Physics 207, Lecture 15, Oct. 22Work and Varying Forces (1D)Slide 3Slide 5Conservative Forces & Potential EnergyConservative Forces and Potential EnergyExercise Work Done by GravityNon-conservative Forces :A Non-Conservative Force, FrictionA Non-Conservative ForceWork & Power:Slide 17Slide 18Exercise Work & PowerChap. 12: Rotational DynamicsRotational Dynamics: A child’s toy, a physics playground or a student’s nightmareOverview (with comparison to 1-D kinematics)System of Particles (Distributed Mass):Slide 29System of Particles: Center of Mass (CM)System of Particles: Center of MassSample calculation:Slide 33Rotational Dynamics: What makes it spin?Slide 36Physics 207: Lecture 15, Pg 1Physics 207, Lecture 15, Oct. 22Goals:Goals:•Chapter 11Chapter 11 Employ conservative and non-conservative forces Use the concept of power (i.e., energy per time)•Chapter 12Chapter 12 Extend the particle model to rigid-bodies Understand the equilibrium of an extended object. Understand rotation about a fixed axis. Employ “conservation of angular momentum” conceptAssignment: HW7 due Oct. 29For Monday: Read Chapter 12, Sections 7, 8 & 11do not concern yourself with the integration process in regards to “center of mass” or “moment of inertia””Physics 207: Lecture 15, Pg 2Work and Varying Forces (1D)Consider a varying force F(x)FxxxArea = Fx xF is increasingHere W = F · r becomes dW = F dx F = 0° StartFinishWork has units of energy and is a scalar!fixxdxxFW )(FxPhysics 207: Lecture 15, Pg 3•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 (vi) on frictionless surface as shown below and xi is the equilibrium position of the spring?xvimtiFspring compressedspring at an equilibrium positionV=0tmfixxdxxFW )(boxfiixxdxxxkW )(-boxfiixxxxkW|221box)( - K 0 )( -221221box kxxkifW2i212 21221v0 - mmxk Example: Hooke’s Law Spring (xi equilibrium)Physics 207: Lecture 15, Pg 5Work signsxvimtiFspring compressedspring at an equilibrium positionV=0tmNotice that the spring force is opposite the displacementFor the mass m, work is negativeFor the spring, work is positive They are opposite, and equal (spring is conservative)Physics 207: Lecture 15, Pg 6Conservative Forces & Potential EnergyFor any conservative force F we can define a potential energy function U in the following way:The work done by a conservative force is equal and opposite to the change in the potential energy function.This can be written as:W = F ·dr = - U U = Uf - Ui = - W = - F • drrirfrirf Uf UiPhysics 207: Lecture 15, Pg 7Conservative 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 15, Pg 8 ExerciseWork Done by GravityAn frictionless track is at an angle of 30° with respect to the horizontal. A cart (mass 1 kg) is released from rest. It slides 1 meter downwards along the track bounces and then slides upwards to its original position. How much total work is done by gravity on the cart when it reaches its original position? (g = 10 m/s2)1 meter30°(A) 5 J (B) 10 J (C) 20 J (D) 0 J h = 1 m sin 30° = 0.5 mPhysics 207: Lecture 15, Pg 13Non-conservative Forces :If the work done does not depend on the path taken, the force involved is said to be conservative.If the work done does depend on the path taken, the force involved is said to be non-conservative.An example of a non-conservative force is friction:Pushing a box across the floor, the amount of work that is done by friction depends on the path taken.and work done is proportional to the length of the path !Physics 207: Lecture 15, Pg 14A 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 15, Pg 15A 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 15, Pg 16Work & 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 15, Pg 17Work & Power:Power is the rate 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 15, Pg 18Work & Power:Power is the rate at which work is done.tWPdtdWP InstantaneousPower:AveragePower:A person of mass 80.0 kg walks up to 3rd floor (12.0m). If he/she climbs in 20.0 sec what is the average power used. Pavg = F h / t = mgh / t = 80.0 x 9.80 x 12.0 / 20.0 WP = 470. W Example:Units (SI) areWatts (W):1 W = 1 J / 1sPhysics 207: Lecture 15, Pg 19Exercise Work & PowerA. TopB. MiddleC. BottomStarting from rest, a car drives up a hill at constant acceleration and then suddenly stops at the top. The instantaneous power delivered by the engine during this drive looks like which of the following,Z3timePowerPowerPowertimetimePhysics 207: Lecture 15, Pg 20Chap. 12: Rotational DynamicsUp until now rotation has been only in terms of circular motion with ac = v2 / R and | aT | = d| v | / dtRotation is common in the world around us.Many ideas developed for translational motion are transferable.Physics 207: Lecture 15, Pg 21Rotational Dynamics: A child’s toy, a physics playground or a student’s nightmare A merry-go-round is spinning and we run and jump on it. What does it do?We are standing on the rim and our “friends” spin it faster. What happens to us?We are standing on the rim a walk towards the center. Does anything change?Physics 207: Lecture 15, Pg 24Overview
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