Physics 07 Lecture 16 Lecture 16 Chapter 11 (Work) v Eploy conservative and non-conservative forces v Relate force to potential energy v Use the concept of power (i.e., energy per tie) Chapter 1 v Define rotational inertia v Define rotational kinetic energy Assignent: l HW7 due Tuesday, Nov. 1 st l For Monday: Read Chapter 1, (skip angular oentu and explicit integration for center of ass, rotational inertia, etc.) Exa 7:15 PM Thursday, Nov. 3 th Physics 07: Lecture 16, Pg 1 Definition of Work, The basics Ingredients: Force ( F ), displaceent ( r ) Work, W, of a constant force F acts through a displaceent r : W F r (Work is a scalar) F θ r displaceent Work tells you soething about what happened on the path! Did soething do work on you? Did you do work on soething? Physics 07: Lecture 16, Pg Page 1
Physics 07 Lecture 16 Units: Force x Distance Work Newton x [M][L] / [T] Meter Joule [L] [M][L] / [T] ks N- or Joule cgs Dyne-c or erg 10-7 J Other BTU 1054 J calorie 4.184 J foot-lb 1.356 J ev 1.6x10-19 J Physics 07: Lecture 16, Pg 3 Net Work: 1-D Exaple (constant force) l A force F 10 N pushes a box across a frictionless floor for a distance x 5. F Start Finish θ 0 x l Net Work is F x 10 x 5 N 50 J l 1 N 1 Joule (energy) l Work reflects positive energy transfer Physics 07: Lecture 16, Pg 4 Page
Physics 07 Lecture 16 Net Work: 1-D nd Exaple (constant force) l A force F 10 N is opposite the otion of a box across a frictionless floor for a distance x 5. Start F Finish θ 180 x l Net Work is F x -10 x 5 N -50 J l Work again reflects negative energy transfer Physics 07: Lecture 16, Pg 5 Work: -D Exaple (constant force) l An angled force, F 10 N, pushes a box across a frictionless floor for a distance x 5 and y 0 F F x Start Finish θ -45 x l (Net) Work is F x x F cos(-45 ) x 50 x 0.71 N 35 J Physics 07: Lecture 16, Pg 6 Page 3
Physics 07 Lecture 16 Exercise Work in the presence of friction and non-contact forces l A box is pulled up a rough (µ > 0) incline by a rope-pulleyweight arrangeent as shown below. v How any forces (including non-contact ones) are doing work on the box? v Of these which are positive and which are negative? v State the syste (here, just the box) v Use a Free Body Diagra v Copare force and path v A. B. 3 C. 4 D. 5 Physics 07: Lecture 16, Pg 7 Work and Varying Forces (1D) l Consider a varying force F(x) F x Area F x x F is increasing Here W F r becoes dw F dx x x W x f x i F ( x) dx Physics 07: Lecture 16, Pg 8 Page 4
Physics 07 Lecture 16 Exaple: Work Kinetic-Energy Theore with variable force How uch will the spring copress (i.e. x x f - x i ) to bring the box to a stop (i.e., v 0 ) if the object is oving initially at a constant velocity (v o ) on frictionless surface as shown below? xf t o v o spring at an equilibriu position x W box W box F ( x) dx xi xf kx dx xi t V0 F spring copressed 1 k x 1 v 0 Physics 07: Lecture 16, Pg 9 Copare work with changes in potential energy l Consider the ball oving up to height h (fro tie 1 to tie ) l How does this relate to the potential energy? Work done by the Earth s gravity on the ball) W F x (-g) -h gh U U f U i g 0 - g h -g h U -W g g h Physics 07: Lecture 16, Pg 10 Page 5
Physics 07 Lecture 16 Conservative Forces & Potential Energy l If a conservative force F we can define a potential energy function U: W F dr - U The work done by a conservative force is equal and opposite to the change in the potential energy function. r f U f (independent of path!) r i U i Physics 07: Lecture 16, Pg 11 A Non-Conservative Force, Friction l Looking down on an air-hockey table with no air flowing (µ > 0). l Now copare two paths in which the puck starts out with the sae speed (K i path 1 K i path ). Path Path 1 Physics 07: Lecture 16, Pg 1 Page 6
Physics 07 Lecture 16 A Non-Conservative Force Path Path 1 Since path distance >path 1 distance the puck will be traveling slower at the end of path. Work done by a non-conservative force irreversibly reoves energy out of the syste. Here W NC E final - E initial < 0 and reflects E theral Physics 07: Lecture 16, Pg 13 A child slides down a playground slide at constant speed. The energy transforation is A. U K B. U E Th C. K U D. K E Th E. There is no transforation because energy is conserved. Physics 07: Lecture 16, Pg 14 Page 7
