Chapter 7 – Potential Energy And Conservation Of Energy

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Chapter 7 – Potential energy and conservation of energyI.Potential energy Energy of configurationII. Work and potential energyIII. Conservative / Non-conservative forcesIV. Determining potential energy values:- Gravitational potential energy- Elastic potential energyI.V. Conservation of mechanical energyVI. External work and thermal energyVII. External forces and internal energy changesVIII. PowerI. Potential energyEnergy associated with the arrangement of a system of objects that exertforces on one another.Units: JExamples:- Gravitational potential energy: associated with the state of separationbetween objects which can attract one another via the gravitational force.- Elastic potential energy: associated with the state ofcompression/extension of an elastic object.II. Work and potential energyIf tomato rises gravitational force transfers energy “from”tomato’s kinetic energy “to” the gravitational potentialenergy of the tomato-Earth system.If tomato falls down gravitational force transfers energy“from” the gravitational potential energy “to” the tomato’skinetic energy.1

U WAlso valid for elastic potential energySpring compressionSpring force does –W on block energytransfer from kinetic energy of the block topotential elastic energy of the spring.fsSpring extensionfsSpring force does W on block energy transfer from potential energyof the spring to kinetic energy of theblock.General:- System of two or more objects.- A force acts between a particle in the system and the rest of the system.- When system configuration changes force does work on the object(W 1) transferring energy between KE of the object and some other form ofenergy of the system.- When the configuration change is reversed force reverses the energytransfer, doing W 2.III. Conservative / Nonconservative forces- If W 1 W 2 always conservative force.Examples: Gravitational force and spring force associated potentialenergies.- If W 1 W 2 nonconservative force.Examples: Drag force, frictional force KEenergy. Non-reversible process.transferred into thermal- Thermal energy: Energy associated with the random movement of atomsand molecules. This is not a potential energy.2

- Conservative force: The net work it does on a particle moving aroundevery closed path, from an initial point and then back to that point is zero.- The net work it does on a particle moving between two points does notdepend on the particle’s path.Conservative force W ab,1 W ab,2Proof:W ab,1 W ba,2 0 W ab,1 -W ba,2W ab,2 - W ba,2 W ab,2 W ab,1IV. Determining potential energy valuesxfW x F ( x)dx UForce F is conservativeiGravitational potential energy:yfyiyfyi U ( mg )dy mg [ y ] mg ( y f yi ) mg yU i 0, yi 0 U ( y ) mgyChange in the gravitationalpotential energy of theparticle-Earth system.Reference configurationThe gravitational potential energy associated with particle-Earth systemdepends only on particle’s vertical position “y” relative to the referenceposition y 0, not on the horizontal position.xfElastic potential energy: U ( kx)dx xik 2x2[ ]xfxi 1 2 1 2kx f kxi22Change in the elastic potential energy of the spring-block system.Reference configuration when the spring is at its relaxed length and theblock is at xi 0.1U i 0, xi 0 U ( x) kx 22Remember! Potential energy is always associated with a system.V. Conservation of mechanical energyMechanical energy of a system: Sum of its potential (U) and kinetic (K)energies.3

Emec U KAssumptions:- Only conservative forces cause energy transfer withinthe system.- The system is isolated from its environment No external force from anobject outside the system causes energy changes inside the system.W KW U K U 0 ( K 2 K1 ) (U 2 U1 ) 0 K 2 U 2 K1 U1 Emec K U 0- In an isolated system where only conservative forces cause energychanges, the kinetic energy and potential energy can change, but theirsum, the mechanical energy of the system cannot change.- When the mechanical energy of a system is conserved, we can relatethe sum of kinetic energy and potential energy at one instant to that atanother instant without considering the intermediate motion and withoutfinding the work done by the forces involved.yEmec constantx Emec K U 0K 2 U 2 K1 U1Potential energy curvesFinding the force analytically: U ( x) W F ( x) x F ( x) dU ( x)(1D motion)dx- The force is the negative of the slope of the curve U(x) versus x.- The particle’s kinetic energy is:K(x) Emec – U(x)4

Turning point: a point x atwhich the particle reversesits motion (K 0).K always 0 (K 0.5mv2 0 )Examples:x x1 Emec 5J 5J K K 0x x1 Emec 5J 5J K K 0 impossibleEquilibrium points: where the slope of the U(x) curve is zero F(x) 0 U -F(x) dx U/dx -F(x) U(x)/dx -F(x) SlopeEquilibrium pointsEmec,1Emec,2Emec,3Example:x x5 Emec,1 4J 4J K K 0 and also F 0 x5 neutral equilibriumx2 x x1, x5 x x4 Emec,2 3J 3J K K 0 Turning pointsx3 K 0, F 0 particle stationary Unstable equilibriumx4 Emec,3 1J 1J K K 0, F 0, it cannot move to x x4 or x x4, since then K 0 Stable equilibrium5

Review: Potential energyW - U- The zero is arbitrary Only potential energy differences havephysical meaning.- The potential energy is a scalar function of the position.- The force (1D) is given by: F -dU/dxP1. The force between two atoms in a diatomic molecule can be represented bythe following potential energy function: a 12 a 6 U ( x) U 0 2 x x i) Calculate the force Fxwhere U0 and a are constants.F ( x) [ a a 11 a a 5 dU ( x) U 0 12 2 2 2 6 dx x x x x ] U 0 12a12 x 13 12a 6 x 7 12U 0a a 13 a 7 x x ii) Minimum value of U(x).U ( x) min if x a137 12U 0 a a dU ( x) F ( x) 0 0dxa x x U (a) U 0 [1 2] U 0U0 is approx. the energy necessary to dissociate the twoatoms.6

