Chapter 8 Conservation of Energy and Potential Energy

Transcription

1 Chapter 8 Conservation o Energy and Potential Energy So ar we have analyzed the motion o point-like bodies under the action o orces using Newton s Laws o Motion. We shall now use the Principle o Conservation o Energy to analyze the change in energy o a system and deduce how the velocity o the constituent components o a system will change between some initial state and some inal state. We begin by recalling the end o Section Conservation o Energy When a system and its surroundings undergo a transition rom an initial state to a inal state, the total change in energy is zero, Δ =Δ +Δ = total E Esystem Esurroundings. (8.1.1) Figure 8.1 (also Figure 7.1) diagram o a system and its surroundings This conservation law is our basic assumption. In any physical application, we irst identiy our system and surroundings, and then attempt to quantiy changes in energy. In order to do this, we need to identiy every type o change o energy in every possible physical process. I we add up all known changes in energy in the system and surroundings and do not arrive at a zero sum, we have an open scientiic problem. y searching or the missing changes in energy, we may uncover some new physical phenomenon. Recently, one o the most exciting open problems in cosmology is the apparent acceleration o the expansion o the universe, which has been attributed to dark energy that resides in space itsel, an energy type without a clearly known source. 1 Energy can change orms inside a system, or example chemical energy stored in the molecular bonds o gasoline can be converted into kinetic energy and heat via combustion. Energy can also low into or out o the system across a boundary. system 1 9/3/8 1

2 in which no energy lows across the boundary is called a closed system. Then the total change in energy o the system is zero, 8. Dissipative Forces: Friction closed Δ E system =. (8.1.) Suppose we consider an object moving on a rough surace. s the object slides it slows down and stops and the object and the surace both increase in temperature. The increase in temperature is due to the molecules inside the materials increasing their kinetic energy. This random kinetic energy is called thermal energy. Kinetic energy associated with the coherent motion o the molecules o the object has been dissipated into kinetic energy associated with the random o motion o the molecules composing the object and surace. I we deine the system to be just the object, then the riction orce acts as an external orce on the system and results in the dissipation o energy into both the block and the surace. Without knowing urther properties o the material we cannot determine the exact changes in the energy o the system. Friction introduces a problem in that the point o contact is not well deined because the surace o contact is constantly deorming as the object moves along the surace. I we considered the object and the surace as the system, then the riction orce is an internal orce, and the decrease in the kinetic energy o the moving object ends up as an increase in the internal random kinetic energy o the constituent parts o the system. When there is dissipation at the boundary o the system, we need an additional model (thermal equation o state) or how the dissipated energy distributes itsel among the constituent parts o the system. We shall return to this problem when we study the thermal properties o matter. Source Energies Consider a person walking. The riction orce between the person and the ground does no work because the point o contact between the person s oot and the ground undergoes no displacement as the person applies a orce against the ground is not displaced (there may be some slippage but that would be opposite the direction o motion o the person). However the kinetic energy o the body increases. Have we disproved the work energy theorem? The answer is no! The chemical energy stored in the body tissue is converted to kinetic energy and thermal energy. Since the person can be treated as an isolated system, we have that closed =Δ E =Δ E +Δ E +ΔK. (8..1) system chemical thermal Thereore some o the internal chemical energy has been transormed into thermal energy and the rest has changed the kinetic energy o the system, 9/3/8

3 Δ E = Δ E + ΔK. (8..) chemical thermal 8.3 Conservative and Non-Conservative Forces Our irst type o energy accounting involves mechanical energy. There are two types o mechanical energy, kinetic energy and potential energy. Our irst task is to deine what we mean by the change o the potential energy o a system. We deined the work done by a orce F, on an object which moves along a path rom an initial point r to a inal point r, as the integral o the component o the orce tangent to the path with respect to the displacement o the point o contact o the orce and the object, W = F d r. (8.3.1) path Does the work done on the object by the orce depend on the path taken by the object? First consider the motion o an object under the inluence o a gravitational orce near the surace o the earth, as was considered in Sections 7.4 and 7.7. The gravitational orce always points downward, so the work done by gravity only depends on the change in the vertical position (we choose the positive y -direction upwards), y y W = F dr = F dy = mg dy = mg( y y ) grav grav,y y path y (8.3.) Thereore when an object alls, ( y y) <, and the work done by gravity is positive. When an object rises, ( y y) >, and the work done by gravity is negative. Suppose an object irst rises and then alls, returning to the original starting height. The positive work done on the alling portion exactly cancels the negative work done on the rising portion, as in Figure 8.. The total work done is zero. Thus the gravitational work done between two points will not depend on the path taken, but only on the initial and inal positions. 9/3/8 3

