rf Determine Net Force acting on object Work Wnet Fnet dl K Conservative forces r0 U Wnet K Emechanical U K 0 Motion from Energy conservation Emechanical, final K f U f Use kinematic equations to determine resulting motion v f at v0 ;... Emechanical, final K i U i Emechanical K f K i U i Emechanical U f vf 2 Ki U i Emechanical U f m Mechanics Lecture 8, Slide 4 Unit 4: Main Points Mechanics Lecture 4, Slide 5 Unit 4: Main Points Mechanics Lecture 4, Slide 6 Unit 4: Main Points Mechanics Lecture 4, Slide 7 Unit 5: Main Points Mechanics Lecture 5, Slide 8 Unit 5: Main Points Mechanics Lecture 5, Slide 9 Unit 5: Main Points Mechanics Lecture 5, Slide 10 Inventory of Forces Weight Normal Force Tension Gravitational Springs …Friction Mechanics Lecture 5, Slide 11 Mechanics Lecture 5, Slide 12 1) FBD m2 N f m2 T g T m2g m1 m1 m1g Mechanics Lecture 6, Slide 13 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 6, Slide 14 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 6, Slide 15 Force Summary Mechanics Lecture 5, Slide 16 Unit 6: Main Points Mechanics Lecture 6, Slide 17 Unit 6: Main Points Mechanics Lecture 6, Slide 18 Friction Mechanics Lecture 6, Slide 19 Friction Mechanics Lecture 6, Slide 20 Block 1 2 2x at a 2 2 t F m g sin f k a net m m 2x f k m g sin m a m g sin 2 t x Mechanics Lecture 5, Slide 21 Pushing Blocks F23net a Fh1 F h1 (m1 m2 m3 m4 ) 4m1 F23 netx (m3 m4 )a 2m1a 2m1 Fh1 Fh1 4m1 2 F23 net y N (m3 m4 ) g 2m1 g 2 F23 net F 2 23 net x F 2 23 net y F 2 h1 2m1 g 2 Mechanics Lecture 5, Slide 22 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 23 Unit 7: Main Points Mechanics Lecture 7, Slide 24 Main Points Mechanics Lecture 7, Slide 25 Summary Unit 8: Potential & Mechanical Energy Mechanics Lecture 8, Slide 26 The Dot Product Mechanics Lecture 7, Slide 27 Vectors!!! Mechanics Lecture 8, Slide 28 Main Points Mechanics Lecture 7, Slide 29 Mechanics Lecture 8, Slide 30 Unit 9:Main Points Mechanics Lecture 8, Slide 31 Main Points Mechanics Lecture 8, Slide 32 Energy Conservation Problems in general For systems with only conservative forces acting Emechanical 0 Emechanical is a constant Emechanical Ki Ui K f U f K (t ) U (t ) Mechanics Lecture 8, Slide 33 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 34 Example Problems Emechanical W friction W friction m k m gx Emechanical, final Emechanical,initial W friction m gh m ghi m k m gx m ghi m k x h hi m k x Mechanics Lecture 8, Slide 35 Example Problems Mechanics Lecture 8, Slide 36 Example Problems Mechanics Lecture 8, Slide 37 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 38 Example: 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? 2.5 kg k=250 N/m h=1.5 m mk = 0.4 d = 0.50 m Mechanics Review 2 , Slide 39 Example: Pendulum vv 22gh gh hh Conserve Energy from initial to final position. 1 2 mgh mv 2 v 2gh Mechanics Review 2 , Slide 40 Example Problems Mechanics Lecture 8, Slide 41 Example Problems Mechanics Lecture 8, Slide 42 Gravitational Potential Energy r rE rE r r rM rM Mechanics Lecture 8, Slide 43 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 44 Trip to the moon Ki U i K f U f 1 2 GM E m GM m m 1 2 GM E m GM m m m vi m vf 2 RE d M E 2 d E M RM 1 GM E GM m GM E GM m v f 2 vi2 RE d M E d E M RM 2 Mechanics Lecture 8, Slide 45