Energy Potential Energy Conservative and Nonconservative Forces

Transcription

1 Energy Potential Energy Conservative and Nonconservative Forces Lana Sheridan De Anza College Oct 26, 2017

2 Last time Work as an integral Kinetic Work-Kinetic theorem Potential

3 Overview Gravitational and elastic potential Conservative and nonconservative forces Potential and force

4 F r 2F uggested by our conceptual argument. Potential Energy Last time we considered lifting a refrigerator with various methods. Does the Ramp Lessen the Work Required? rator onto a truck using Figure He claims d to load the truck if the ased. Is his claim valid? is wheeled on a hand t speed. In this case, for nd the hand truck, DK 5 the ramp on the system lacement of its point of k on the system. Because y theorem gives W ext 5 W by man 1 W by gravity 5 0 ional force equals the product of the weight mg of the system, the distance L through ced, and cos (u 1 908). Therefore, W by man 52W by gravity 521mg21L23cos 1u mgl sin u 5mgh of the ramp. Therefore, the man must do the same amount of work mgh on the system amp. The work depends only on the height of the ramp. Although less force is required f application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. We found in each case, the work done to raise the fridge was the same, W = mgh. If we drop the fridge, the amount of kinetic it gains was also K = 1 2 mv 2 = mgh. L u

5 F r 2F uggested by our conceptual argument. Potential Energy Last time we considered lifting a refrigerator with various methods. Does the Ramp Lessen the Work Required? rator onto a truck using Figure He claims d to load the truck if the ased. Is his claim valid? is wheeled on a hand t speed. In this case, for nd the hand truck, DK 5 the ramp on the system lacement of its point of k on the system. Because y theorem gives W ext 5 W by man 1 W by gravity 5 0 ional force equals the product of the weight mg of the system, the distance L through ced, and cos (u 1 908). Therefore, W by man 52W by gravity 521mg21L23cos 1u We 5 mgl concluded sin u5mghthat the raised fridge stored some. of the ramp. Therefore, the man must do the same amount of work mgh on the system amp. The work depends only on the height of the ramp. Although less force is required f application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. We found in each case, the work done to raise the fridge was the same, W = mgh. If we drop the fridge, the amount of kinetic it gains was also K = 1 2 mv 2 = mgh. We called this potential for kinetic potential, U. L u

6 Potential Energy Notice that the refrigerator has potential once it is lifted because of its configuration in space. It was a height h above the ground: we changed how far it was from the (center of the) earth.

7 Potential Energy: What s in the System? In order for the configuration of the refrigerator to be a meaningful concept, we have to say where it is in relation to the Earth. We must now include the Earth in our system. The system consists of the refrigerator and the Earth. Their gravitational interaction is now internal forces.

8 Potential Energy: What s in the System? In order for the configuration of the refrigerator to be a meaningful concept, we have to say where it is in relation to the Earth. We must now include the Earth in our system. The system consists of the refrigerator and the Earth. Their gravitational interaction is now internal forces. External mean an influence from something in the environment; internal means an influence from something else in our system.

9 The work done by the agent on the gravitational force. We do some work on the system rest through a vertical displacement D S r 5 1y f 2 mgy f mgy i. to our discussion of work as an transfer, th appear as an increase in of the system. The Physics the Generally, work and is when at rest lifting after we a mass perform near the the work. the Earth s kinetic surface: of the system. S F app Because the change of the system is no S yf r the work-kinetic W = theorem F y does not apply h y Physics f appear as some form of y storage other than book, we could release i yf it and let it fall back to the S mg y i (and therefore, the = system) (mgj) now has (dyj) kinetic that was done in lifting y i the book. While the book tem had the potential = to mg(y possess f kinetic y i ), but allowed to fall. Therefore, we call the stor Figure The work 7.15 we do becomes stored is released, potential so we can. equate We will it tofind thethat the An external agent lifts change a book itslowly potential from a height. y i only be associated with specific types of forces acti to a height y f. The amount of potential in the system is d Gravitational potential the for system. a uniform Moving gravitational members of field the system of to diff may change the configuration of the system and th strength, g: Let us now derive an expression for the potentia at a given location above the surface of the Earth Uing g = an mg object y of mass m from an initial height y i ab Gravitational book Earth system Potential is Energy, Uniform G-Field Pitfall Prevention 7.7 Potential Energy The phrase potential does not refer to something that has the potential to become. Potential is. y f as in Figure We assume the lifting is done sl applied force from the agent is equal in magnitude

