Energy Energy Diagrams and Equilibrium Conservation Laws Isolated and Nonisolated Systems

Transcription

1 Energy Energy Diagrams and Equilibrium Conservation Laws Isolated and Nonisolated Systems Lana Sheridan De Anza College Oct 30, 2017

2 Last time gravitational and spring potential energies conservative and nonconservative forces

3 Overview force and potential stable and unstable equilibrium conservation laws isolated and nonisolated systems

4 Conservative Forces and Potential Energy In general, for a conservative force F the particle might move along an arbitrary path s: U = F dr s and 1 F = U where: U = x U i + y U j + z U k 1 If you are not yet familiar with this vector calculus notation, you will not need it for this course.

5 Energy Diagrams Potential can be plotted as a function of position. eg. potential of200 a spring: Chapter 7 Energy of a System U s a 1 U 2 kx 2 s x max 0U s 1 2 a kx 2 U s The restoring force exerted by the spring always acts toward x 0, x the x position max of stable 0 equilibrium. E x max The restoring force x 0 exerted x max by the spring b always acts toward x 0, the Figure position 7.20 of stable equilibrium. (a) Potential as a function of x for the frictionless block spring system shown F in s S (b). For a given E of the sys- m E x max S F s F s,x = du s dx = kx x function for a block plotted versus x in Figure 7.2 exerted by the spring on the As we saw in Quick Quiz 7.8 tive of the slope function of the for U-ve a equilibrium plotted versus position x of in the Figs some exerted external by force the spring F ext acts o equilibrium, x is positive an exerted by the spring is nega released. If the external forc negative; As we therefore, saw in Quick F s is posi Q release. tive of the slope of th From this analysis, we con equilibrium position o tem is one of stable equilibr results some in external a force directed force F ba e tem equilibrium, in stable equilibrium x is posit co minimum. exerted by the spring i released. If the block If in the Figure extern 7.20 from negative; rest, its total therefore, init F s As release. the block starts to move,. The block oscillates From this analysis, 2x max and x 5 1x max, called

6 Energy Diagrams and Equilibrium System is in equilibrium when F net = F s = 0. This happens when the slope of U(x) 200 is zero. Chapter 7 Energy of a System 1 U 2 kx 2 s U s E function for plotted versus x in F exerted by the spring a x max 0 x max The restoring force exerted by the spring always acts toward x 0, the position of stable equilibrium. x As we saw in Quick tive of the slope of equilibrium position some external force equilibrium, x is po exerted by the spring released. If the exter negative; therefore, F release. In this case, the force is always back toward the x = 0 point, so this is a stable equilibrium. Examples: spring force ball inside a bowl

7 neral, configurations of a system in unstable equilibrium correspond hich U(x) for the system is a maximum. Energy Diagrams and Equilibrium onfiguration called neutral equilibrium arises when U is constant ion. Small displacements of an object from a position in this region er restoring Systemnor is in disrupting equilibrium forces. when A ball F net lying = F s on = a 0. flat, This horizontal happens when xample the of slope an object of U(x) in neutral is zero. equilibrium. gure 7.21 A plot of U versus or a particle that has a position unstable equilibrium located x 5 0. For any finite displacent of the particle, the force on e particle is directed away from 0. U Positive slope Negative slope x 0 x 0 x 0 In this case, the force is always away from the x = 0 point, so this is a unstable equilibrium. Examples: the L1 Lagrange point between the Sun and Earth ball on upside-down a bowl

8 Neutral Equilibrium A system can also be in neutral equilibrium. In this case, no forces act, even when the system is displaced left or right. Example: ball on a flat surface

9 dx 5 4P dx ca x b 2 a x b d 5 4P c Equilibrium Example: The Lennard-Jones Potential derivative of the two The Lennard-Jones13 x eq potential function describes the force between two neutral atoms in a molecule. x is the atomic separation distance. x eq / nm m on both sides of iagram as shown large when the imum when the d then increases x) is a minimum, icating that the urs at this point. x 13 4P c 212s12 1 6s6 x d 5 0 S x 7 eq /6 s eq U (10 23 J) x 7 d x (10 10 m) x eq 20 Figure 7.22 (Example 7.9) [ Potential (σ ) 12 ( curve associated with a molecule. The distance x is the separation σ ) ] 6 between the two U(x) = 4ɛ atoms making up the molecule. x x σ and ɛ are constants: typical values are σ = nm and ɛ = J. What value does du dx give when at equilibrium?