4 Figure 8. Gravitational work sums to zero in a closed loop. This is also true or projectile motion. The displacement o the projectile has both a horizontal component and a vertical component. However the gravitational orce is only in the vertical direction, so the horizontal motion does not contribute to the work done. Now consider the motion o an object on a surace with a kinetic rictional orce between the object and the surace and denote the coeicient o kinetic riction by μ k. Let s compare two paths rom an initial point x to a inal point x. The irst path is a straight-line path. long this path the work done is just W = F dr = F dx= μ Ns = μ NΔ x< riction x k 1 k path 1 path 1 where the length o the path is equal to the displacement, s 1 (8.3.3) = Δ x. Note that the act that the kinetic riction orce is directed opposite to the displacement is relected in the minus sign in Equation (8.3.3). The second path goes past x some distance and them comes back to x (Figure 8.3). Since the orce o riction always opposes the motion, the work done by riction is negative, W = F dr = F dx= μ Ns <. (8.3.4) riction x k path path The total work depends on the total distance traveled s, and is greater than the displacement s >Δx. The magnitude o the work done along the second path is greater than the magnitude o the work done along the irst path. 9/3/8 4

5 Figure 8.3 Two dierent paths rom x to x. These two examples typiy two undamentally dierent types o orces and their contribution to work. The gravitation orce near the surace o the earth does the same amount o work regardless o the path taken between the initial and inal points. In the case o sliding riction, the work done depends on the path taken. Deinition: Conservative Force Whenever the work done by a orce in moving an object rom an initial point to a inal point is independent o the path, the orce is called a conservative orce. The work done by a conservative orce F c in going around a closed path is zero. Consider the two paths shown in Figure 8.4 that orm a closed path starting and ending at the point with Cartesian coordinates ( 1, ). The work done along path 1 (the upper path in the igure, blue i viewed in color) rom point to point with coordinates (1, ) is given by W = F (1) d r. (8.3.5) path 1 c 1 The work done along path (the lower path, green in color) rom to is given by W = F () d r. (8.3.6) path c For the explicit calculation o the integrals in the igure, where the orce is F= yˆi+ xˆ j (8.3.7) and the paths are quarter-circles, see ppendix 8.. 9/3/8 5

6 Figure 8.4 Two paths in the presence o a conservative orce. The work done around the closed path is just the sum o the work along paths 1 and, W W W d closed path = path 1 + path = Fc () 1 r1 + Fc ( ) dr. (8.3.8) I we reverse the endpoints o path, then the integral changes sign, W = F () dr = F () dr. (8.3.9) path c c We can then substitute Equation (8.3.9) into Equation (8.3.8) to ind that the work done around the closed path is closed path c 1 c W = F (1) dr F () dr. (8.3.1) Since the orce is conservative, the work done between the points o the path, so to is independent F r F r. (8.3.11) c(1) d 1 = c() d 9/3/8 6