Physics 07 Lecture 16 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 Cobining these two, U - F x x l Letting sall quantities go to infinitesials, l Or, du - F x dx F x -du / dx Physics 07: Lecture 16, Pg 15 l Exaple Equilibriu v Spring: F x 0 > du / dx 0 for xx eq The spring is in equilibriu position l In general: du / dx 0 for ANY function establishes equilibriu U U stable equilibriu unstable equilibriu Physics 07: Lecture 16, Pg 16 Page 8
Physics 07 Lecture 16 Work & Power: l Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the sae ass. l Assuing identical friction, both engines do the sae aount of work to get up the hill. l Are the cars essentially the sae? l NO. The Corvette can get up the hill quicker l It has a ore powerful engine. Physics 07: Lecture 16, Pg 17 Work & Power: l Power is the rate, J/s, at which work is done. l Average Power is, l Instantaneous Power is, l If force constant, W F x F (v 0 t + ½ a t ) and W P t dw P dt P W / t F (v 0 + a t) Physics 07: Lecture 16, Pg 18 Page 9
Physics 07 Lecture 16 Work & Power: Exaple P Average Power: W t l A person, ass 80.0 kg, runs up floors (8.0 ) at constant speed. If they clib it in 5.0 sec, what is the average power used? l P avg F h / t gh / t 80.0 x 9.80 x 8.0 / 5.0 W l P 150 W Exaple: 1 W 1 J / 1s Physics 07: Lecture 16, Pg 19 Exercise Work & Power l Starting fro 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, A. Top Power tie B. Middle C. Botto Power Power tie tie Z3 Physics 07: Lecture 16, Pg 0 Page 10
Physics 07 Lecture 16 Exercise Work & Power l P dw / dt and W F d (µ g cos θ g sin θ) d and d ½ a t (constant accelation) So W F ½ a t P F a t F v l (A) Power tie l (B) l (C) Power Power tie tie Z3 Physics 07: Lecture 16, Pg 1 Chap. 1: Rotational Dynaics l Up until now rotation has been only in ters of circular otion with a c v / R and a T d v / dt l Rotation is coon in the world around us. l Many ideas developed for translational otion are transferable. Physics 07: Lecture 16, Pg Page 11
Physics 07 Lecture 16 Rotational otion has consequences Katrina How does one describe rotation (agnitude and direction)? Physics 07: Lecture 16, Pg 3 Rotational Variables l l Rotation about a fixed axis: v Consider a disk rotating about an axis through its center: θ Recall : dθ π (rad/s) vtangential /R dt T (Analogous to the linear case v dx ) dt Physics 07: Lecture 16, Pg 4 Page 1
Physics 07 Lecture 16 Syste of Particles (Distributed Mass): l Until now, we have considered the behavior of very siple systes (one or two asses). l But real objects have distributed ass! l For exaple, consider a siple rotating disk and equal ass plugs at distances r and r. 1 l Copare the velocities and kinetic energies at these two points. Physics 07: Lecture 16, Pg 5 For these two particles 1 K ½ v K ½ (v) 4 ½ v l Twice the radius, four ties the kinetic energy K + 1 4 1 1 v + v ) ( r1 ) ( r K 1 [ r 1 + r ] Physics 07: Lecture 16, Pg 6 Page 13
Physics 07 Lecture 16 For these two particles 1 K ½ v K K K ½ (v) 4 ½ v 1 [ r 1 + r ] 1 [ ] All ass points r Physics 07: Lecture 16, Pg 7 For these two particles K [ 1 All ass points r ] K Rot 1 I (where I is the oent of inertia) Physics 07: Lecture 16, Pg 8 Page 14
Physics 07 Lecture 16 Rotation & Kinetic Energy... l The kinetic energy of a rotating syste looks siilar to that of a point particle: Point Particle K 1 v v is linear velocity is the ass. Rotating Syste I K 1 I i is angular velocity I is the oent of inertia about the rotation axis. ir i Physics 07: Lecture 16, Pg 9 Calculating Moent of Inertia I N i 1 i r i where r is the distance fro the ass to the axis of rotation. Exaple: Calculate the oent of inertia of four point asses () on the corners of a square whose sides have length L, about a perpendicular axis through the center of the square: L Physics 07: Lecture 16, Pg 30 Page 15
Physics 07 Lecture 16 Calculating Moent of Inertia... l For a single object, I depends on the rotation axis! l Exaple: I 1 4 R 4 ( 1/ L / ) I 1 L I L I L L Physics 07: Lecture 16, Pg 31 Lecture 16 Assignent: l HW7 due Tuesday Nov. 1 st Physics 07: Lecture 16, Pg 3 Page 16