10 Equilibrium Example: The Lennard-Jones Potential Find a value for the equilibrium distance of the two atoms, x eq, in terms of σ. [ (σ ) 12 ( σ ) ] 6 U(x) = 4ɛ x x

11 with 40.0 J of kinetic. Force and Potential Energy 49. A potential function for a system in which a two-dimensional force acts is of the form U 5 3x 3 y 2 7x. Find the force that acts at the point (x, y). Page 208, # A single conservative force acting on a particle within a system varies as F S 5 12Ax 1 Bx 2 2i^, where A and B are constants, F S is in newtons, and x is in meters. (a) Calculate the potential function U(x) associated with this force for the system, taking U 5 0 at x 5 0. Find (b) the change in potential and (c) the change in kinetic of the system as the particle moves from x m to x m. 51. A single conservative force acts on a 5.00-kg particle M within a system due to its interaction with the rest of the system. The equation F x 5 2x 1 4 describes the force, where F x is in newtons and x is in meters. As the particle moves along the x axis from x m to x m, calculate (a) the work done by this force on the particle, (b) the change in the potential of the Q/C

12 Conservation Laws A general comment on conservation laws... A number of important conservation laws come up in this course (specifically, and linear and angular momentum). Where do they come from? Why do they hold?

13 Conservation Laws A general comment on conservation laws... A number of important conservation laws come up in this course (specifically, and linear and angular momentum). Where do they come from? Why do they hold? Noether s Theorem: conservation laws are the direct result of symmetries in the equations of motion of a system.

14 Conservation Laws A general comment on conservation laws... A number of important conservation laws come up in this course (specifically, and linear and angular momentum). Where do they come from? Why do they hold? Noether s Theorem: conservation laws are the direct result of symmetries in the equations of motion of a system. Energy conservation comes from a time symmetry.

15 Isolated and Nonisolated Systems Isolated systems do not exchange with the environment. Nonisolated systems do. Non-isolated systems can lose to the environment or gain from it. Note: in these lectures, I mean the system is isolated or not with respect to specifically.

16 e, describe the system from the same set of choices. (iii) If the system nd Nonisolated the surface, describe Systems the system from the same set of choices. Energy) zed ial, nged of six e total e total undis n: (8.1) System boundary The change in the total amount of in the system is equal to the total amount of that crosses the boundary of the system. Work Matter transfer Heat Kinetic Potential Internal Electrical transmission Mechanical waves Electromagnetic radiation 1 Figures from Serway & Jewett.

17 system by applying a force to the system such that the point of application of the force undergoes a displacement (Fig. 8.1a). Nonisolated Systems: Energy Transfer Mechanisms Energy is transferred to the block by work. Energy leaves the radio from the speaker by mechanical waves. Energy transfers to the handle of the spoon by heat. Cengage Learning/George Semple Cengage Learning/George Semple Cengage Learning/George Semple a b c Energy enters the automobile gas tank by matter transfer. Energy enters the hair dryer by electrical transmission. Energy leaves the lightbulb by electromagnetic radiation. ergy trans- In each into which nergy is dicated. Cocoon/Photodisc/Getty Images d Cengage Learning/George Semple e Cengage Learning/George Semple f

18 Isolated Systems: No Energy Transfers l Isolated System (Energy) Compare to an isolated system: e System m boundary nd Kinetic Potential stem Internal gy can m in etic, The total amount of tera Energy transforms among in the system is constant. h no the three possible types. e sysod. Then, the system is isolated; transforms Exam p e f t D w t t

19 Isolated Systems: No Energy Transfers For an isolated system: If we choose the system so that it does not have the internal degrees of freedom, then the internal can be neglected and is conserved. (It is a constant.) Or equivalently: E mech = K + U K + U = 0

20 Nonisolated Systems: Energy Transfer For a nonisolated system we can write: More explicitly: where E system = ( Etransfers ) K + U + E int = W + Q + T E int is the change in in internal degrees of freedom within the system (eg. molecular vibrations) W is work done on the system Q is heat transferred to the system T includes transfer by all of the other transfer mechanisms

21 Nonisolated System Special Case K + U + E int = W + Q + T Suppose U = E int = Q = T = 0. Or in other words, that work was done on a system, but the only change in the system was its speed, then:

22 Nonisolated System Special Case K + U + E int = W + Q + T Suppose U = E int = Q = T = 0. Or in other words, that work was done on a system, but the only change in the system was its speed, then: K = W The Work-Kinetic Energy theorem.

