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Last Chapter: Kinetic energy and work Chapter 8 Potential Energy and Conservation of Energy 8-1 Potential Energy Five sections S-1: Potential Energy S-2: Conservation of Mechanical Energy: S-3: Reading a Potential Energy Curve S-4: Work Done on a System by an External Force S-5: Conservation of Energy 8.1 Potential energy Potential energy is energy that can be associated with the configuration (arrangement) of a system of objects that exert forces on one another. Some forms of potential energy: 1. Gravitational Potential Energy Interaction: gravity Configuration: relative position 2. Elastic Potential Energy • Interaction: EM interaction between molecules • Configuration: Relative position among molecules 8.2 Work and potential energy: Start from an example F Change in potential energy Case-(a) Case-(b) 0+ = + +0 = - Work = Fd F + Always have opposite sign 8.2 Work and potential energy The change ΔU in potential energy (gravitational, elastic, etc) is defined as being equal to the negative of the work done on the object by the force (gravitational, elastic, etc) Do we care about the kinetic energy in this workenergy process? No: If only the potential energy changes = If we keep the speed of the object unchanged Yes: Regarding the total energy that has a variable kinetic energy 8.2 Conservative and non-conservative forces A popular scenario in reality 1. A system consists of two or more objects. 2. A force acts between a particle-like object in the system and the rest of the system. 3. When the system configuration changes, the force does work (call it W1) on the object-1 Transferring energy between the kinetic energy of the object, K, and some other type of energy of the system. 4. When the configuration change is reversed, the force reverses the energy transfer, doing work W2 in the process. 8.2 Conservative and non-conservative forces In a situation in which (1) W1= -W2 is always true (2) the other type of energy is a potential energy Then, the force is said to be a conservative force. Or, the work done by the force in moving a particle between two points is independent of the path taken: • gravitational force • spring force • magnetic force • electric force A force that is not conservative is called a non-conservative force: • • • The kinetic frictional force Air/fluid drag force non-elastic scattering force 8.3 Path Independence of Conservative Forces The net work done by a conservative force on a particle moving around any closed path is zero. If the work done from a to b along path 1 as Wab,1 and the work done from b back to a along path 2 as Wba,2. If the force is conservative, then the net work done during the round trip must be zero If the force is conservative, Sample problem, slippery cheese Sample problem, slippery cheese Calculations: Let us choose the dashed path in Fig. 8-5b; it consists of two straight segments. Along the horizontal segment, the angle φ is a constant 90°. Even though we do not know the displacement along that horizontal segment, the work Wh done there is Along the vertical segment, the displacement d is 0.80 m and, with and both downward, the angle is a constant = 0°. Thus, for the work Wv done along the vertical part of the dashed path, The total work done on the cheese by Fg as the cheese moves from point a to point b along the dashed path is then This is also the work done as the cheese slides along the track from a to b. Sample problem, slippery cheese Calculations: Let us choose the dashed path in Fig. 8-5b; it consists of two straight segments. Along the horizontal segment, the angle φ is a constant 90°. Even though we do not know the displacement along that horizontal segment, the work Wh done there is Along the vertical segment, the displacement d is 0.80 m and, with and both downward, the angle is a constant = 0°. Thus, for the work Wv done along the vertical part of the dashed path, The total work done on the cheese by Fg as the cheese moves from point a to point b along the dashed path is then This is also the work done as the cheese slides along the track from a to b. Sample problem, slippery cheese Calculations: Let us choose the dashed path in Fig. 8-5b; it consists of two straight segments. Along the horizontal segment, the angle φ is a constant 90°. Even though we do not know the displacement along that horizontal segment, the work Wh done there is Along the vertical segment, the displacement d is 0.80 m and, with and both downward, the angle is a constant = 0°. Thus, for the work Wv done along the vertical part of the dashed path, The total work done on the cheese by Fg as the cheese moves from point a to point b along the dashed path is then This is also the work done as the cheese slides along the track from a to b. Sample problem, slippery cheese: Let’s try this together Sample problem, slippery cheese: Let’s try this together 2 kg 0.8 m α And we need this α to write down the calculations March 27 8.4: Determining Potential Energy values: For the most general case, in which the force may vary with position, we may write the work W: We can use it to derive “Potential Energy values” corresponding different forces! 