7 We now use path independence o work or a conservative orce (Equation (8.3.11) in Equation (8.3.1)) to conclude that the work done by a conservative orce around a closed path is zero, W closed path Deinition: Non-conservative Force = = F c d. (8.3.1) closed path Whenever the work done by a orce in moving an object rom an initial point to a inal point depends on the path, the orce is called a non-conservative orce. 8.4 Changes in Potential Energies o a System Consider an isolated body near the surace o the earth as a system that is initially given a velocity directed upwards. Once the body is released, the gravitation orce, acting as an external orce, does a negative amount o work on the body, and the kinetic energy decreases until the body reaches its highest point, at which its kinetic energy is zero. The gravitation then orce does positive work until the body returns to its initial starting point with a velocity directed downward. I we ignore any eects o air resistance, the descending body will then have the identical kinetic energy as when it was thrown. ll the kinetic energy was completely recovered. Now consider both the earth and the body as a system and assume that there are no other external orces acting on the system. Then the gravitation orce is an internal conservative orce, and does work on both the body and the earth during the motion. s the body moves upward, the kinetic energy o the system decreases, primarily because the body slows down, but there is also an imperceptible increase in the kinetic energy o the earth. The change in kinetic energy o the earth must also be included because the earth is part o the system. When the body returns to its original height (vertical distance rom the surace o the earth), all the kinetic energy in the system is recovered, even though a very small amount has been transerred to the Earth. I we included the air as part o the system, and the air resistance as a non-conservative internal orce, then the kinetic energy lost due to the work done by the air resistance is not recoverable. This lost kinetic energy, which we have called thermal energy, is distributed as random kinetic energy in both the air molecules and the molecules that compose the body (and, to a smaller extent, the earth). We shall deine a new quantity, the change in the internal potential energy o the system, which measures the amount o lost kinetic energy that can be recovered during an interaction. When only internal conservative orces act on the system, the sum o the changes o the kinetic and potential energies o the system is zero. Consider a system consisting o two bodies with masses m1 and m respectively. ssume that there is one conservative orce (internal orce) that is the source o the interaction between two bodies. We denote the orce on body 1 due to the interaction 9/3/8 7

8 with body by F 1, and the orce on body due to the interaction with body 1 by F,1. From Newton s Third Law, F = F 1,,1 The orces acting on the bodies are shown in Figure (8.4.1) Figure 8.5 Internal orces acting on two bodies Choose a coordinate system (Figure 8.6) in which the position vector o body 1 is given by r 1 and the position vector o body is given byr. The relative position o body 1 with respect to body is given by r 1, = r 1 r. During the course o the interaction, body 1 is displaced by d r 1 and body is displaced by d r 1, so the relative displacement o the two bodies during the interaction is given by dr 1, = dr 1 dr. Figure 8.6 Coordinate system or two bodies with relative position vector r = r 1 r In Chapter 7 we observed that the change in the total kinetic energy o a body is equal to the work done by the orces in displacing the body. For two bodies displaced rom an initial state to a inal state, Δ K =Δ K +Δ K = W = F dr + F dr. (8.4.) system 1 c 1, 1,1 9/3/8 8

9 (In Equation (8.4.), the labels and reer to initial and inal states, not paths as in Section 8.3.) From Newton s Third Law, Equation (8.4.1), the sum in Equation (8.4.) becomes Δ K = W = F dr F dr = F ( dr dr ) = F dr system c 1, 1 1, 1, 1 1, 1, where dr = dr dr is the relative displacement o the two bodies. Note that since F = F 1,,1 1, 1 and dr = dr, 1, 1, F dr = F dr 1, 1,,1,1 Deinition: Change in Potential Energy or Two odies Consider a system consisting o two bodies interacting through a conservative orce. Let F 1, denote the orce on body 1 due to the interaction with body and let dr 1, = dr 1 dr be the relative displacement o the two bodies. The change in internal potential energy o the system is deined to be the negative o the work done by the conservative orce when the bodies undergo a relative displacement rom the initial state to the inal state along any displacement that changes the initial state to the inal state, Δ U = W = F dr = F dr system c 1, 1,,1,1. (8.4.3). (8.4.4) Our deinition o potential energy only holds or conservative orces, because the work done by a conservative orce does not depend on the path but only on the initial and inal positions. Since the work done by the conservative orce is equal to the change in kinetic energy, we have that Δ U = Δ K. (8.4.5) system system Recall that the work done by a conservative orce in going around a closed path is zero (Equation (8.3.1)); thereore the change in kinetic energy when a system returns to its initial state is zero. This means that the kinetic energy is completely recoverable. In ppendix 8., we show that the work done by an internal conservative orce in changing a system o two particles o masses m1 and m respectively rom an initial state to a inal state is equal to 9/3/8 9