23 Nonisolated System Special Case K + U + E int = W + Q + T Suppose U = E int = Q = T = 0. Or in other words, that work was done on a system, but the only change in the system was its speed, then: K = W The Work-Kinetic Energy theorem. It can be viewed as a special case of conservation in a non-isolated system. Or it is the case all forces on a system are external forces.

24 Question: Isolated vs. Nonisolated System In this problem, assume molecular vibrations within a system are part of the system description, E int. Quick Quiz Consider a block sliding over a horizontal surface with friction. Ignore any sound the sliding might make. (i) If the system is only the block, this system is (A) isolated (B) nonisolated (C) impossible to determine 1 Serway & Jewett, page 214.

25 Question: Isolated vs. Nonisolated System In this problem, assume molecular vibrations within a system are part of the system description, E int. Quick Quiz Consider a block sliding over a horizontal surface with friction. Ignore any sound the sliding might make. (ii) If the system is only the surface, this system is (A) isolated (B) nonisolated (C) impossible to determine 1 Serway & Jewett, page 214.

26 Question: Isolated vs. Nonisolated System In this problem, assume molecular vibrations within a system are part of the system description, E int. Quick Quiz Consider a block sliding over a horizontal surface with friction. Ignore any sound the sliding might make. (iii) If the system is the block and the surface, this system is (A) isolated (B) nonisolated (C) impossible to determine 1 Serway & Jewett, page 214.

27 Isolated Systems l Isolated System (Energy) e System m boundary nd Kinetic Potential stem Internal gy can m in etic, The total amount of tera Energy transforms among in the system is constant. h no the three possible types. e sysod. Then, the system is isolated; transforms Exam p e f t D w t t e

28 of thermodynamics, DE int 5 W 1 Q (Chapter 20) : DE int 5 T ET 1 T ER (Chapter 27) Isolated Systems DK 1 DU 5 Pick a system such that does not flow across the system boundary. eg. consider a book-earth system: solated System (Energy) ry common scenario in physics problems: a syscrosses the system boundary by any method. We l situation. Think about the book Earth system pter. After we have lifted the book, there is gravthe system, which can be calculated from the n the system, using W 5 DU g. (Check to see that ore, is contained within Eq. 8.2 above.) work done on the book alone by the gravitational ck to its original height. As the book falls from y i ional force on the book is 12mg j^2? 31y f 2 y i 2j^4 5 mgy i 2 mgy f (8.3) rem of Chapter 7, the work done on the book is of the book: on book 5 DK book ns for the work done on the book: ook 5 mgy i 2 mgy f (8.4) quation to the system of the book and the Earth. y i y f r S The book is held at rest here and then released. Physics Physics At a lower position, the book is moving and has kinetic K. Figure 8.2 A book is released from rest and falls due to work done by the gravitational force on the book. The Earth s gravitational force does work on the book. 2(mgy f 2 mgy i ) 5 2DU g potential of the system. For the left-hand ook is the only part of the system that is moving, K book = U g

29 Isolated Systems In that case we have K book + U g = 0 and the mechanical is conserved: E mech = 0 This holds when only conservative forces act in an isolated system. If non-conservative forces are allowed to act in an isolated system, we must include the internal degrees of freedom in our system and E mech 0 ; E system = 0

30 Question Quick Quiz A rock of mass m is dropped to the ground from a height h. A second rock, with mass 2m, is dropped from the same height. When the second rock strikes the ground, what is its kinetic? (A) twice that of the first rock (B) four times that of the first rock (C) the same as that of the first rock (D) half as much as that of the first rock 2 Serway & Jewett, page 216.