8.4: Determining Potential Energy values: Gravitational Potential Energy A particle with mass m moving vertically along a y axis (the positive direction is upward). As the particle moves from point yi to point yf, the gravitational force does work on it. The corresponding change in the gravitational potential energy of the particle–Earth system is: Reference point The gravitational potential energy associated with a particle–Earth system depends only on the vertical position y (or height) of the particle relative to the reference position y =0, not on the horizontal position. So, 8.4: Determining Potential Energy values: Elastic Potential Energy In a block–spring system, the block is moving on the end of a spring of spring constant k. As the block moves from point xi to point xf , the spring force Fx =- kx does work on the block. The corresponding change in the elastic potential energy of the block–spring system is Reference point If the reference configuration is when the spring is at its relaxed length, and the block is at xi = 0. Sample problem: gravitational potential energy Sample problem: gravitational potential energy Reference point with U = 0 Sample problem: gravitational potential energy 8.5: Conservation of Mechanical Energy Principle of Conservation of Energy In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy Emec of the system, cannot change. The mechanical energy Emec of a system is the sum of its potential energy U and the kinetic energy K of the objects within it: With the absence of Potential Energy With the absence of Kinetic Energy With both Potential Energy and Kinetic Energy 8.5: Conservation of Mechanical Energy: An example Potential energy: Reference point for zero U In stages (a) and (e), all the energy is kinetic energy. In stages (c) and (g), all the energy is potential energy. In stages (b), (d), ( f ), and (h), half the energy is kinetic energy and half is potential energy. Note: If the swinging involved a frictional force then Emec would not be conserved, and eventually the pendulum would stop. March 30 8.5: Conservation of Mechanical Energy Principle of Conservation of Energy In an ISOLATED system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy Emec of the system, cannot change. Sample problem: water slide Possible ways we can try (1) Newton’s Laws: Motion, Force, Acceleration, … - Will it work? - No: Force, track, … (2) Energy conservation: U, K - Will it work? - Yes • Two forces: (a) mg does work, (b) normal force doesn’t do work Sample problem: water slide 8.6: Reading a Potential Energy Curve: Potential Energy & Force Negative slope! A plot of U(x): the potential energy function of a system containing a particle confined to move along an x axis. (Since there is no friction, the mechanical energy is conserved.) A plot of the force F(x) acting on the particle, derived from the potential energy plot by taking its slope at various points. 8.6: Reading a Potential Energy Curve: Potential & kinetic energy Escape: With K>= 0 outside the potential well (x infinity). 8.6: Potential Energy Curve: Equilibrium Points (assuming K=0 on the line) Case-3: An particle at any point to the right of x5 • The system’s mechanical energy is equal to its potential energy, and so it must be stationary. A particle at such a position is said to be in neutral equilibrium. Case-1: An particle at x2 or x4 • It will be stuck there, cannot move left or right on its own because to do so would require a negative kinetic energy. • If we push it slightly left or right, a restoring force appears that moves it back to x2 or x4. A particle at such a position is said to be in stable equilibrium. Case-2: An particle at x3 When the particle at x3 has K =0. • If the particle is located exactly there, the force on it is also zero, and the particle remains stationary. • If it is displaced even slightly in either direction, a nonzero force pushes it farther in the same direction, and the particle continues to move. A particle at such a position is said to be in unstable equilibrium. Useful in describing processes in subatomic world Sample problem: reading a potential energy graph Sample problem: reading a potential energy graph Sample problem: reading a potential energy graph April 1 Sample problem: reading a potential energy graph Principle of Conservation of Energy In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy Emec. of the system, remains as a constant. How about a non-isolated system? – A system subject to external forces 8.7: Work done on a System by an External Force Work is energy transferred to or from a system by means of an external force acting on that system. 8.7: Work done on a System by an External Force 1. When FRICTION is NOT involved: Attention: The conservative gravitational force: internal force: ΔEmec=ΔK+ΔU=0 8.7: Work done on a System by an External Force 2. When FRICTION is involved: Net external force except frictions Next slide To include conservative force Fdx ! fk dx = ma " dx d d d # F dx ! # f 0 k 0 dx = # ma " dx 0 d dv Fd = fk d + # m " dx = dt 0 d # 0 vd dx fk dx + # m " dv dt v0 vd Fd = fk d + # mv " dv v0 1 1 1 Fd = fk d + (mvd2 ! mv02 ) = fk d + ( mvd2 ! mv02 ) 2 2 2 Fd = fk d + $K (8-34) !Fdx ! fk dx = !ma " dx !Fd = !