10 Work, Kinetic Energy, Potential Energy Doing work (positive) on a system will increase its potential. That potential can be converted to kinetic. This illustrates the conservation of. Energy is not created or destroyed, but it is exchanged and transformed.

11 stone at the top edge of the well, what is the gravitational potential of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential of the system from release to reaching the bottom of the well? Gravitational Potential Energy re or on nt raec- e- w- ct rts so es. of nd arnd ur ge he Page 207, # A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. Section 7.7 Conservative and Nonconservative Forces 43. A 4.00-kg particle moves M from the origin to position, having coordi- Q/C nates x m and y m (Fig. P7.43). One force on the particle is y (m) (5.00, 5.00)

12 stone at the top edge of the well, what is the gravitational potential of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential of the system from release to reaching the bottom of the well? Gravitational Potential Energy re or on nt raec- e- w- ct rts so es. of nd arnd ur ge he Page 207, # A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. (a) Section U = (mg)y 7.7 Conservative = (400 N)(2and m) Nonconservative = 800J Forces 43. A 4.00-kg particle moves y (m) M from the origin to position, having coordi- Q/C (5.00, 5.00) nates x m and y m (Fig. P7.43). One force on the particle is

13 stone at the top edge of the well, what is the gravitational potential of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential of the system from release to reaching the bottom of the well? Gravitational Potential Energy re or on nt raec- e- w- ct rts so es. of nd arnd ur ge he Page 207, # A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. (a) Section U = (mg)y 7.7 Conservative = (400 N)(2and m) Nonconservative = 800J Forces 43. A 4.00-kg particle moves y (m) (b) U = (mg)y = (400 N)(2 m)(1 cos 30 from the origin to position, having coordi- ) = 107J M Q/C (5.00, 5.00) nates x m and y m (Fig. P7.43). One force on the particle is

14 stone at the top edge of the well, what is the gravitational potential of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential of the system from release to reaching the bottom of the well? Gravitational Potential Energy re or on nt raec- e- w- ct rts so es. of nd arnd ur ge he Page 207, # A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. (a) Section U = (mg)y 7.7 Conservative = (400 N)(2and m) Nonconservative = 800J Forces 43. A 4.00-kg particle moves y (m) (b) U = (mg)y = (400 N)(2 m)(1 cos 30 from the origin to position = 0., having coordi- ) = 107J M (c) Q/CU (5.00, 5.00) nates x m and y m (Fig. P7.43). One force on the particle is

15 Elastic Potential Energy Springs also can store potential when they are compressed or extended. U = W = 1 2 kx f kx i 2 so we define: Spring potential U s = 1 2 kx 2 where x is the amount by which the spring is compressed or extended from its natural length.

16 Work, Kinetic Energy, Potential Energy 7.6 Potential Energy of a Sy x 0 m Before the spring is compressed, there is no in the spring block system. % Kinetic Potential Total a x m When the spring is partially compressed, the total of the system is elastic potential. % Kinetic Potential Total Wor on t syste of th b x max m The spring is compressed by a maximum amount, and the block is held steady; there is elastic potential in the %

17 Work, Kinetic Energy, Potential Energy 7.6 Potential Energy of a Sys x 0 m Before the spring is compressed, there is no in the spring block system. % Kinetic Potential Total a x m When the spring is partially compressed, the total of the system is elastic potential. % Kinetic Potential Total Work on th syste of th b x max m The spring is compressed by a maximum amount, and the block is held steady; there is elastic potential in the system and no kinetic. % Kinetic Potential Total c x S v After the block is released, the elastic potential in % No w sprin