10 1 W ( c = μ v v ) (8.4.6) where v is the square o the relative velocity in state, v is the square o the relative velocity in state, and μ = mm / m + m is a quantity known as the reduced mass o the system. ( ) 1 1 Change in Potential Energy or Several Conservative Forces When there are several internal conservative orces acting on the closed system we deine a separate change in potential energy or the work done by each conservative orce, where F c, i bodies on which Δ U = W = F dri. (8.4.7) system, i i c, i is a conservative internal orce and dr i a change in the relative positions o the F when the system is changed rom state to state. c, i The total work done is the sum o the work done by the individual conservative orces, W = W + W +. (8.4.8) total c c,1 c, Hence, the sum o the changes in potential energies or the system is the sum total Δ U = Δ U +Δ U +. (8.4.9) system system,1 system, Thereore the total change in potential energy o the system is equal to the negative o the total work done Once again, i the external orces do no work, 8.5 Mechanical Energy total total Δ U = W = d r. (8.4.1) F system c c, i i i Δ K system Deinition: Change in Mechanical Energy = ΔU total system. (8.4.11) The total change in the mechanical energy o the system is deined to be the sum o the changes o the kinetic and the potential energies, 9/3/8 1

11 Δ Emechanical =Δ Ksystem +ΔUsystem. (8.5.1) From Equation (8.4.11), or a closed system (no external orces) with only conservative internal orces, the total change in the mechanical energy is zero, Δ Emechanical =Δ Ksystem +Δ Usystem =. (8.5.) Equation (8.5.) is the symbolic statement o what is called the conservation o mechanical energy. Recall that the work done by a conservative orce in going around a closed path is zero (Equation (8.3.1)), thereore the both the changes in kinetic energy and potential energy are zero when a system returns to its initial state. Throughout the process, the kinetic energy may change into internal potential energy but i the system returns to its initial state, the kinetic energy is completely recoverable. We shall reer to closed system in which processes take place in which only conservative orces act as completely reversible processes. Change o Mechanical Energy or a Closed System with Internal non-conservative Forces Consider a closed system that undergoes a transormation rom an initial state to a inal state by a prescribed set o changes. Suppose the internal orces are both conservative and non-conservative. The total work done by the internal orces is a sum o the internal total conservative work, which is path-independent, and the internal non-conservative work total W nc, internal W c, internal, which is path-dependent, W = W + W total total total internal c, internal nc, internal. (8.5.3) The work done by the internal conservative orces is equal to the negative o the change in the total internal potential energy Substituting Equation (8.5.4) into Equation (8.5.3) yields total total Δ U = W c, internal. (8.5.4) W = Δ U + W total total total internal nc, internal. (8.5.5) Since the system is closed, the total work done is equal to the change in the kinetic energy, Δ K = W = Δ U + W total total total system internal nc, internal. (8.5.6) 9/3/8 11

12 We can now substitute Equation (8.5.6) into our expression or the change in the mechanical energy, Equation (8.5.1), with the result ΔE Δ K +Δ U = W total total total mechanical system system nc, internal. (8.5.7) The mechanical energy is no loner constant. The total change in energy o the system is zero, Δ E =ΔE W = total total total system mechanical nc, internal. (8.5.8) W nc, internal Energy is conserved but some mechanical energy has been transerred into nonrecoverable energy. We shall reer to processes in which there is non-zero total nonrecoverable energy as irreversible processes. Change o Mechanical Energy or a System in Contact with Surroundings When the system is no longer closed but in contact with its surroundings, the total change in energy is zero, so rom Equation (8.1.1) the total change in energy o the system is equal to the negative o the change in energy rom the surroundings, Δ E = Δ E (8.5.9) system surroundings The energy rom the surroundings can be the result o external work done by the surroundings on the system or by the system on the surroundings, W total ext total = F ext d r. (8.5.1) This work will result in the system undergoing coherent motion. I the system is in thermal contact with the surroundings, then thermal energy Δ Ethermal can low into or out o the system. This will result in either an increase or decrease in random thermal motion o the molecules inside the system. There may also be other orms o energy that enter the system, or example radiative energy Δ E. radiative Several questions naturally arise rom this set o deinitions and physical concepts. Is it possible to identiy all the conservative orces and calculate the associated changes in potential energies? How do we account or non-conservative orces such riction that act at the boundary o the system? 8.6 Examples: Change in Potential Energy In Chapter 7, Examples 7.4.4, and 7.6., we calculated the work done by dierent conservative orces: constant gravity near the surace o the earth, the spring orce, and 9/3/8 1

