1. Physics 207: Lecture 10, Pg 1 Physics 207, Lecture 10, Oct. 9 Exams will be returned in your next discussion section Regrades: Write down, on a separate sheet, what you want regraded and why. Mean: 64.6 Median: 67 Std. Dev.: 19.0 Range: High 100 Low 5 Solution posted on http://my.wisc.edu Nominal curve (conservative): 86-100 A 70-85 B or A/B 40-69 C or B/C 35-40 marginal 25-35 D Below 25 F l MidTerm I

2. Physics 207: Lecture 10, Pg 2 Physics 207, Lecture 10, Oct. 9 Agenda: Chapter 7, Work and Energy Transfer Assignment: For Wednesday read Chapter 8 l WebAssign Problem Set 4 due Tuesday next week (start now) Definition of Work (a scalar quantity) Variable force devices (e.g., Hooke’s Law spring) Work/Energy Theorem  W =  K Kinetic Energy  K = 1/2 mv 2 Power (on Wednesday) = =  P = dW / dt = F · v

3. Physics 207: Lecture 10, Pg 3 Work & Energy l One of the most important concepts in physics.  Alternative approach to mechanics. l Many applications beyond mechanics.  Thermodynamics (movement of heat or particles).  Quantum mechanics... l Very useful tools.  You will learn a complementary approach (often much easier) way to solve problems. But there is no free lunch….easier but there are fewer details that are explicitly known.

4. Physics 207: Lecture 10, Pg 4 Energy Conservation l Energy cannot be destroyed or created.  Just changed from one form to another. energy is conserved l We say energy is conserved !  True for any isolated system. work energy l Doing “work” on an otherwise isolated system will change it’s “energy”... See text: 7-1

5. Physics 207: Lecture 10, Pg 5 Definition of Work, The basics Ingredients: Fr Ingredients: Force ( F ), displacement (  r ) “Scalar or Dot Product” See text: 7-1  r rr r displacement F F Work, W, of a constant force F r acting through a displacement  r is: F r (Work is a scalar) W = F ·  r (Work is a scalar) Work tells you something about what happened on the path! Did something do work on you? Did you do work on something? Simplest case (no frictional forces and no non-contact forces) Did your speed change? ( what happened to | v | !!!)

6. Physics 207: Lecture 10, Pg 6 Remember that a path evolves with time and acceleration implies a force acting on an object l A tangetial force is the important one for work!  How long (time dependence) gives the kinematics  The distance over which this force Tang is applied: Work a v path and time t a a a = 0 Two possible options: Change in the magnitude of v Change in the direction of v a = 0 a a a a = + a radial a tang a = + F radial F tang F = +

7. Physics 207: Lecture 10, Pg 7 Definition of Work... l Only the component of F along the path (i.e. “displacement”) does work. The vector dot product does that automatically.  Example: Train on a track. r rr r F  F cos  If we know the angle the force makes with the track, the dot product gives us F cos  and  r

8. Physics 207: Lecture 10, Pg 8 l Useful for performing projections. A  î = A x î A A x A y  A  B = (A x )(B x ) + (A y )(B y ) + (A z )(B z ) l Calculation is simple in terms of components. Calculation also in terms of magnitudes and relative angles. Review: Scalar Product (or Dot Product) 7.3 A  B ≡ | A | | B | cos  You choose the way that works best for you!

9. Physics 207: Lecture 10, Pg 9 Work: 1-D Example (constant force) x = 10 x 5 N m = 50 J Work is A  B ≡ | A | | B | cos  = F  x = 10 x 5 N m = 50 J l 1 Nm is defined to be 1 Joule and this is a unit of energy l Work reflects energy transfer xxxx See text: 7-1 See example 7-1: Pushing a trunk. F x A force F = 10 N pushes a box across a frictionless floor for a distance  x = 5 m. F  = 0° Start Finish

10. Physics 207: Lecture 10, Pg 10 Units: N-m (Joule) Dyne-cm (erg) = 10 -7 J BTU= 1054 J calorie= 4.184 J foot-lb= 1.356 J eV= 1.6x10 -19 J cgsOthermks Force x Distance = Work Newton x [M][L] / [T] 2 Meter = Joule [L] [M][L] 2 / [T] 2 See text: 7-1

11. Physics 207: Lecture 10, Pg 11 Work: 1-D 2 nd Example (constant force) x (-1) = -10 x 5 N m = -50 J Work is A  B ≡ | A | | B | cos  = F  x (-1) = -10 x 5 N m = -50 J l Work reflects energy transfer xxxx F See text: 7-1 See example 7-1: Pushing a trunk. F x A force F = 10 N pushes a box across a frictionless floor for a distance  x = 5 m.  = 180° Start Finish

12. Physics 207: Lecture 10, Pg 12 Work: 1-D 3 rd Example (constant force) x 0.71= 50 x 0.71 Nm = 35 J Work is A  B ≡ | A | | B | cos  = F  x 0.71= 50 x 0.71 Nm = 35 J l Work reflects energy transfer xxxx F See text: 7-1 See example 7-1: Pushing a trunk. F x A force F = 10 N pushes a box across a frictionless floor for a distance  x = 5 m.  = -45° Start Finish