31 8.2 Analysis Model: Isolated System (Energy) Isolated and Nonisolated system example 8.3 continued y a S v y b y 0 % c % d % e % f Kinetic Kinetic Kinetic Kinetic Elastic pot. Elastic pot. Elastic pot. Elastic pot. Grav. pot. Grav. pot. Grav. pot. Grav. pot. Total Total Total Total Nonisolated system: total changes Isolated system: total constant Figure 8.6 (Example A spring-loaded popgun firing and (b) when the extends to its relaxed le (c) An bar chart popgun projectile Ear before the popgun is lo The in the system (d) The popgun is load means of an external ag work on the system to p spring downward. Ther the system is nonisolate this process. After the p loaded, elastic potentia stored in the spring and tational potential energ system is lower because jectile is below point. projectile passes throug, all of the of t system is kinetic. (f) Wh projectile reaches point the of the isolate gravitational potential. Categorize We identify the system as the projectile, the spring, and the Earth. We ignore both air resistan

32 8.2 Analysis Model: Isolated System (Energy) Isolated and Nonisolated system example 8.3 continued y a S v y b y 0 % c % d % e % f Kinetic Kinetic Kinetic Kinetic Elastic pot. Elastic pot. Elastic pot. Elastic pot. Grav. pot. Grav. pot. Grav. pot. Grav. pot. Total Total Total Total Nonisolated system: total changes Isolated system: total constant Figure 8.6 (Example A spring-loaded popgun firing and (b) when the extends to its relaxed le (c) An bar chart popgun projectile Ear before the popgun is lo The in the system (d) The popgun is load means of an external ag work on the system to p spring downward. Ther the system is nonisolate this process. After the p loaded, elastic potentia stored in the spring and tational potential energ system is lower because jectile is below point. projectile passes throug, all of the of t system is kinetic. (f) Wh projectile reaches point the of the isolate gravitational potential. Categorize We identify the system as the projectile, the spring, and the Earth. We ignore both air resistan

33 Energy of an Isolated System For the gravitational situation of the falling book, Equa 1 2mv f 2 1 mgy f 5 1 2mv i 2 1 mgy i As the book falls to the Earth, the book Earth system gains kinetic such that the total of the two type Quick Quiz Three identical balls are thrown from the top of a building, all with the same constant: initial speed. E total,i 5 As E total,f shown,. the first is thrown horizontally, the second at some angle above the horizontal, and the third at some angle below the horizontal. Neglecting air resistance, rank the speeds of the balls at the of the system as instant each hits the ground, from largest to smallest. The total of an isolated system is conserved. 2 1 If there are nonconservative forces acting within the is transformed to internal as discussed in Sect forces act in an isolated system, the total of the sy the mechanical is not. In that case, we can e DE system 5 0 where E system includes all kinetic, potential, and interna the most general statement of the version of the equivalent to Equation 8.2 with all terms on the right-ha 3 (A) 2, 1, 3 (B) 3, 1, 2 Q uick Quiz 8.3 A rock of mass m is dropped to the gro second rock, with mass 2m, is dropped from the same rock strikes (C) 1, the 2, ground, 3 what is its kinetic? (a) (b) four times that of the first rock (c) the same as tha as much (D) as that all the of the same first rock (e) impossible to dete Q uick Quiz 8.4 Three identical balls are thrown from with the same initial speed. As shown in Figure 8.3, t Figure 8.3 (Quick Quiz 8.4) zontally, the second at some angle above the horizont 2 Three identical balls are thrown Adapted from Serway & Jewett, angle page below 216. the horizontal. Neglecting air resistance,

34 Isolated and Nonisolated system example Page 237, #7 Figure P Two objects are connected M by a light string passing over a light, frictionless pulley as shown in Figure P8.7. The object of mass m kg is released from rest at a height h m above the table. Using the isolated system model, (a) determine the speed of the object of mass m kg just as the 5.00-kg object hits the table and (b) find the maximum height above the table to which the 3.00-kg object rises m 8. Two objects are connected by a light string passing S over a light, frictionless pulley as shown in Figure P8.7. The object of mass m 1 is released from rest at height m 2 m 1 h Figure P8.7 Problems 7 and 8. blocks block B leys. T rest so height are the block separa Section A sled kick im The co is the sle 13. A sled S kick im ficient Use en moves 14. A crate M an init paralle with th

35 Summary isolated and nonisolated systems Next Test Friday, Nov 3, Chapters 6-8, and friction/pulleys from Ch 5. (Uncollected) Homework Serway & Jewett, (Set at the end of the lecture Friday) Ch 7, onward from page 207. Probs: 49, 52 Read Chapter 8. Ch 8, onward from page 236. Probs: 5, 9, 13, 15, 17