PHYS 1443 – Section 501Lecture #13AnnouncementsKinetic Energy and Work-Kinetic Energy TheoremKE ExampleKE Example with FrictionKinetic Energy at High SpeedPotential EnergyGravitational Potential EnergyPotential Energy ExampleElastic Potential EnergyConservative and Non-conservative ForcesMore Conservative and Non-conservative ForcesConservative Forces and Potential EnergyPHYS 1443 – Section 501Lecture #13Monday, Mar. 8, 2004Dr. Andrew Brandt1. Kinetic Energy2. Potential Energy3. Conservation of Mechanical Energy4. Conservative and non-conservative forcesMonday, Mar. 8, 2004 PHYS 1443-501, Spring 2004Dr. Andrew Brandt1Announcements• HW#5 on Ch. 6 is due tonight at midnight• HW#6 on Ch. 7+part of Ch. 8 is due Weds 3/24 at midnight (note HW#7 will be due 3/29)• Interim grades posted on website:If you don’t like your grade there is still time to change it; it starts with the HW’s, spend more time on them and everything else will followMonday, Mar. 8, 2004 2PHYS 1443-501, Spring 2004Dr. Andrew BrandtKinetic Energy and Work-Kinetic Energy Theorem• Some problems are hard to solve using Newton’s second law– If forces exerted on the object during the motion are complicated– Relate the work done on the object by the net force to the change of the speed of the objectMΣFMdvivfSuppose a net force ΣF was exerted on an object over a displacement d to increase its speed from vito vf. The work on the object by the net force ΣF is()()()dmadmadFW ==⋅=∑0cos222fivvad−=()222211222fifivvW ma d m d mv mvd⎡⎤⎛⎞−== =−⎢⎥⎜⎟⎜⎟⎢⎥⎝⎠⎣⎦AccelerationKinetic Energy221mvKE=WorkMonday, Mar. 8, 2004 3PHYS 1443-501, Spring 2004Dr. Andrew BrandtKEKEKEmvmvWifif∆=−=−=222121The work done by the net force causes a change of object’s kinetic energy.WorkKE Example A 6.0kg block initially at rest is pulled to the East along a horizontal, frictionless surface by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m.Work done by the force F isMFMdvi=0vf()JdFdFW 360cos0.312cos =×==⋅=θ222121ifmvmvW −=From the work-kinetic energy theorem, we know221fmvW =Since initial speed is 0, the above equation becomes smmWvf/5.30.63622=×==Solving the equation for vf, we obtain Monday, Mar. 8, 2004 4PHYS 1443-501, Spring 2004Dr. Andrew BrandtKE Example with FrictionA 6.0kg block initially at rest is pulled to East along a horizontal surface with coefficient of kinetic friction µk=0.15 by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m.Work done by the force F is()JdFdFWF360cos0.312cos =×==⋅=θMFMd=3.0mvi=0vfWork done by friction FkisFkθµcosdmgdFWkkk=⋅=()J26180cos0.38.90.615.0−=×××=Thus the total work is)(102636 JWWWkF=−=+=Using work-kinetic energy theorem and the fact that initial speed is 0, we obtainSolving the equation for vf, we obtain smmWvf/8.10.61022=×==221fkFmvWWW =+=Monday, Mar. 8, 2004 5PHYS 1443-501, Spring 2004Dr. Andrew BrandtWhat’s another way to solvethis problem?Kinetic Energy at High SpeedThe laws of Newtonian mechanics is no longer valid for object moving at the speed close to that of light, c. It must be generalized for these special cases. Theory of relativity. ()⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−−= 11122cvmcKThe kinetic energy must be modified to reflect the fact that the object is moving very high speed.What does this expression tell you?The speed of an object cannot be faster than light in vacuum. Have not seen any particle that goes faster than light, yet.()⎟⎟⎠⎞⎜⎜⎝⎛−+⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−−= 1...8321111142222cvcvmccvmcKHowever this equation must satisfy the Newtonian expression for smaller speeds!!Monday, Mar. 8, 2004 6PHYS 1443-501, Spring 2004Dr. Andrew Brandt2222221211211 mvcvmccvmcK =⎟⎠⎞⎜⎝⎛×=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎠⎞⎜⎝⎛+=Potential EnergyEnergy associated with a system of objects Stored energy which has Potential or possibility to work or to convert to kinetic energyThe concept of potential energy can only be used under the special class of forces called, conservative forces which results in principle of conservation of mechanical energy.ffiiMPEKEPEKEE+=+≡What are some other forms of energy?Mechanical Energy Biological EnergyElectromagnetic Energy Nuclear EnergyChemical EnergyIf one takes into account ALL forms of energy, the total energy in the entire universe is conserved. It just transforms from one form to another.Monday, Mar. 8, 2004 7PHYS 1443-501, Spring 2004Dr. Andrew BrandtGravitational Potential EnergyPotential energy given to an object by gravitational field due to its height above the surface of the EarthWhen an object is falling, a gravitational force, Mg, performs work on the object, increasing its kinetic energy. The potential energy of an object at a height y (its potential to do work) is expressed asmyfmmgyiyFUgg⋅=()()jyjmg −⋅−=mgyUg≡mgy=What does this mean?figUUW−=The work performed on the object by the gravitational force as the brick goes from yito yfis:Monday, Mar. 8, 2004 8PHYS 1443-501, Spring 2004Dr. Andrew BrandtgfiUmgymgy∆−=−=Work by the gravitational force as the brick goes from yito yfis negative of the change in the system’s potential energy Potential energy was lost in order for gravitational force to increase the brick’s kinetic energy.Potential Energy ExampleA bowler drops bowling ball of mass 7kg on his toe. Choosing floor level as y=0, estimate the total work done on the ball by the gravitational force as the ball falls. Let’s assume the top of the toe is 0.03m from the floor and the hand was 0.5m above the floor.MJmgyUff06.203.08.97=××==JmgyUii3.345.08.97=××==()JJUUUif3024.32≅=−−=∆b) Perform the same calculation using the top of the bowler’s head as the origin.Assuming the bowler’s height is 1.8m, the ball’s original position is –1.3m, and the toe is at –1.77m.What has to change?First we must re-compute the positions of ball at the hand and of the toe. ()JmgyUii2.893.18.97−=−××==()JmgyUff4.12177.18.97−=−××==()JJUUUif302.32≅=−−=∆Monday, Mar. 8, 2004 9PHYS 1443-501, Spring 2004Dr. Andrew BrandtElastic Potential EnergyElastic Potential EnergyThe Potential energy given to an object by a spring or an object with elasticity (the system consists of the object and the spring)The force the spring exerts on an object
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