# Review D: Potential Energy and the Conservation of Mechanical Energy

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physics 8. Review D: Potential Energy and the Conservation o Mechanical Energy D.1 Conservative and Non-conservative Force... D.1.1 Introduction... D.1. Conservative and Non-Conservative Forces... 3 D. Potential Energy... 4 D..1 Change in Potential Energy... 4 D.. Deinition: Change in Potential Energy... 5 D..3 Several Conservative Forces... 6 D..4 Change in Mechanical Energy and Conservation o Mechanical Energy.. 6 D..5 Non-Conservation o Mechanical Energy... 7 D.3 Worked Examples: Calculation o the Change in Potential Energy... 8 D.3.1 Example 1: Change in Gravitational Potential Energy Uniorm Gravity 8 D.3. Example : Hooke s Law Spring-mass system... 9 D.3.3 Example 3: Inverse Square Gravitational Force... 1 D.4 Energy Diagrams: One-Dimensional Example: Spring Forces D-1

2 Potential Energy and the Conservation o Mechanical Energy D.1 Conservative and Non-conservative Force D.1.1 Introduction A physical system consists o a well-deined set o bodies that are interacting by means o orces. Any bodies that lie outside the boundary o the system reside in the surroundings. A state o the system is a set o measurable physical quantities that completely characterize the system. Figure 1 shows this division into system, boundary, and surroundings. Figure D.1.1: system, boundary, and surroundings. Up to now we have analyzed the dynamical evolution in time o our system under the action o orces using Newton s Laws o Motion. We shall now introduce the concept o Conservation o Energy in order to analyze the change o state o a system. Deinition: Change o Energy The change in energy o a system and its surroundings between the inal state and the initial state is zero, E = Esystem + Esurroundings = (D.1.1) Our quest is then to identiy experimentally every type o change o energy or all physical processes and veriy that energy is conserved. Can we really play this zero sum game? Is there any physical content to this concept o change o energy? The answer is that experimentally we can identiy all the changes in energy. One important point to keep in mind is that i we add up all the changes in energy and do not arrive at a zero sum then we have an open scientiic problem: ind the missing change in energy! Our irst example o this type o energy accounting involves mechanical energy. There will be o 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. D-

3 D.1. Conservative and Non-Conservative Forces We have deined the work done by a orce F on an object which moves rom an initial point x to a inal point x (with displacement x x x ), as the product o the component o the orce with the displacement x, Fx W = Fx x (D.1.) Does the work done on the object by the orce depend on the path taken by the object? For a orce such as riction we expect that the work should depend on the path. Let s compare two paths rom an initial point x to a inal point x. The irst path is a straightline path with no reversal o direction. The second path goes past x some distance and them comes back to x (Figure ). Since the orce o riction always opposes the motion, the work done by riction is negative, W = N x< (D.1.3) riction µ k Thereore the work depends on the distance traveled, not the displacement. Hence, the magnitude o the work done along the second path is greater than the magnitude o the work done along the irst path. Figure D.1. Two dierent paths rom x to x Let s consider the motion o an object under the inluence o a gravitational orce near the surace o the earth. The gravitational orce always points downward, so the work done by gravity only depends on the change in the vertical position, W = F y = mg y (D.1.4) grav grav,y Thereore, when an object alls, the work done by gravity is positive and when an object rises 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, (Figure 3). The 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. D-3

4 Figure D.1.3 Gravitational work cancels. These two examples typiy two undamentally dierent types o orces and their contribution to work. In the irst example o sliding riction, the work done depends on the path. Deinition: Non-conservative Force Whenever the work done by a orce in moving an object rom an initial point to a inal point depends on the path, then the orce is called a non-conservative orce. In the second example o ree all, the work done by the gravitational orce is independent o the path. 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, then the orce is called a conservative orce. D. Potential Energy D..1 Change in Potential Energy Let s consider an example in which only conservative orces are acting on a system, or example the gravitational orce considered above. The key concept or a conservative orce such as gravitation is that the work done by the gravitational orce on an object only depends on the change o height o the object. Instead o analyzing the orces acting on an object, we introduce the idea o changes in the kinetic and potential energy o a physical system. This allows us use introduce conservation o energy to describe motion. In the case o an object alling near the surace o the earth, our system consists o the earth and the object and the state is described by the height o the object above the surace o the earth. The initial state is at a height above the surace o the earth and the inal state is at a height y y above the surace o the earth. The change in kinetic energy between the initial and inal states or both the earth and the object is D-4