13 the universal gravitational orce. We chose the system in each case so that the conservative orce was an external orce. In each case, there was no change o potential energy and the work done was equal to the change o kinetic energy, W total ext =ΔK system. (8.6.1) We now treat each o these conservative orces as internal orces and calculate the change in potential energy according to our deinition Δ U = W = F dr. (8.6.) system c c We shall also choose a zero reerence potential or the potential energy o the system, so that we can consider all changes in potential energy relative to this reerence potential Example: Change in Gravitational Potential Energy Near the Surace o the Earth Let s consider the example o a body alling near the surace o the earth. Choose our system to consist o the earth and the body. The gravitational orce is now an internal conservative orce acting inside the system. The initial and inal state are speciied by the distance separating the body and the center o mass o the earth, and the velocities o the earth and the body. The change in kinetic energy between the initial and inal states or the system is Δ Ksystem =Δ Kearth +ΔKbody, (8.6.3) Δ K = m v m v + m v m v ( ) ( ) ( ) ( ) system e earth, e earth, b body, b body,.(8.6.4) The change o kinetic energy o the earth due to the gravitational interaction between the earth and the body is negligible. The total change in kinetic energy o the system is approximately equal to the change in kinetic energy o the body, 1 1 ΔKsystem Δ Kbody = mb ( vbody, ) mb ( vbody, ). (8.6.5) Let s choose a coordinate system with the origin on the surace o the earth and the + y - direction pointing away rom the center o the earth. Since the displacement o the earth is negligible, we need only consider the displacement o the body in order to calculate the change in potential energy o the system. brie outline o a proo o this obvious statement is given in ppendix 8.C. 9/3/8 13

14 Suppose the body starts at an initial height y above the surace o the earth and ends at inal height y. The gravitation orce on the body is given by F gravity = mg= F ˆj= mg ˆj. (8.6.6) gravity, y The work done by the gravitational orce on the body is then W = F Δ y = mgδ y, (8.6.7) gravity gravity, y which is o course the same result as ound in Equation (7.4.16). The change in potential energy is then given by Δ U = W = mgδ y = mg y mg y system gravity Choice o Zero Point Reerence or Potential Energy We introduce a potential energy unction U so that. (8.6.8) ΔUsystem U U. (8.6.9) Only dierences in the unction U have a physical meaning. We can choose a zero reerence point or the potential energy anywhere we like, since change in potential energy only depends on the displacement, Δ y (in general, the change in coniguration). We have some lexibility to adapt our choice o zero or the potential energy to best it a particular problem. In the above expression or the change o potential energy, let y y be an arbitrary point and y = be the origin. In addition, we choose the zero reerence potential or the potential energy to be at the surace o the earth corresponding to our origin, U( y = ) =. Then Δ U = U( y) U( y = ) = U( y) = mg y, (8.6.1) U( y) = mg y, with U( y = ) =. (8.6.11) We now deine the mechanical energy unction or the system 1 E ( ) mechanical = K + U = mb vbody + mg y, with U( y = ) =, (8.6.1) where K is the kinetic energy and U is the potential energy. The change in mechanical energy is then ( ) ( ) ΔE E E = K + U K +U. (8.6.13) mechanical mechanical, mechanical, 9/3/8 14

15 When the work done by the external orces is zero and there are no internal nonconservative orces, the total mechanical energy o the system is constant, E mechanical, = E, (8.6.14) mechanical, or equivalently ( K U ) ( K U) + = +. (8.6.15) 8.6. Example: Hooke s Law Spring-ody System Consider a spring-body system lying on a rictionless horizontal surace with one end o the spring ixed to a wall and the other end attached to a body o mass m (Figure 8.6). The spring orce is an internal conservative orce. The wall exerts an external orce on the spring-body system but since the point o contact o the wall with the spring undergoes no displacement this external orce does no work. Figure 8.6 spring-body system. Choose the origin at the position o the center o the body when the spring is relaxed (the equilibrium position). Let x be the displacement o the body rom the origin. We choose the +i ˆ unit vector to point in the direction the body moves when the spring is being stretched (to the right o x = in the igure). The spring orce on a mass is then given by The work done by the spring orce on the mass is x= x F= F i = kx ˆ i. (8.6.16) x ˆ 1 W = kx dx= k x x. (8.6.17) ( ) ( ) spring x= x 9/3/8 15

