1. Physics 207: Lecture 15, Pg 1 Lecture 15 Goals: Chapter 11 (Work) Chapter 11 (Work)  Employ conservative and non-conservative forces  Relate force to potential energy  Use the concept of power (i.e., energy per time) Assignment: l HW7 due Tuesday, Nov. 1 st l For Monday: Read Chapter 12, (skip angular momentum and explicit integration for center of mass, rotational inertia, etc.) Exam 2 7:15 PM Thursday, Nov. 3 th

2. Physics 207: Lecture 15, Pg 2 Definition of Work, The basics Ingredients: Force ( F ), displacement (  r )  r rr r displacement F Work, W, of a constant force F acts through a displacement  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?

3. Physics 207: Lecture 15, Pg 3 Net Work: 1-D Example (constant force) = 10 x 5 N m = 50 J Net Work is F  x = 10 x 5 N m = 50 J l 1 N m ≡ 1 Joule (energy) l Work reflects energy transfer xx A force F = 10 N pushes a box across a frictionless floor for a distance  x = 5 m. F  = 0° Start Finish

4. Physics 207: Lecture 15, Pg 4 Net Work: 1-D 2 nd Example (constant force) = -10 x 5 N m = -50 J Net Work is F  x = -10 x 5 N m = -50 J l Work again reflects energy transfer xx F A force F = 10 N is opposite the motion of a box across a frictionless floor for a distance  x = 5 m.  = 180° Start Finish

5. Physics 207: Lecture 15, Pg 5 Work: “2-D” Example (constant force) = F cos( = 50 x 0.71 Nm = 35 J ( Net) Work is F x  x = F cos( -45°)  x = 50 x 0.71 Nm = 35 J l Work reflects energy transfer xx F An angled force, F = 10 N, pushes a box across a frictionless floor for a distance  x = 5 m and  y = 0 m  = -45° Start Finish FxFx

6. Physics 207: Lecture 15, Pg 6 Exercise Work in the presence of friction and non-contact forces A. 2 B. 3 C. 4 D. 5 A box is pulled up a rough (  > 0) incline by a rope-pulley- weight arrangement as shown below.  How many forces (including non-contact ones) are doing work on the box ?  Of these which are positive and which are negative?  State the system (here, just the box)  Use a Free Body Diagram  Compare force and path v

7. Physics 207: Lecture 15, Pg 7 Work and Varying Forces (1D) l Consider a varying force F(x) FxFx x xx Area = F x  x F is increasing Here W = F ·  r becomes dW = F dx

8. Physics 207: Lecture 15, Pg 8 How much will the spring compress (i.e.  x = x f - x i ) to bring the box 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x vovo m toto F spring compressed spring at an equilibrium position V=0 t m Example: Work Kinetic-Energy Theorem with variable force

9. Physics 207: Lecture 15, Pg 9 Compare work with changes in potential energy l Consider the ball moving up to height h (from time 1 to time 2) l How does this relate to the potential energy? Work done by the Earth’s gravity on the ball) W = F   x = (-mg) -h = mgh  U = U f – U i = mg 0 - mg h = -mg h  U = -W h mg

10. Physics 207: Lecture 15, Pg 10 Conservative Forces & Potential Energy l If a conservative force F we can define a potential energy function U: The work done by a conservative force is equal and opposite to the change in the potential energy function. l or W = F ·dr = -  U   U = U f - U i = - W = - F dr  riri rfrf riri rfrf UfUf UiUi

11. Physics 207: Lecture 15, Pg 11 A Non-Conservative Force, Friction Looking down on an air-hockey table with no air flowing (  > 0). l Now compare two paths in which the puck starts out with the same speed (K i path 1 = K i path 2 ). Path 2 Path 1

12. Physics 207: Lecture 15, Pg 12 A Non-Conservative Force Path 2 Path 1 Since path 2 distance >path 1 distance the puck will be traveling slower at the end of path 2. Work done by a non-conservative force irreversibly removes energy out of the “system”. Here W NC = E final - E initial < 0  and reflects E thermal