18 m Work, Kinetic Energy, Potential Energy a in the spring block system. 0 Kinetic Potential Total x m When the spring is partially compressed, the total of the system is elastic potential. % Kinetic Potential Total Work on th syste of th b x max m The spring is compressed by a maximum amount, and the block is held steady; there is elastic potential in the system and no kinetic. % Kinetic Potential Total c x m S v After the block is released, the elastic potential in the system decreases and the kinetic increases. % Kinetic Potential Total No w sprin the s total stays d x 0 S v After the block loses contact with the spring, the total %

19 Figure 7.16 A spring on a frictionless, horizontal surface is compressed a distance x max when a block of mass m is pushed against it. The block is then released and the spring pushes it to the right, where the block eventually loses contact with the spring. Parts (a) through (e) show various instants in the process. Energy bar charts on the right of each part of the figure help keep track of the in Work, Kinetic Energy, Potential Energy b of the system is elastic potential. 0 Kinetic Potential Total syste of th x max m The spring is compressed by a maximum amount, and the block is held steady; there is elastic potential in the system and no kinetic. % Kinetic Potential Total c x m S v After the block is released, the elastic potential in the system decreases and the kinetic increases. % Kinetic Potential Total No w sprin the s total stays d x 0 m S v After the block loses contact with the spring, the total of the system is kinetic. % Kinetic Potential Total e

20 m Work, Kinetic Energy, Potential Energy c elastic potential in the system and no kinetic. 0 Kinetic Potential Total x m S v After the block is released, the elastic potential in the system decreases and the kinetic increases. % Kinetic Potential Total No w sprin the s total stays d x 0 m S v After the block loses contact with the spring, the total of the system is kinetic. % Kinetic Potential Total e Figure 7.16 A spring on a frictionless, horizontal surface is compressed a distance x max when a block of mass m is pushed against it. The block is then released and the spring pushes it to the right, where the block eventually loses contact with the spring. Parts (a) through (e) show various instants in the process. Energy bar charts on the right of each part of the figure help keep track of the in the system. either stretched or compressed. Because the elastic potential is proportional to x 2, we see that U s is always positive in a deformed spring. Everyday examples of the storage of elastic potential can be found in old-style clocks or watches that operate from a wound-up spring and small wind-up toys for children.

21 Mechanical Energy The mechanical of a system is the that can be used to do work. It is defined as the sum of the system s kinetic and potential : E mech = K + U

22 ebassign Work, Kinetic Energy, Potential Energy Example S als, the oils you just way hile nds of me oor. for calthe Page 236, #3 Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass kg is placed on top of a light, vertical spring of force constant N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle W speed of m/s at an angle of with the hor- izontal. A second ball is fired at an angle of Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical of the ball Earth system at the maximum height for each ball. Let y 5 0 at the cannon. 5. Review. A bead slides without frich R

23 ebassign Work, Kinetic Energy, Potential Energy Example S als, the oils you just way hile nds of me oor. for calthe Page 236, #3 Section 8.2 Analysis Model: Isolated System (Energy) 3. A block of mass kg is placed on top of a light, vertical spring of force constant N/m and pushed W downward so that the spring is compressed by m. After the block is released from rest, it travels upward and then leaves the spring. To what maximum height above the point of release does it rise? 4. A 20.0-kg cannonball is fired from a cannon with muzzle speed of m/s at an angle of with the hor- W izontal. U A second ball is fired at an angle of Use s = 1 the isolated 2 kx 2 K.E. & Grav P.E. U g = mgh system model to find (a) the maximum height reached by each ball and (b) the total mechanical of the ball Earth system at 2 (5000)(0.1)2 U U s = 1 the maximum height for g = 25 J = (.25)(9.8)h each = ball. 25 J Let y 5 0 at the cannon. h = 10.2 m h 5. Review. A bead slides without fric- R

