Lecture 16Definition of Work, The basicsNet Work: 1-D Example (constant force)Net Work: 1-D 2nd Example (constant force)Work: “2-D” Example (constant force)Exercise Work in the presence of friction and non-contact forcesWork and Varying Forces (1D)Slide 9Compare work with changes in potential energyConservative Forces & Potential EnergyA Non-Conservative Force, FrictionA Non-Conservative ForceSlide 14Conservative Forces and Potential EnergyEquilibriumWork & Power:Slide 18Work & Power: ExampleExercise Work & PowerSlide 21Chap. 12: Rotational DynamicsSlide 23Rotational VariablesSystem of Particles (Distributed Mass):For these two particlesSlide 27Slide 28Rotation & Kinetic Energy...Calculating Moment of InertiaCalculating Moment of Inertia...Slide 32Physics 207: Lecture 16, Pg 1Lecture 16•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: HW7 due Tuesday, Nov. 1st For Monday: Read Chapter 12, (skip angular momentum and explicit integration for center of mass, rotational inertia, etc.)Exam 2 Exam 2 7:15 PM Thursday, Nov. 37:15 PM Thursday, Nov. 3thth •Chapter 12 Chapter 12 Define rotational inertia Define rotational kinetic energyPhysics 207: Lecture 16, Pg 2Definition of Work, The basicsIngredients: Force ( F ), displacement ( r ) rrdisplacementFFWork, W, of a constant force F acts through a displacement r :W = F · r (Work is a scalar)(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?Physics 207: Lecture 16, Pg 4Net Work: 1-D Example (constant force)Net Work is F x = 10 x 5 N m = 50 J= 10 x 5 N m = 50 J1 N m ≡ 1 Joule (energy)Work reflects positive energy transferxA force F = 10 N pushes a box across a frictionless floor for a distance x = 5 m.F = 0° StartFinishPhysics 207: Lecture 16, Pg 5Net Work: 1-D 2nd Example (constant force)Net Work is F x = -10 x 5 N m = -50 J= -10 x 5 N m = -50 JWork again reflects negative energy transferxFA force F = 10 N is opposite the motion of a box across a frictionless floor for a distance x = 5 m. = 180° StartFinishPhysics 207: Lecture 16, Pg 6Work: “2-D” Example (constant force)(Net) Work is Fx x = F cos(= F cos(-45°) x = 50 x 0.71 Nm = 35 J = 50 x 0.71 Nm = 35 J xFAn angled force, F = 10 N, pushes a box across a frictionless floor for a distance x = 5 m and y = 0 m = -45° StartFinishFxPhysics 207: Lecture 16, Pg 7ExerciseWork in the presence of friction and non-contact forcesA. 2B. 3C. 4D. 5A 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 vPhysics 207: Lecture 16, Pg 8Work and Varying Forces (1D)Consider a varying force F(x)FxxxArea = Fx xF is increasingHere W = F · r becomes dW = F dx fixxdxxFW )(Physics 207: Lecture 16, Pg 9•How much will the spring compress (i.e. x = xf - xi) to bring the box to a stop (i.e., v = 0 ) if the object is moving initially at a constant velocity (vo) on frictionless surface as shown below ?xvomtoFspring compressedspring at an equilibrium positionV=0tmfixxdxxFW )(boxfixxdxkxW box2021221v mxk Example: Work Kinetic-Energy Theorem with variable forcePhysics 207: Lecture 16, Pg 10Compare work with changes in potential energyConsider the ball moving up to height h(from time 1 to time 2) How does this relate to the potential energy? Work done by the Earth’s gravity on the ball)W = F - x = (-mg) -h = mghU = Uf – Ui = mg 0 - mg h = -mg hU = -WhmgmgPhysics 207: Lecture 16, Pg 11Conservative Forces & Potential EnergyIf 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.(independent of path!)W = F ·dr = - U rirf Uf UiPhysics 207: Lecture 16, Pg 12A Non-Conservative Force, FrictionLooking down on an air-hockey table with no air flowing ( > 0). Now compare two paths in which the puck starts out with the same speed (Ki path 1 = Ki path 2) .Path 2Path 1Physics 207: Lecture 16, Pg 13A Non-Conservative ForcePath 2Path 1Since path2 distance >path1 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 WNC = Efinal - Einitial < 0 and reflects EthermalPhysics 207: Lecture 16, Pg 14A. U K B. U EThC. K UD. K EThE. There is no transformation because energy is conserved.A child slides down a playground slide at constant speed. The energy transformation isPhysics 207: Lecture 16, Pg 15Conservative Forces and Potential EnergySo we can also describe work and changes in potential energy (for conservative forces)U = - WRecalling (if 1D)W = Fx xCombining these two,U = - Fx xLetting small quantities go to infinitesimals,dU = - Fx dxOr,Fx = -dU / dxPhysics 207: Lecture 16, Pg 16EquilibriumExample Spring: Fx = 0 => dU / dx = 0 for x=xeqThe spring is in equilibrium positionIn general: dU / dx = 0 for ANY function establishes equilibriumstable equilibriumunstable equilibriumUUPhysics 207: Lecture 16, Pg 17Work & Power:Two cars go up a hill, a Corvette and a ordinary Chevy Malibu. Both cars have the same mass. Assuming identical friction, both engines do the same amount of work to get up the hill.Are the cars essentially the same ?NO. The Corvette can get up the hill quickerIt has a more powerful engine.Physics 207: Lecture 16, Pg 18Work & Power:Power is the rate, J/s, at which work is done.Average Power is, Instantaneous Power is,If force constant, W= F x = F (v0 t + ½ at2)and P = W / t = F (v0 + at) tWPdtdWP Physics 207: Lecture 16, Pg 19Work & Power: ExampleAveragePower:A person, mass 80.0 kg, runs up 2 floors (8.0 m) at constant speed. If they climb it in 5.0 sec, what is the average power used?Pavg = F h / t = mgh / t = 80.0 x 9.80 x 8.0 / 5.0 WP = 1250 W
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