13. Physics 207: Lecture 15, Pg 13 A. U  K B. U  E Th C. K  U D. K  E Th E. There is no transformation because energy is conserved. A child slides down a playground slide at constant speed. The energy transformation is

14. Physics 207: Lecture 15, Pg 14 Conservative Forces and Potential Energy l So we can also describe work and changes in potential energy (for conservative forces)  U = - W l Recalling (if 1D) W = F x  x l Combining these two,  U = - F x  x l Letting small quantities go to infinitesimals, dU = - F x dx l Or, F x = -dU / dx

15. Physics 207: Lecture 15, Pg 15 Equilibrium l Example  Spring: F x = 0 => dU / dx = 0 for x=x eq The spring is in equilibrium position l In general: dU / dx = 0  for ANY function establishes equilibrium stable equilibrium unstable equilibrium U U

16. Physics 207: Lecture 15, Pg 16 Work & Power: l Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass. l Assuming identical friction, both engines do the same amount of work to get up the hill. l Are the cars essentially the same ? l NO. The Corvette can get up the hill quicker l It has a more powerful engine.

17. Physics 207: Lecture 15, Pg 17 Work & Power: l Power is the rate, J/s, at which work is done. l Average Power is, l Instantaneous Power is, If force constant, W= F  x = F (v 0  t + ½ a  t 2 ) and P = W /  t = F (v 0 + a  t)

18. Physics 207: Lecture 15, Pg 18 Work & Power: Average Power: l A person, mass 80.0 kg, runs up 2 floors (8.0 m). If they climb it in 5.0 sec, what is the average power used? P avg = F h /  t = mgh /  t = 80.0 x 9.80 x 8.0 / 5.0 W l P = 1250 W Example: 1 W = 1 J / 1s

19. Physics 207: Lecture 15, Pg 19 Exercise Work & Power A. Top B. Middle C. Bottom l Starting from rest, a car drives up a hill at constant acceleration and then suddenly stops at the top. l The instantaneous power delivered by the engine during this drive looks like which of the following, Z3 time Power time

20. Physics 207: Lecture 15, Pg 20 Chap. 12: Rotational Dynamics l Up until now rotation has been only in terms of circular motion with a c = v 2 / R and | a T | = d| v | / dt l Rotation is common in the world around us. l Many ideas developed for translational motion are transferable.

21. Physics 207: Lecture 15, Pg 21 Conservation of angular momentum has consequences How does one describe rotation (magnitude and direction)? Katrina

22. Physics 207: Lecture 15, Pg 22 Rotational Variables l Rotation about a fixed axis:  Consider a disk rotating about an axis through its center: l Recall : (Analogous to the linear case )  

23. Physics 207: Lecture 15, Pg 23 System of Particles (Distributed Mass): l Until now, we have considered the behavior of very simple systems (one or two masses). l But real objects have distributed mass ! l For example, consider a simple rotating disk and 2 equal mass m plugs at distances r and 2r. l Compare the velocities and kinetic energies at these two points. 1 2 

24. Physics 207: Lecture 15, Pg 24 For these two particles l Twice the radius, four times the kinetic energy 1 K= ½ m v 2 2 K= ½ m (2v) 2 = 4 ½ m v 2 

25. Physics 207: Lecture 15, Pg 25 For these two particles 1 K= ½ m v 2 2 K= ½ m (2v) 2 = 4 ½ m v 2 

26. Physics 207: Lecture 15, Pg 26 For these two particles 

27. Physics 207: Lecture 15, Pg 27 Calculating Moment of Inertia where r is the distance from the mass to the axis of rotation. Example: Calculate the moment of inertia of four point masses (m) on the corners of a square whose sides have length L, about a perpendicular axis through the center of the square: mm mm L

28. Physics 207: Lecture 15, Pg 28 Calculating Moment of Inertia... For a single object, I depends on the rotation axis! Example: I 1 = 4 m R 2 = 4 m (2 1/2 L / 2) 2 L I = 2mL 2 I 2 = mL 2 mm mm I 1 = 2mL 2

29. Physics 207: Lecture 15, Pg 29 Lecture 16 Assignment: l HW7 due Tuesday Nov. 1 st