#K + fk d Fd = #K ! fk d W = #Emec ! #Eth (8-34) Work by the external force !Emec = W + !Eth Mechanical energy “Work/Energy” source of the changes in energies Sample problem: change thermal energy F m m d How could this happen? Sample problem: change in thermal energy How all types of energies vary in the process? • • • • • • Force F // d Work done by F to the crate-floor system: + Change in the kinetic energy of the crate: Change in the potential energy of the crate: 0 What is missing? Thermal energy changes due to friction! Sample problem: change in thermal energy (8-34) 8.8: Conservation of Energy: with the presence of non-conservative external forces & internal structure Law of Conservation of Energy The total energy E of a system can change only by amounts of energy that are transferred to or from the system. Emec: any change in the mechanical energy of the system Eth: any change in the thermal energy of the system Eint: any change in any other type of internal energy of the system No energy or material exchanges with environment! The total energy E of an isolated system cannot change. Sample problem: energy, friction, spring, and tamales Sample problem: energy, friction, spring, and tamales System: The package–spring–floor–wall system includes all these forces and energy transfers in one isolated system. From conservation of energy, Forces: (1) The normal force on the package from the floor does no work on the package. (2) The gravitational force on the package does no work. (3) As the spring is compressed, a spring force does work on the package. (4) The spring force also pushes against a rigid wall does no work. (5) There is friction between the package and the floor increases their thermal energies. April 6 8.8: Conservation of Energy: Attention !! An external force can change the kinetic energy or potential energy of an object without doing work on the object! That is, without transferring energy to the object. Change in potential energy 8.8: Conservation of Energy: Attention!! Should not treat the skater as a simple point. Instead, the internal “structure” needs to be taken into account. What actually did the work are internal force(s)! 8.8: Conservation of Energy: Power In general, power P is the rate at which energy is transferred by a force from one type to another. If an amount of energy E is transferred in an amount of time t, the average power due to the force is Pavg !E = !t and the instantaneous power due to the force is dE P= dt The most important conclusions and relations Conservative Forces Potential Energy Net work on a particle over a closed path is 0 Energy associated with the configuration of a system and a conservative force Eq. (8-6) Gravitational Potential Energy Energy associated with Earth + a nearby particle Eq. (8-9) Elastic Potential Energy Energy associated with compression or extension of a spring Eq. (8-11) The most important conclusions and relations Potential Energy Curves Mechanical Energy Eq. (8-22) Eq. (8-12) For only conservative forces within an isolated system, mechanical energy is conserved Work Done on a System by an External Force Without/with friction: Eq. (8-26) Eq. (8-33) At turning points a particle reverses direction At equilibrium, slope of U(x) is 0 Conservation of Energy The total energy can change only by amounts transferred in or out of the system Eq. (8-35) The most important conclusions and relations Power The rate at which a force transfers energy Average power: Eq. (8-40) Instantaneous power: Eq. (8-41) 8.7.2. A rubber ball is dropped from rest from a height h. The ball bounces off the floor and reaches a height of 2h/3. How can we use the principle of the conservation of mechanical energy to interpret this observation? a) During the collision with the floor, the floor did not push hard enough on the ball for it to reach its original height. b) Some of the ball s potential energy was lost in accelerating it toward the floor. c) The force of the earth s gravity on the ball prevented it from returning to its original height. d) Work was done on the ball by the gravitational force that reduced the ball s kinetic energy. e) Work was done on the ball by non-conservative forces that resulted in the ball having less total mechanical energy after the bounce. 8.7.2. A rubber ball is dropped from rest from a height h. The ball bounces off the floor and reaches a height of 2h/3. How can we use the principle of the conservation of mechanical energy to interpret this observation? a) During the collision with the floor, the floor did not push hard enough on the ball for it to reach its original height. b) Some of the ball s potential energy was lost in accelerating it toward the floor. c) The force of the earth s gravity on the ball prevented it from returning to its original height. d) Work was done on the ball by the gravitational force that reduced the ball s kinetic energy. e) Work was done on the ball by non-conservative forces that resulted in the ball having less total mechanical energy after the bounce. 