5 K = K + K = mv mv + mv mv system earth object earth, earth, object, object, (D..1) The change o kinetic energy o the earth due to the gravitational interaction between the earth and the object is negligible. So the change in kinetic energy o the system is approximately equal to the change in kinetic energy o the object, 1 1 K K = mv mv system object object, object, (D..) We will enlarge our idea o change o energy by deining a new type o change o energy, a change o gravitational potential energy U, so that the change o both kinetic and potential energy is zero, system K + U = (D..3) system system In what ollows, we ll look at arbitrary conservative orces to obtain a general result, and then return to the example o gravity, along with other perhaps amiliar examples. D.. Deinition: Change in Potential Energy The change in potential energy o a body associated with a conservative orce is the negative o the work done by the conservative orce in moving the body along any path connecting the initial position to the inal position. U = W = F dr system cons cons A (D..4) where A and are use to represent the initial and inal positions o the body. This deinition only holds or conservative orces because the work done by a conservative orce does not depend on the path and only depends on the initial and inal positions. The work-kinetic energy theorem states that i the work done by the conservative orce is the only work done, and hence is the work, then that work is equal to the change in kinetic energy o the system; = = Ksystem W Wcons (D..5) The change in mechanical energy is deined to be the sum o the changes o the kinetic and the potential energies, D-5

6 Emechanical = Ksystem + Usystem (D..6) From both the deinition o change o potential energy, Usystem = Wcons, and the workkinetic energy theorem, the change in the mechanical energy is zero, E = W + ( W ) = mechanical cons cons (D..7) This expression is known as the conservation o mechanical energy. D..3 Several Conservative Forces When there are several conservative orces acting on the system we deine a separate change in potential energy or the work done by each conservative orce, Usystem, i = Wi = F cons, i dr A (D..8) The work done is just the sum o the individual work, W = W + W + (D..9) cons cons,1 cons, The sum o the change in potential or the system is U = U + U + system system,1 system, (D..1) Thereore, the change in potential energy o the system is equal to the negative o the work done U = = dr system Wcons F cons, i i A (D..11) D..4 Change in Mechanical Energy and Conservation o Mechanical Energy We now deine the mechanical energy unction or a system, where K is the kinetic energy and U is the potential energy. Emechanical = K + U (D..1) Then the change in mechanical energy when the system goes rom an initial state to a inal state is D-6

7 E E E = ( K + U ) ( K + U ) mechanical mechanical, mechanical, (D..13) When there are no non-conservative orces acting on the system, the mechanical energy o the system is conserved, and E mechanical, = E (D..14) mechanical, or equivalently ( K + U ) = ( K + U ) (D..15) D..5 Non-Conservation o Mechanical Energy The orce acting on a system may, in general, consist o a vector sum o conservative orces and non-conservative orces, = + j F F F cons, i non-cons, i j (D..16) The work done by the non-conservative orce, F non-cons = F j non-cons, j (D..17) between two points A and is a path-dependent quantity which we denote by Wnon-cons [ A, ] = F non-cons d r Then the work done by the orce is W [ A, ] = F dr = Fcons, i + Fnon-cons, j dr A A i j = F dr+ W A, i A cons, i A non-cons [, ] U Wnon-cons A = + [ ] (D..18) (D..19) The work-kinetic energy theorem states that the work done by the orce is equal to the change in kinetic energy, 1 1 W = K = mv mv (D..) D-7

8 Thus [ ] = + (D..1) K U Wnon-cons A, We deine the change in mechanical energy o the system, between the states represented by A and, [ ] Emechanical A, K + U (D..) as the sum o the change o the kinetic energy and change o the potential energy. E A is independent o the path between the states A and. This change [ ] mechanical, Thereore the change in the mechanical energy o the system between the states A and is equal to the work done by the non-conservative orces along the chosen path connecting the points A and, [ ] E = W A (D..3) mechanical non-cons, When there are no non-conservative orces acting on the system, then the mechanical energy o the system is conserved, E mechanical = (D..4) D.3 Worked Examples: Calculation o the Change in Potential Energy There are our examples o conservative orces or which we will calculate the work done by the orce; constant gravity near the surace o the earth, the ideal spring orce, the universal gravitational orce between two point-like objects, and the electric orce between charges (this last calculation will not be presented in this review). For each o these orces we can describe and calculate the change in potential energy according to our deinition U = W = F dr system cons cons A (D.3.1) In addition, we would like to choose a zero point or the potential energy. This is a point where we deine potential energy to be zero and all changes in potential energy are made relative to this point. This point will in general be dierent or dierent orces. D.3.1 Example 1: Change in Gravitational Potential Energy Uniorm Gravity Let s choose a coordinate system with the origin at the surace o the earth and the + y - direction pointing away rom the center o the earth. Suppose an object o mass m moves D-8

9 y rom an initial point to a inal point y. When an object is placed in the gravitational ield near the surace o the earth, the gravitational orce on the object is given by F = mg= F ˆj= mgĵ grav grav, y The work done by the gravitational orce on the object is then grav grav, y (D.3.) W = F y = mg y (D.3.3) and the change in potential energy is given by U = W = mg y = mg y mg y system grav (D.3.4) We will introduce a potential energy unction U given by Usystem U U. Suppose the mass starts out at the surace o the earth with y =. Then the change in potential energy is U U( y ) U( y = ) = mg y mg y system (D.3.5) We are ree to choose the zero point or the potential energy anywhere we like since change in potential energy only depends on the displacement, y. 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. Then we can choose the zero reerence point or the potential energy to be at the origin, ( ) ( ) U( y) = mg y or U y = mgy with U y = = (D.3.6) D.3. Example : Hooke s Law Spring-mass system Consider a spring-object system lying on a rictionless horizontal surace with one end o the spring ixed to a wall and the other end attached to a object o mass m. Choose the origin at the position o the center o the object when the spring is unstretched, the equilibrium position. Let x be the displacement o the object rom the origin. We choose the +i ˆ unit vector to point in the direction the object moves when the spring is being stretched. Then the spring orce on a mass is given by F = F i = kxi ˆ x ˆ (D.3.7) D-9