16 This is o course the result obtained in Equation (7.6.6). We then deine the change in potential energy in the spring-body system in moving the body rom an initial position x rom equilibrium to a inal position x rom equilibrium by ΔUspring Uspring( x) Uspring ( x) = Wspring = k( x x ) 1. (8.6.18) Thereore an arbitrary stretch or compression o a spring-body system rom equilibrium x = to a inal position x = x changes the potential energy by 1 ( ) ( )= Δ U = Uspring x Uspring x kx. (8.6.19) For the spring-body system, there is an obvious choice o position where the potential energy is zero, the equilibrium position o the spring- body, U ( x=). (8.6.) spring Then with this choice o zero reerence potential, the potential energy unction is given by 1 Uspring( x)= kx, with Uspring( x=). (8.6.1) Example: Inverse Square Gravitational Force Consider a system consisting o two bodies o masses m1 and m that are separated by a center-to-center distance r. The internal gravitational orce between the two bodies is given by F Gm m = ˆ r. (8.6.) r 1 grav The work done by this gravitational orce in moving the two bodies rom an initial position in which the center o mass o the two bodies are a distance apart to a inal position in which the center o mass o the two bodies are a distance r r apart is given by W r r Gm1m = F d r = r r r dr. (8.6.3) Evaluating this integral, we have 9/3/8 16

17 W = dr = = Gm m r. (8.6.4) r r Gm1m Gm1m 1 1 r 1 r r r r Thereore the change in potential energy o the system is 1 1 Δ Ugravity = Wgravity = Gm1 m r r (8.6.5) analogous to the result given in Equation (7.6.9). We now choose our reerence point or the zero o the potential energy to be at ininity, r =, with the choice that Ugravity( r = ). y making this choice, the term 1/ r in the expression or the change in potential energy vanishes when r =. The gravitational potential energy unction or the two bodies when their center o mass to center o mass distance is r = r becomes Gm1m Ugravity( r) =, with Ugravity( r = ). (8.6.6) r There s a subtle aspect to the above calculation. In Equation (8.6.), our convention is that d r is the dierential change in the relative separation o the bodies, but we don t speciy which body moves, or i they both do. From the inal result, Equation (8.6.6), we might correctly iner that it doesn t matter. lso, we have used the act that ˆr dr = dr; the component o d r in the radial ˆr -direction is the scalar change dr in the separation. Lastly, watch all the minus signs. In Equation (8.6.3), as r, it might appear that the improper integral should need one more minus sign, but in this case dr < and the calculus does the sign accounting or us. For a derivation o the gravitational potential energy or a spherical shell, one that avoids vector calculus altogether, see ppendix 8.D. 8.7 Spring Force Energy Diagram The spring orce on a body is a restoring orce F= F i = kxi ˆ x ˆ where we choose a coordinate system with the equilibrium position at x = and x is the amount the spring has been stretched ( x > ) or compressed ( x < ) rom its equilibrium position. The negative o the work done by the spring orce in moving the body rom x to x deines the potential energy dierence 9/3/8 17

18 x 1 U( x) U( x) = Fx dx= k( x x ). (8.7.1) x The irst undamental theorem o calculus states that x du U( x) U( x) = dx. (8.7.) x dx Comparing Equation (8.7.1) with Equation (8.7.) shows that the orce is the negative derivative (with respect to position) o the potential energy, F x du ( x) =. (8.7.3) dx Choose the zero reerence point or the potential energy to be at the equilibrium position, U( x =). Then the potential energy unction becomes U( x) From this, we obtain the spring orce law as 1 = kx. (8.7.4) F du ( x) d 1 dx dx x = = k = x kx. (8.7.5) In Figure 8.7 we plot the potential energy unction or the spring orce as unction o x with U( x =) (the units are arbitrary). Figure 8.7 Graph o potential energy unction or the spring. The minimum o the potential energy unction occurs at the point where the irst derivative vanishes 9/3/8 18