8.5.5. You are investigating the safety of a playground slide. You are interested in finding out what the maximum speed will be of children sliding on it when the conditions make it very slippery (assume frictionless). The height of the slide is 2.5 m. What is that maximum speed of a child if she starts from rest at the top? a) 1.9 m/s b) 2.5 m/s c) 4.9 m/s d) 7.0 m/s e) 9.8 m/s 8.5.5. You are investigating the safety of a playground slide. You are interested in finding out what the maximum speed will be of children sliding on it when the conditions make it very slippery (assume frictionless). The height of the slide is 2.5 m. What is that maximum speed of a child if she starts from rest at the top? a) 1.9 m/s b) 2.5 m/s c) 4.9 m/s d) 7.0 m/s e) 9.8 m/s - Maximum speed … frictionless - Initial velocity = 0 1 2 mgh = mv 2 v = 2gh = 2 ! 9.8 ! 2.5 = 7.0m / s 8.5.6. A quarter is dropped from rest from the fifth floor of a very tall building. The speed of the quarter is v just before striking the ground. From what floor would the quarter have to be dropped from rest for the speed just before striking the ground to be approximately 2v? Ignore all air resistance effects to determine your answer. a) 14 b) 25 c) 20 d) 7 e) 10 8.5.6. A quarter is dropped from rest from the fifth floor of a very tall building. The speed of the quarter is v just before striking the ground. From what floor would the quarter have to be dropped from rest for the speed just before striking the ground to be approximately 2v? Ignore all air resistance effects to determine your answer. - Initial velocity = 0 - One floor … H a) 14 b) 25 c) 20 d) 7 e) 10 1 2 mv (1) 2 1 mg(nH ) = m(2v)2 (2) 2 (2) : (1) n =4 5 n = 20 mg(5H ) = 8.5.7. Two identical balls are thrown from the same height from the roof of a building. One ball is thrown upward with an initial speed v. The second ball is thrown downward with the same initial speed v. When the balls reach the ground, how do the kinetic energies of the two balls compare? Ignore any air resistance effects. a) The kinetic energies of the two balls will be the same. b) The first ball will have twice the kinetic energy as the second ball. c) The first ball will have one half the kinetic energy as the second ball. d) The first ball will have four times the kinetic energy as the second ball. e) The first ball will have three times the kinetic energy as the second ball. 8.5.7. Two identical balls are thrown from the same height from the roof of a building. One ball is thrown upward with an initial speed v. The second ball is thrown downward with the same initial speed v. When the balls reach the ground, how do the kinetic energies of the two balls compare? Ignore any air resistance effects. a) The kinetic energies of the two balls will be the same. b) The first ball will have twice the kinetic energy as the second ball. c) The first ball will have one half the kinetic energy as the second ball. d) The first ball will have four times the kinetic energy as the second ball. e) The first ball will have three times the kinetic energy as the second ball. 8.7.1. A car is being driven along a country road on a dark and rainy night at a speed of 20 m/s. The section of road is horizontal and straight. The driver sees that a tree has fallen and covered the road ahead. Panicking, the driver locks the brakes at a distance of 20 m from the tree. If the coefficient of friction between the wheels and road is 0.8, determine the outcome. a) The car stops 5.5 m before the tree. b) The car stops just before reaching the tree. c) As the car crashes into the tree, its speed is 18 m/s. d) As the car crashes into the tree, its speed is 9.3 m/s. e) This problem cannot be solved without knowing the mass of the car. initial energy 1 2 mvi 2 8.7.1. A car is being driven along a country road a darkby and friction rainy night workondone mgµatk D a speed of 20 m/s. The section of road is horizontal and straight. The 1 the 1 2ahead. Panicking, driver sees that a tree has fallen and covered mvi2 ! road mv f = mgµ k D 2 20 m from 2 the driver locks the brakes at a distance of the tree. If the 2 2 ! v f road = 2gis µk D coefficient of friction between the wheelsvi and 0.8, determine the outcome. v 2f = vi2 ! 2gµ k D = 400 ! 313.6 = 86.4 a) The car stops 5.5 m before the tree. v f = 9.29m / s b) The car stops just before reaching the tree. c) As the car crashes into the tree, its speed is 18 m/s. d) As the car crashes into the tree, its speed is 9.3 m/s. e) This problem cannot be solved without knowing the mass of the car. 8.7.3. The Jensens decided to spend their family vacation white water rafting. During one segment of their trip down a horizontal section of the river, the raft (total mass = 544 kg) has an initial speed of 6.75 m/s. The raft then drops a vertical distance of 14.0 m, ending with a final speed of 15.2 m/s. How much work was done on the raft by non-conservative forces? a) -12 100 J b) -18 200 J c) -24 200 J d) -36 300 J e) -48 400 J 8.7.3. The Jensens decided to spend their family vacation white water rafting. During one segment of their trip down a horizontal section of the river, the raft (total mass = 544 kg) has an initial speed of 6.75 m/s. The raft then drops a vertical distance of 14.0 m, ending with a final speed of 15.2 m/s. How much work was done on the raft by non-conservative forces? a) -12 100 J b) -18 200 J c) -24 200 