10 The work done by the spring orce on the mass is W x= x 1 = ( kx ) dx = ( ) x x k x x (D.3.8) = Thereore the change in potential energy in the spring-object system in moving the object rom an initial position x rom equilibrium to a inal position x rom equilibrium is 1 Uspring Uspring ( x) Uspring ( x) = Wspring = k( x x ) (D.3.9) For the spring- object system, there is an obvious choice o position where the potential energy is zero, the equilibrium position o the spring- object, U ( x=) (D.3.1) spring Then with this choice o zero point x =, with x = x, an arbitrary extension or compression o a spring- object system, the potential energy unction is given by 1 ( )= Uspring x k x (D.3.11) D.3.3 Example 3: Inverse Square Gravitational Force Consider two objects o masses m 1 and m that are separated by a distance r. The gravitational orce between the two objects is given by F Gm m r 1 ˆ m1, m = r center-to-center (D.3.1) where ˆr is the unit vector directed along the line joining assumes that the objects may be approximated as spherical. the masses. This expression The work done by this gravitational orce in moving the two objects rom an initial position in which the center o mass o the two objects are a distanc e r apart to a inal position in which the center o mass o the two objects are a distance r apart is given by W r r Gm1m = F dr = r r dr r (D.3.13) Upon evaluation o this integral, we have or the work D-1

11 r r Gm1m Gm1m 1 1 W = dr Gm r = = 1m r r r r r (D.3.14) Thereore the change in potential energy o the system is 1 1 U grav Ugrav(r ) Ugrav(r ) = W = gravity Gmm 1 r r (D.3.15) We now choose our reerence point or the zero o the potential energy to be at ininity, r =, U g rav (r = ) (D.3.16) The reason or this choice is that the term 1 r in the expression or the change in potential energy vanishes when r =. Then the gravitational potential energy unction or the two objects wh en their center o mass to center o mass distance is r = r, becomes 1 U, grav (r) = Gm m r D.4 Energy Diagrams: One-Dimensional Example: Spring Forces From the above derivation o the potential energy o a compressed or extended ideal sp ring o spring constant k, we have 1 Usprin g ( x) Uspring ( x) = Wspring = k( x x ) (D.4.1) Taking x = and setting Uspring( x)=, and dropping the subscript spring, we have the simpliied It ollows immediately that U( x) 1 = k x (D.4.) d U( x) = k x= Fx (D.4.3) dx D-11

12 and indeed, this expression ollows immediately rom the deinition o potential energy in terms o the orce in one dimension, regardless o the unctional orm o F x In Figure D.4.1 we plot the potential energy unction or the spring orce as unction o with U( x =). x Figure D.4.1: graph o potential energy unction or the spring The minimum o the potential energy unction occurs at the point where the irst derivative vanishes For the spring orce du ( x) = (D.4.4) dx du ( x) = = kx (D.4.5) dx implies that the minimum occurs at x = (the second derivative that this point is a minimum). d U/ dx = k > shows Since the orce is the negative derivative o the potential energy, and this derivative necessarily vanishes at the minimum o the potential, we have that the spring orce is zero at the minimum x = agreeing with our orce law, F = k x =. x x= x= In general, i the potential energy unction has a minimum at some point then the orce is zero at that point. I the object is extended a small distance x > away rom equilibrium, the slope o the poten tial energy unction is positive, du ( x) dx > ( d U/ dx > or a minimum); hence the component o the orce is negative since Fx = du( x) dx<. Thus, the object experiences a restoring orce directed towards the minimum point o the potential. I the object is compresses with x < then du ( x) dx < ; the component o D-1

13 the orce is positive, Fx = du( x) dx>, and the object again experiences a orce back towards the minimum o the potential energy (Figure D.4.). Figure D.4. Stability Diagram Suppose our spring-object system has no loss o mechanical energy due to dissipative orces such as riction or air resistance. The energy at any time is the sum o the kinetic energy K( x ) and the potential energy U( x ) E = K( x) + U( x) (D.4.6) oth the kinetic energy and the potential energy are unctions o the position o the object with respect to equilibrium. The energy is a constant o the motion and with our choice o U( x =) the energy can be positive or zero. When the energy is zero, the object is at rest at its equilibrium position. In Figure 5, we draw a straight line corresponding to a positive value or the energy on the graph o potential energy as a unction o x. The line corresponding to the energy intersects the potential energy unction at two points x, x with x max >. These points correspond to the maximum compression and { } max max maximum extension o the spring. They are called the classical turning points. The kinetic energy is the dierence between the energy and the potential energy, at the turning points, where ( ) E U( x) K x ( ) = (D.4.7) 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. The object can never reach these regions classically. In quantum mechanics, there is a very small probability that the object can be ound in the classically orbidden regions. D-13

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