24 Dxr 5 F r Dx 5 2F Dx 5 2 Dx Conservative and Non-Conservative Forces rter as suggested by our conceptual argument. le 7.7 Remember the refrigerator that we wanted to lift into the truck? Does the Ramp Lessen the Work Required? a refrigerator onto a truck using hown in Figure He claims required to load the truck if the ere increased. Is his claim valid? igerator is wheeled on a hand constant speed. In this case, for erator and the hand truck, DK 5 erted by the ramp on the system he displacement of its point of s no work on the system. Because tic theorem gives W ext 5 W by man 1 W by gravity 5 0 gravitational force equals the product of the weight mg of the system, the distance L through is displaced, and cos (u 1 908). Therefore, there W by man 52W was by the gravity 521mg21L23cos same change 1u in potential U g = mgh. 5 mgl sin u5mgh e height of the ramp. Therefore, the man must do the same amount of work mgh on the system of the ramp. The work depends only on the height of the ramp. Although less force is required e point of application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. (We assumed there was no friction.) We found that whether we lifted it up directly or used a ramp we were required to do the same amount of work (W = mgh) and The path we took with the fridge didn t matter: the only thing that mattered was the where the fridge started from y = 0 and where it ended up y = h. L u

25 Conservative Forces Forces like this are called Conservative Forces.

26 Conservative Forces Forces like this are called Conservative Forces. The have the property that the work done by or against these forces is independent of the path taken. All that needs to be known is the starting and ending points. For external forces that act on a part of the system following any closed path (one that ends back at the starting point) zero work is done on the system by the force.

27 Conservative Forces Forces like this are called Conservative Forces. The have the property that the work done by or against these forces is independent of the path taken. All that needs to be known is the starting and ending points. For external forces that act on a part of the system following any closed path (one that ends back at the starting point) zero work is done on the system by the force. Examples of conservative forces: Gravity Spring force

28 Conservative Forces Only for conservative forces internal to the system can we define associated potential energies. U = W int where W int is the work done by an internal force of the system and U is the change in the potential of the system associated to that force.

29 Conservative Forces Only for conservative forces internal to the system can we define associated potential energies. U = W int where W int is the work done by an internal force of the system and U is the change in the potential of the system associated to that force. Note: by the system, not on the system! (W int indicates the work done is internal to the system.)

30 GY AND CONSERVATION OF ENERGY Conservative Forces ty on W = 0 Side View orie two work ppork closed D mg C W = mgh W = mgh h mg W = 0 mg A B mg The W s in this diagram are internal-force works W int : work done by gravity. but does positive work from D to A (displacement and force are in the s direction). Hence, W = -mgh and W = mgh. As a result, the total work d

31 Non-Conservative Forces Some forces are not path independent. These forces are dissipative and do not conserve mechanical (mechanical decreases).

32 Non-Conservative Forces Some forces are not path independent. These forces are dissipative and do not conserve mechanical (mechanical decreases). These forces are non-conservative forces.

33 Non-Conservative Forces Some forces are not path independent. These forces are dissipative and do not conserve mechanical (mechanical decreases). These forces are non-conservative forces. This does not mean that overall is not conserved. Examples of non-conservative forces: Friction Air resistance Mechanical is converted to heat or other inaccessible forms.