13. Physics 207: Lecture 10, Pg 13 Work and Varying Forces l Consider a varying force F(x) Text : 7.3 FxFx x xx Area = F x  x F is increasing F r Here W = F ·  r F becomes dW = F dx F  = 0° Start Finish Work is a scalar, the catch is that there is no time/position info on hand F xx

14. Physics 207: Lecture 10, Pg 14 Lecture 10, Exercise 1 Work in the presence of friction and non-contact forces A box is pulled up a rough (  > 0) incline by a rope- pulley-weight arrangement as shown below.  How many forces are doing work on the box ?  Of these which are positive and which are negative?  Use a Force Body Diagram  Compare force and path (A) (A) 2 (B) (B) 3 (C) (C) 4 v

15. Physics 207: Lecture 10, Pg 15 A variable force device: A Hooke’s Law Spring l Springs are everywhere, (probe microscopes, DNA, an effective interaction between atoms) l In this spring, the magnitude of the force increases as the spring is further compressed (a displacement). l Hooke’s Law, F S = - k  x  x is the amount the spring is stretched or compressed from it resting position. Text : 7.3 F (work done on spring) xx Active Figure Rest or equilibrium position

16. Physics 207: Lecture 10, Pg 16 Lecture 10, Exercise 2 Hooke’s Law l Remember Hooke’s Law, F x = -k  x What are the units for the constant k ? (A)(B)(C)(D) F is in kg m/s 2 and dividing by m gives kg/s 2 or N/m

17. Physics 207: Lecture 10, Pg 17 Lecture 10, Exercise 3 Hooke’s Law 0.50 kg 9 cm 8 cm What is the spring constant “k” ? (A) 50 N/m(B) 100 N/m(C) 400 N/m (D) 500 N/m

18. Physics 207: Lecture 10, Pg 18 Lecture 10, Exercise 3 Hooke’s Law 0.50 kg 9 cm 8 cm What is the spring constant “k” ?  F = 0 = F s – mg = k  x - mg Use k = mg/  x = 5 N / 0.01 m (A) 50 N/m(B) 100 N/m(C) 400 N/m (D) 500 N/m mg F spring

19. Physics 207: Lecture 10, Pg 19 F-x relation for a foot arch: Force (N) Displacement (mm)

20. Physics 207: Lecture 10, Pg 20 F-x relation for a single DNA molecule

21. Physics 207: Lecture 10, Pg 21 Measurement technique: optical tweezers

22. Physics 207: Lecture 10, Pg 22 Work & Kinetic Energy: Energy transfer involving changes in speed F x A force, F = 10 N, pushes a box across a frictionless floor for a distance  x = 5m. l The speed of the box is v 1 before the push, and v 2 after the push. l Consider only this force and the box l Relate the work to the kinetic energy of the box xxxx F’F’F’F’ v1v1 v2v2 i m F

23. Physics 207: Lecture 10, Pg 23 Work Kinetic-Energy Theorem: NetWork {Net Work done on object} = changekinetic energy {change in kinetic energy of object} See text: 7-4 (final – initial)

24. Physics 207: Lecture 10, Pg 24 xxxx vovo m toto F Example: Work Kinetic-Energy Theorem x How much will the spring compress (i.e.  x ) to bring the object to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (v o ) on frictionless surface as shown below ? spring compressed spring at an equilibrium position V=0 t m Notice that the spring force is opposite to the displacemant. For the mass m, work is negative For the spring, work is positive

25. Physics 207: Lecture 10, Pg 25 Example: Work Kinetic-Energy Theorem x = x f - x i How much will the spring compress (i.e.  x = x f - x i ) to bring the object to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (v o ) on frictionless surface as shown below ? xxxx vovo m toto F spring compressed spring at an equilibrium position V=0 t m

26. Physics 207: Lecture 10, Pg 26 Lecture 10, Exercise 4 Kinetic Energy l To practice your pitching you use two baseballs. The first time you throw a slow curve and clock the speed at 50 mph (~25 m/s). The second time you go with high heat and the radar gun clocks the pitch at 100 mph. What is the ratio of the kinetic energy of the fast ball versus the curve ball ? (A) 1/4 (B) (C) (A) 1/4 (B) 1/2 (C) 1 (D) 2 (E) 4

27. Physics 207: Lecture 10, Pg 27 Lecture 10, Exercise 5 Work & Friction Two blocks having mass m 1 and m 2 where m 1 > m 2. They are sliding on a frictionless floor and have the same kinetic energy when they encounter a long rough stretch (i.e.  > 0) which slows them down to a stop. l Which one will go farther before stopping? l Hint: How much work does friction do on each block ? (A) (B) (C) (A) m 1 (B) m 2 (C) They will go the same distance m1m1 m2m2 v2v2 v1v1

28. Physics 207: Lecture 10, Pg 28 Physics 207, Lecture 10, Recap Agenda: Chapter 7, Work and Energy Transfer Assignment: For Wednesday read Chapter 8 l WebAssign Problem Set 4 due Tuesday next week (start now) Definition of Work (a scalar quantity) Variable force devices (e.g., Hooke’s Law spring) Work/Energy Theorem  W =  K Kinetic Energy  K = 1/2 mv 2 Power (on Wednesday) = =  P = dW / dt = F · v