19 From Equation (8.7.4), the minimum occurs at x =, du ( x) =. (8.7.6) dx du ( x) = = kx. (8.7.7) dx Since the orce is the negative derivative o the potential energy, and this derivative vanishes at the minimum, we have that the spring orce is zero at the minimum x = agreeing with our orce law, F = kx =. x x= x= Suppose the potential energy unction has positive curvature in the neighborhood o the minimum. I the body is extended a small distance x > away rom equilibrium, the slope o the potential energy unction is positive, du ( x) dx > ; hence the component o the orce is negative since Fx = du( x) dx<. Thus the body experiences a restoring orce towards the minimum point o the potential. I the body is compresses with x < then du ( x) dx < ; the component o the orce is positive, Fx = du( x) dx>, and the body again experiences a orce back towards the minimum o the potential energy as in Figure 8.8. Figure 8.8 Stability diagram or the spring orce. Suppose our spring-body system has no loss o mechanical energy due to dissipative orces such as riction or air resistance. The total mechanical energy at any time will then be the sum o the kinetic energy K( x ) and the potential energy U( x) 9/3/8 19

20 E = K( x) + U( x). (8.7.8) oth the kinetic energy and the potential energy are unctions o the position o the body with respect to equilibrium. The energy is a constant o the motion and with our choice o U( x =), the energy can be either a positive value or zero. When the energy is zero, the body is at rest at the equilibrium position. In Figure 8.8, we draw a straight horizontal line corresponding to a non-zero positive value or the energy on the graph o potential energy as a unction o x. The energy intersects the potential energy unction at two points { xmax, x max} with x max >. These points correspond to the maximum compression and maximum extension o the spring, which are called the turning points. The kinetic energy is the dierence between the energy and the potential energy, K( x) = E U( x). (8.7.9) t the turning points, where E = U( x), the kinetic energy is zero. Regions where the kinetic energy is negative, x < xmax or x > xmax are called the classically orbidden regions, which the body can never reach i subject to the laws o classical mechanics. In quantum mechanics, there is a very small probability that the body can be ound in the classically orbidden regions. 8.8 Worked Examples Example Escape Velocity Toro The asteroid Toro, discovered in 1964, has a radius o about R = 5.km and a mass o about m Toro = kg. Let s assume that Toro is a perectly uniorm sphere. What is the escape velocity or an object o mass this speed (on earth) by running? Solution: m on the surace o Toro? Could a person reach The only potential energy in this problem is the gravitational potential energy. We choose the zero point or the potential energy to be when the object and Toro are an ininite distance apart, U gravity (r = ). With this choice, the potential energy when the object and Toro are a inite distance r apart is given by U GmToro m gravity ()= r (8.8.1) r 9/3/8

21 with U gravity (r = ). The expression escape velocity reers to the minimum speed necessary or an object to escape the gravitational interaction o the asteroid and move o to an ininite distance away. I the object has a speed less than the escape velocity, it will be unable to escape the gravitational orce and must return to Toro. I the object has a speed greater than the escape velocity, it will have a non-zero kinetic energy at ininity. The condition or the escape velocity is that the object will have exactly zero kinetic energy at ininity. We choose our initial state, at time t, when the object is at the surace o the asteroid with speed equal to the escape velocity. We choose our inal state, at time t, to occur when the separation distance between the asteroid and the object is ininite. Initial Energy: The initial kinetic energy is K = 1 mv esc. The initial potential energy is U = G m m Toro, and so the initial mechanical energy is R E = K + U = 1 mv esc Final Energy: The inal kinetic energy is G m Toro m R. (8.8.) K =, since this is the condition that deines the escape velocity. The inal potential energy is zero, U = since we chose the zero point or potential energy at ininity. The inal mechanical energy is then E = K + U =. (8.8.3) There is no non-conservative work, so the change in mechanical energy = W =Δ E, (8.8.4) nc mech is then = 1 mv esc This can be solved or the escape velocity, G m Toro m R. (8.8.5) 9/3/8 1

