Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu11. Potential Energy• Elastic Potential Energy2. Conservative and non-conservative forces3. Potential Energy and Conservative Force4. Conservation of Mechanical Energy5. Work Done by Non-conservative ForcesPHYS 1443 – Section 003Lecture #13Monday, Oct. 11, 2004Dr. Jaehoon YuQuiz next Monday, Oct. 18!!Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu2Work and Kinetic EnergyWork in physics is done only when the sum of forces exerted on an object made a motion to the object.What does this mean?However much tired your arms feel, if you were just holding an object without moving it you have not done any physical work to the object.Mathematically, work is written in a product of magnitudes of the net force vector, the magnitude of the displacement vector and the angle between them.Kinetic Energy is the energy associated with motion and capacity to perform work. Work causes change of energy after the completion Work-Kinetic energy theoremW=221mvK =KKKWif∆=−=∑Nm=Joule()iFd=∑i()cosiFdθ∑ Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu3Potential EnergyEnergy associated with a system of objects Stored energy which has the potential or the possibility to work or to convert to kinetic energyWhat does this mean?In order to describe potential energy, U,a system must be defined.What are other forms of energies in the universe?The 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.Mechanical Energy Biological EnergyElectromagnetic Energy Nuclear EnergyChemical EnergyME≡These different types of energies are stored in the universe in many different forms!!!If one takes into account ALL forms of energy, the total energy in the entire universe is conserved. It just transforms from one form to the other.iiKE PE+=ffKE PE+Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu4Gravitational Potential EnergyWhen an object is falling, gravitational force, Mg, performs work on the object, increasing its kinetic energy. The potential energy of an object at a height y which is the potential to work is expressed asPotential energy given to an object by gravitational field in the system of Earth due to its height from the surfacemyfmmgyigU =What does this mean?gW=Work performed on the object by the gravitational force as the brick goes from yito yfis:mgyUg≡mgy=ifmgy mgy=−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.cosgFyθ=gFygU=−∆ifUU−gFy⋅= Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu5Conservative and Non-conservative ForcesWhen directly falls, the work done on the object by the gravitation force isThe work done on an object by the gravitational force does not depend on the object’s path.gW=How about if we lengthen the incline by a factor of 2, keeping the height the same??gW=Still the same amount of work☺The forces like gravitational or elastic forces are called conservative forcesSo the work done by the gravitational force on an object is independent on the path of the object’s movements. It only depends on the difference of the object’s initial and final position in the direction of the force.mghWg=hlmθmgWhen sliding down the hill of length l, the work islmg×=θsinNmgh=Total mechanical energy is conserved!!ffiiMPEKEPEKEE+=+≡g inclin eFl−×()sinmg lθ=mgh1. If the work performed by the force does not depend on the path.2. If the work performed on a closed path is 0.Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu6Conservative Forces and Potential EnergyThe work done on an object by a conservative force is equal to the decrease in the potential energy of the systemcW=What else does this statement tell you?The work done by a conservative force is equal to the negative of the change of the potential energy associated with that force.U∆=We can rewrite the above equation in terms of potential energy U()ixxxfUdxFxUfi+−=∫So the potential energy associated with a conservative force at any given position becomesOnly the changes in potential energy of a system is physically meaningful!!Potential energy functionWhat can you tell from the potential energy function above?Since Uiis a constant, it only shifts the resulting Uf(x) by a constant amount. One can always change the initial potential so that Uican be 0.fiUU−=fixxxFdx−∫fixxxFdx=∫U−∆Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu7Example for Potential EnergyA 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. iU=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.MLet’s assume the top of the toe is 0.03m from the floor and the hand was 0.5m above the floor.What has to change? First we must re-compute the positions of ball at the hand and of the toe. fU=gW=iU =fU=gW =imgy=7 9.8 0.5 34.3J××=fmgy=7 9.8 0.03 2.06J××=()fiUU−−=32.24 30JJ≅U−∆=imgy=()7 9.8 1.3 89.2J××− =−fmgy=()7 9.8 1.77 121.4J××− =−U−∆ =()fiUU−−=32.2 30JJ≅Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu8Elastic Potential EnergyThe force spring exerts on an object when it is distorted from its equilibrium by a distance x isPotential energy given to an object by a spring or an object with elasticity in the system consists of the object and the spring without friction.kxFs−=What do you see from the above equations?The work performed on the object by the spring isThe work done on the object by the spring depends only on the initial and final position of the distorted spring.Where else did you see this trend?The potential energy of this system is221kxUs≡The gravitational potential energy, Ug()dxkxWfixxs∫−=fixxkx⎥⎦⎤⎢⎣⎡−=221222121ifkxkx +−=222121fikxkx −=So what does this tell you about the elastic force?A conservative force!!!Monday, Oct. 11, 2004 PHYS 1443-003, Fall 2004Dr. Jaehoon Yu9Conservation of Mechanical
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