34 Non-conservative force: Friction Two paths for moving from A to B are not equivalent: The work done in moving the book is greater along the brown path than along the blue path. Physics Figure 7.19 The work done against the force of kinetic friction depends on the path taken as the book is moved from to. where K includes the kinetic includes all types of potentia action of the gravitational fo remains fixed; gravitational the total of the syste within a system, however, ca For example, for a book sent (Fig. 7.18a), the mechanical internal as we discus transformed to internal ene in the surface. (When you tr the skin on your knees warm friction transforms the mech nonconservative force. As an example of the path consider Figure Suppose the book is displaced in a str in Figure 7.19, you do a ce to keep the book moving at a along the brown semicircula friction along this curved pa path is longer. The work don force cannot be conservative. If s is the path length and n is the magnitude of the normal force: W against-friction = µ k n s

35 Non-conservative force: Friction Two paths for moving from A to B are not equivalent: The work done in moving the book is greater along the brown path than along the blue path. Physics Figure 7.19 The work done against the force of kinetic friction depends on the path taken as the book is moved from to. where K includes the kinetic includes all types of potentia action of the gravitational fo remains fixed; gravitational the total of the syste within a system, however, ca For example, for a book sent (Fig. 7.18a), the mechanical internal as we discus transformed to internal ene in the surface. (When you tr the skin on your knees warm friction transforms the mech nonconservative force. As an example of the path consider Figure Suppose the book is displaced in a str in Figure 7.19, you do a ce to keep the book moving at a along the brown semicircula friction along this curved pa path is longer. The work don force cannot be conservative. If s is the path length and n is the magnitude of the normal force: W against-friction = µ k n s

36 ates 2.80 cm points. (b) Repeat part (a), setting the zero configuration 5.00 with m the and car y at 5 point m as shown in Figure P7.43. Calculate the work done by S 41. A 0.20-kg stone is held 1.3 m above F on the the top particle edge of as a it moves water along well (a) and the then purple dropped path, (b) the red path, and (c) the blue path. (d) Is S into it. The well has a depth of 5.0 m. Relative to the Fconfiguration conservative with or nonconservative? (e) the Explain top edge your of the answer well, what to part is the (d). gravita- the stone tional potential of the stone Earth system 46. (a) An before object the moves stone is in released the xy and plane (b) when in Figure it reaches P7.43 and Q/C the experiences bottom of the a friction well? (c) What force is with the change constant in gravitational 3.00 N, potential always acting of in the the system direction from release opposite to the magnitude reaching object s the velocity. bottom Calculate of the well? the work that you must do 42. A to 400-N slide child the object is in a swing at constant that is attached speed against to a pair the friction ropes force 2.00 as m long. the object Find the moves gravitational along potential (a) the purple W of of the child Earth system relative to the child s path O to followed by a return purple path to O, lowest position when (a) the ropes are horizontal, (b) (b) the the ropes purple make path a O angle to with followed the vertical, by a return and blue (c) path the to child O, is and at the (c) bottom the blue of the path circular O to arc. followed by a return blue path to O. (d) Each of your three answers Section should 7.7 Conservative be nonzero. and Nonconservative What is the significance Forces of this 43. A observation? 4.00-kg particle moves y (m) M from the origin to posi- Section Q/C 7.8, Relationship having coordinates Between Conservative (5.00, 5.00) Forces and x 5 Potential 5.00 m and Energy y m (Fig. P7.43). One 47. force The potential on the particle is of a system of two particles separated gravitational by a distance force r is given by U(r) 5 A/r, where A S the acting is a constant. in the negative Find the y radial force S Fr that each x (m) particle direction. Using Equa- O exerts on the other. electron in Friction and Work Example of light over nergy of the ctrons carry creen where it glow. For the electron he constant the acceleraval the elec thellel to Newences affect, solve parts ) and (e) so wo theories. o a speed of m. (a) Find ves the barrem to find (c) Use your the average it was in the rticle under acceleration s a speed of Page 208, #46 positive zero a points (b) Ind of stab and neu rium. ( curve fo 53. A right horizon three e position Additional P 54. The po is given tion of force F x values o sus x an unstabl 55. Review ball at a

37 Summary Gravitational and elastic potential Conservative and nonconservative forces Next Test Friday, Nov 3, Chapters 6-8, and friction/pulleys from Ch 5. (Uncollected) Homework Serway & Jewett, Ch 7, onward from page 207. Probs: 41, 43 Read ahead in chapter 8.