22 v esc = Gm R Toro ( )( ) 3 ( 5. 1 m) N m kg. 1 kg = = 7.3m s 1. (8.8.6) Considering that Olympic sprinters typically reach velocities o1 m s 1, this is an easy speed to attain by running on earth. It may be harder on Toro to generate the acceleration necessary to reach this speed by pushing o the ground, since any slight upward orce will raise the runner s center o mass and it will take substantially more time than on earth to come back down or another push o the ground. Example 8.8. Spring-Loop-the-Loop small block o mass m is pushed against a spring with spring constant k and held in place with a catch. The spring compresses an unknown distance x. When the catch is removed, the block leaves the spring and slides along a rictionless circular loop o radius r. When the block reaches the top o the loop, the orce o the loop on the block (the normal orce) is equal to twice the gravitational orce on the mass. a) Using conservation o energy, ind the kinetic energy o the block at the top o the loop. b) Using Newton s Second Law, derive the equation o motion or the block when it is at the top o the loop. Speciically, ind the speed in terms o the gravitational constant g and the loop radius r. c) How much (what distance) was the spring compressed? Solution: a) Initial Energy: Choose or the initial state the instant beore the catch is released. The initial kinetic energy is K =. The initial potential energy is non-zero, U = ( 1) kx. The initial mechanical energy is then v top 9/3/8

23 1 ( ) E = K + U = kx. (8.8.7) Final Energy: Choose or the inal state the instant the block is at the top o the loop. The 1 inal kinetic energy is K = mvtop ; the mass is in motion with speed v top. The inal potential energy is non-zero, U = mg R. The inal mechanical energy is then ( )( ) 1 E = K + U = ( mg) R+ mvtop. (8.8.8) Non-conservative Work: Since we are assuming the track is rictionless, there is no nonconservative work. Change in Mechanical Energy: The change in mechanical energy is thereore zero,. (8.8.9) = W nc =ΔE mechanical = E E Mechanical energy is conserved,, or E = E 1 1 ( mg ) R + mvtop = k x. (8.8.1) You may note that we pulled a ast one, sort o, in that both Equations (8.8.7) and (8.8.8) assumed that the gravitational potential energy is zero at the bottom o loop. I we had set the vertical height h above the bottom o the track to correspond to zero gravitational potential energy to be someplace else, we would have needed to subtract mgh rom both equations, but this term would cancel in Equation (8.8.9) and Equation (8.8.1) remains the same. From Equation (8.8.1), the kinetic energy at the top o the loop is 1 1 mvtop = k x ( mg) R. (8.8.11) b) t the top o the loop, the orces on the block are the gravitational orce o magnitude mg and the normal orce o magnitude N, both directed down. Newton s Second Law in the radial direction, which is the downward direction, is mv top mg N =. (8.8.1) R 9/3/8 3

24 In this problem, we are given that when the block reaches the top o the loop, the orce o the loop on the block (the normal orce, downward in this case) is equal to twice the weight o the block, N = mg. The Second Law, Equation (8.8.1), then becomes mvtop 3mg =. (8.8.13) R We can rewrite Equation (8.8.13) in terms o the kinetic energy as 3 1 mg R = mvtop. (8.8.14) c) Combing Equations (8.8.11) and (8.8.14) yields 7 mg R = 1 kx. (8.8.15) Thus the initial displacement o the spring rom equilibrium was Example Mass-Spring on a Rough Surace 7mg R x =. (8.8.16) k block o mass m slides along a horizontal table with speed v. t x = it hits a spring with spring constant k and begins to experience a riction orce. The coeicient o riction is variable and is given by μ = bx, where b is a positive constant. Find the loss in mechanical energy when the block irst momentarily comes to rest. Solution: From the model given or the rictional orce, we could ind the nonconservative work done, which is the same as the loss o mechanical energy, i we knew the position x 9/3/8 4

25 where the block irst comes to rest. The most direct (and easiest) way to ind the work-energy theorem. x is to use The initial mechanical energy is E = mv / and the inal mechanical energy is E = kx / (note that there is no potential energy term in E and no kinetic energy term in E ). The dierence between these two mechanical energies is the nonconservative work done by the riction orce, x= x x= x x= x W = F dx = F dx = μ N dx nc nc riction x= x= x= x 1 = b x mg dx = bmg x. (8.8.17) We then have that Δ E = W mech E E = W nc nc kx mv = bmgx. (8.8.18) Solving the last o these equations or x gives x = mv k + bmg (8.8.19) and substitution into Equation (8.8.17) gives the result W bmg mv mv k = = + k+ bmg bmg. (8.8.) nc 1 1 It is worth checking that the above result is dimensionally correct. From the model, the parameter b must have dimensions o inverse length (the coeicient o riction μ must be dimensionless), and so the product bmg has dimensions o orce per length, as does the spring constant k ; the result is dimensionally consistent. 9/3/8 5

26 9/3/8 6