Wednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu1PHYS 1441 – Section 004Lecture #13Wednesday, Mar. 10, 2004Dr. Jaehoon Yu• Conservation of Mechanical Energy• Work Done by Non-conservative forces•Power• Energy Loss in Automobile• Linear Momentum• Linear Momentum ConservationToday’s homework is homework #8, due 1pm, Wednesday, Mar. 24!!Wednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu2Announcements• Spring break: Mar. 15 – 19 • Second term exam on Monday, Mar. 29– In the class, 1:00 – 2:30pm – Sections 5.6 – 8.8– Mixture of multiple choices and numeric problemsWednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu3Conservation of Mechanical EnergyTotal mechanical energy is the sum of kinetic and potential energiesmghUg=Let’s consider a brick of mass m at a height h from the groundfiUU U∆=−The brick gains speedgtv=The lost potential energy converted to kinetic energyWhat does this mean?The total mechanical energy of a system remains constant in any isolated system of objects that interacts only through conservative forces: Principle of mechanical energy conservationPrinciple of mechanical energy conservationmmghWhat is its potential energy?What happens to the energy as the brick falls to the ground?mh1By how much?So what?The brick’s kinetic energy increasedK=And?UKE+≡ifEE=iiffKUK U+=+∑∑212mv=2212mg tmgh=Wednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu4Example for Mechanical Energy ConservationA ball of mass m is dropped from a height h above the ground. Neglecting air resistance determine the speed of the ball when it is at a height y above the ground.ffiiUKUK+=+b) Determine the speed of the ball at y if it had initial speed viat the time of release at the original height h.mghmymUsing the principle of mechanical energy conservationffiiUKUK+=+Again using the principle of mechanical energy conservation but with non-zero initial kinetic energy!!!This result look very similar to a kinematicexpression, doesn’t it? Which one is it?mgh+0()yhmgmv −=221()yhgv −=∴2PE KEmghmgy00mv2/2mgymv +=221mghmvi+221()2212fimv v−=()yhgvvif−+=∴22mvi2/2mvi2/2mgymvf+=221()mg h y−Reorganize the termsWednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu5Example 6.8If the original height of the stone in the figure is y1=h=3.0m, what is the stone’s speed when it has fallen 1.0 m above the ground? Ignore air resistance. 21112mv mgy+=22212mv mgy+=2211.02mv mg+=2212.02mv mg=2212.02vg=2v=mgh=3.0mgAt y=3.0mAt y=1.0m2211.02mv mg+Since Mechanical Energy is conserved3.0mgCancel mSolve for v4.0g = 4.0 9.8 6.3 /ms×=Wednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu6Work Done by Non-conservative ForcesMechanical energy of a system is not conserved when any one of the forces in the system is a non-conservative force.Two kinds of non-conservative forces:Applied forces: Forces that are external to the system. These forces can take away or add energy to the system. So the mechanical energy of the system is no longer conserved.If you were to carry around a ball, the force you apply to the ball is external to the system of ball and the Earth. Therefore, you add kinetic energy to the ball-Earth system.fric tio nW=Kinetic Friction: Internal non-conservative force that causes irreversible transformation of energy. The friction force causes the kinetic and potential energy to transfer to internal energy ;KWWgyou∆=+E∆=UWg∆−=youW=appW=KU∆+∆fric tio nK∆=kfd−fiEE−=KU∆+∆ =kfd−Wednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu7Example for Non-Conservative ForceA skier starts from rest at the top of frictionless hill whose vertical height is 20.0m and the inclination angle is 20o. Determine how far the skier can get on the snow at the bottom of the hill with a coefficient of kinetic friction between the ski and the snow is 0.210.mghME=What does this mean in this problem?Don’t we need to know mass?K∆=Compute the speed at the bottom of the hill, using the mechanical energy conservation on the hill before friction starts working at the bottomh=20.0mθ=20oThe change of kinetic energy is the same as the work done by kinetic friction. Since we are interested in the distance the skier can get to before stopping, the friction must do as much work as the available kinetic energy. ;ikKfd−=−Since 0=fKWell, it turns out we don’t need to know mass. What does this mean?No matter how heavy the skier is he will get as far as anyone else has gotten.nfkkµ= mgkµ=mgKdkiµ=mgmvkµ221=gvkµ22=()m2.9580.9210.028.192=××=221mv=ghv 2=smv /8.190.208.92 =××=fiKK−=kfd− kifdK=Wednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu8Energy Diagram and the Equilibrium of a SystemOne can draw potential energy as a function of position Î Energy Diagram=sULet’s consider potential energy of a spring-ball systemA ParabolaWhat shape would this diagram be? xUs-xmxm221kxU =What does this energy diagram tell you?1. Potential energy for this system is the same independent of the sign of the position. 2. The force is 0 when the slope of the potential energy curve is 0 at the position.3. x=0 is one of the stable or equilibrium of this system where the potential energy is minimum.Position of a stable equilibrium corresponds to points where potential energy is at a minimum.Position of an unstable equilibrium corresponds to points where potential energy is a maximum. MinimumÎ Stable equilibrium MaximumÎ unstable equilibrium 221kxWednesday, Mar. 10, 2004 PHYS 1441-004, Spring 2004Dr. Jaehoon Yu9General Energy Conservation and Mass-Energy EquivalenceGeneral Principle of Energy ConservationThe total energy of an isolated system is conserved as long as all forms of energy are taken into account.Friction is a non-conservative force and causes mechanical energy to change to other forms of energy.What about friction?Principle of Conservation of MassEinstein’s Mass-Energy equality.However, if you add the new form of energy altogether, the system as a whole did not lose any energy, as long as it is self-contained or isolated.In the grand scale of the universe, no energy can be destroyed or created but just transformed or transferred from one place to another. Total energy of universe is constant.In any physical or chemical process, mass is neither created nor destroyed. Mass before a process is identical to the mass after the process.2mcER=How many joules does
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