J d) -36 300 J e) -48 400 J !K + !U + W = 0 1 1 "W = mgH " ( mv 2f " mvi2 ) 2 2 1 1 "W = (544)(9.8)(14.0) "[ (544)(15.2)2 " (544)(6.75)2 ] 2 2 "W = 74636.8 "[62842.88 "12393.0] W = "24186.92J 8.8.1. A dam blocks the passage of a river and generates electricity. Approximately, 57 000 kg of water fall each second through a height of 19 m. If one half of the gravitational potential energy of the water were converted to electrical energy, how much power would be generated? a) 2.7 × 106 W b) 5.3 × 106 W c) 1.1 × 107 W d) 1.3 × 108 W e) 2.7 × 108 W 8.8.1. A dam blocks the passage of a river and generates electricity. Approximately, 57 000 kg of water fall each second through a height of 19 m. If one half of the gravitational potential energy of the water were converted to electrical energy, how much power would be generated? a) 2.7 × 106 W b) 5.3 × 106 W c) 1.1 × 107 W d) 1.3 × 108 W e) 2.7 × 108 W !E 0.5" mgH P= = !t 1s P = (0.5)(57000)(9.8)(19) P = 5,306, 700Watts 8.8.2. If the amount of energy needed to operate a 100 W light bulb for one minute were used to launch a 2-kg projectile, what maximum height could the projectile reach, ignoring any resistive effects? a) 20 m b) 50 m c) 100 m d) 200 m e) 300 m 8.8.2. If the amount of energy needed to operate a 100 W light bulb for one minute were used to launch a 2-kg projectile, what maximum height could the projectile reach, ignoring any resistive effects? a) 20 m b) 50 m c) 100 m d) 200 m e) 300 m E = conserved (P)(1min) = mgH (100J / s)(60s) = (2)(9.8)H H = (100J / s)(60s) /[(2)(9.8)] H = 306.12m 8.8.3. A 65-kg hiker eats a 250 C-snack. Assuming the body converts this snack with an efficiency of 25%, what change of altitude could this hiker achieve by hiking up the side of a mountain before completely using the energy in the snack? One food calorie (C) is equal to 4186 joules. a) 270 m b) 410 m c) 650 m d) 880 m e) 1600 m 8.8.3. A 65-kg hiker eats a 250 C-snack. Assuming the body converts this snack with an efficiency of 25%, what change of altitude could this hiker achieve by hiking up the side of a mountain before completely using the energy in the snack? One food calorie (C) is equal to 4186 joules. a) 270 m b) 410 m c) 650 m d) 880 m e) 1600 m E = conserved E food = mgH (25%)(250)(4186J) = (65)(9.8)H H = (25%)(250)(4186J) /[(65)(9.8)] H = 410.71m Five sections 8-1 Potential Energy + Major learning objectives S-1: Potential Energy 1. Distinguish a conservative force from a non-conservative force. 2. For a particle moving between two points, identify that the work done by a conservative force: It does not depend on which path the particle takes. 3. Calculate the gravitational potential energy of a particle (or, more properly, a particle-Earth system). 4. Calculate the elastic potential energy of a block-spring system. 8-2 Conservation of Mechanical Energy S-2: Conservation of Mechanical Energy: 5. After first clearly defining which objects form a system, identify that the mechanical energy of the system is the sum of the kinetic energies and potential energies of those objects. 6. For an isolated system in which only conservative forces act, apply the conservation of mechanical energy to relate the initial potential and kinetic energies to the potential and kinetic energies at a later instant. 8-2 Conservation of Mechanical Energy S-3: Reading a Potential Energy Curve 7. Given a particle's potential energy as a function of position x, determine the force on the particle. 8. Given a graph of potential energy versus x, determine the force on a particle. 9. On a graph of potential energy versus x, superimpose a line for a particle's mechanical energy and determine kinetic energy for any given value of x. 10. If a particle moves along an x axis, use a potential-energy graph for that axis and the conservation of mechanical energy to relate the energy values at one position to those at another position. 11. On a potential-energy graph, identify any turning points and any regions where the particle is not allowed because of energy requirements. 12. Explain neutral equilibrium, stable equilibrium, and unstable equilibrium. 8-2 Conservation of Mechanical Energy S-4: Work Done on a System by an External Force 13. When work is done on a system by an external force with no friction involved, determine the changes in kinetic energy and potential energy. 14. When work is done on a system by an external force with friction involved, relate that work to the changes in kinetic energy, potential energy, and thermal energy. 8-2 Conservation of Mechanical Energy S-5: Conservation of Energy 15. For an isolated system (no net external force), apply the conservation of energy to relate the initial total energy (energies of all kinds) to the total energy at a later instant. 16. For a non-isolated system, relate the work done on the system by a net external force to the changes in the various types of energies within the system. 17. Apply the relationship between average power, the associated energy transfer, and the time interval in which that transfer is made. 18. Given an energy transfer as a function of time (either as an equation or graph), determine the instantaneous power (the transfer at any given instant).