VI. Work done on a system by an external forceWork is energy transfer “to” or “from” a system by means of an externalforce acting on that system.When more than one force acts on a system their network is the energy transferred to or from the system.No Friction:W Emec K U Ext. force Emec K U 0 only when:Remember!- System isolated.- No ext. forces act on a system.- All internal forces are conservative.Friction:F f k mav 2 v02 2ad a 0.5(v 2 v02 ) / dm 2 2111(v v0 ) Fd f k d m(v 2 v02 ) Fd mv 2 mv02 f k d2d222W Fd K f k dF fk General:W Fd Emec f k dExample: Block sliding up a ramp.Thermal energy: E f dthkFriction due to cold welding between twosurfaces. As the block slides over the floor,the sliding causes tearing and reforming of thewelds between the block and the floor, whichmakes the block-floor warmer.Work done on a system by an external force, friction involvedW Fd Emec Eth7

VI. Conservation of energyTotal energy of a system E mechanical E thermal E internal- The total energy of a system can only change by amounts of energytransferred “from” or “to” the system.W E Emec Eth Eint Experimental law-The total energy of an isolated system cannot change. (There cannot beenergy transfers to or from it).Isolated system: Emec Eth Eint 0In an isolated system we can relate the total energy at one instant to thetotal energy at another instant without considering the energies atintermediate states.Example: Trolley pole jumper1) Run Internal energy (muscles)gets transferred into kinetic energy.2) Jump/Ascent Kinetic energytransferred to potential elasticenergy (trolley pole deformation)and to gravitational potential energy3) Descent Gravitational potentialenergy gets transferred into kineticenergy.8

VII. External forces and internal energy changesExample: skater pushes herself away from a railing. There is a force Fon her from the railing that increases her kinetic energy.i) One part of an object (skater’s arm) does notmove like the rest of body.ii) Internal energy transfer (from one part of thesystem to another) via the externalforce F. Biochemical energy from musclestransferred to kinetic energy of the body.WF ,ext K F (cos ϕ )dNon isolated system K U WF ,ext Fd cos ϕ Emec Fd cos ϕProof:Change in system’s mechanical energyby an external forcev 2 v02 2a x d ( 0.5M )11Mv 2 Mv02 Ma x d22 K ( F cos ϕ )dVIII. PowerAverage power:Pavg Instantaneous power: E tP dEdt9

61. In the figure below, a block slides along a path that is without friction until theblock reaches the section of length L 0.75m, which begins at height h 2m. In thatsection, the coefficient of kinetic friction is 0.4. The block passes through point Awith a speed of 8m/s. Does it reach point B (where the section of friction ends)? Ifso, what is the speed there and if not, what greatest height above point A does itreach?N mg cos 30 8.5mf k µ k N (0.4)(8.5m) 3.4mNA C Only conservative forces Emec 0fC K A U A KC UCmg1 2 1 2mv A mvc mghc vc 5m / s22The kinetic energy in C turns into thermal and potential energy Block stops.K c 0.5mvc2 12.4mK c mgy f k d 12.4m mg (d sin 30 ) 3.4md d 1.49 metersd L 0.75m Block reaches BIsolated system E 0 Emec U Eth K C U C K B U B f k L12.4m 0.5mvB2 mg ( y B yc ) µ k mgL cos 30 0.5mvB2 mgL sin 30 µ k mgL cos 30 12.4m 0.5mvB2 3.67m 2.5m vB 3.5m / s129. A massless rigid rod of length L has a ball of mass m attached to one end. The other end is pivoted insuch a way that the ball will move in a vertical circle. First, assume that there is no friction at the pivot.The system is launched downward from the horizontal position A with initial speed v0. The ball justbarely reaches point D and then stops. (a) Derive an expression for v0 in terms of L, m and g. (b) What isthe tension in the rod when the ball passes through B? (c) A little girl is placed on the pivot to increasethe friction there. Then the ball just barely reaches C when launched from A with the same speed asbefore. What is the decrease in mechanical energy during this motion? (d) What is the decrease in mechanical energy by the time the ball finally comes to rest at B after several oscillations?(a) Emec 0 K f U f Ki U i(b) Fcent mac T mgK D 0; U A 0v B2 1 T mg T m v B2 g L L U A K A U B KB1mgL mv02 v0 2 gL2(c) vc 0W E Emec Eth Eth f k dm1 21mv0 mgL mvB2 22112 gL gL v B2 v B 2 gL22DyALCxv0TBT 5mgFcmgThe difference in heights or in gravitational potential energies between the positionsC (reached by the ball when there is friction) and D during the frictionless movementIs going to be the loss of mechanical energy which goes into thermal energy.(c) Eth mgL(d) The difference in height between B and D is 2L. The total loss of mechanical energy(which all goes into thermal energy) is: Emec 2mgL10

101. A 3kg sloth hangs 3m above the ground. (a) What is the gravitationalpotential energy of the sloth-Earth system if we take the reference point y 0 to beat the ground? If the sloth drops to the ground and air drag on it is assumed to benegligible, what are (b) the kinetic energy and (c) the speed of the sloth just beforeit reaches the ground?( a ) Emec 0 K f U f K i U i(b) K f 94.1JU f ( ground ) 0; K i 0( c) K f 2K f1 2mv f v f 7.67 m / s2mU i mgh (3.2kg )(9.8m / s 2 )(3m) 94.1J130. A metal tool is sharpen by being held against the rim of a wheel on agrinding machine by a force of 180N. The frictional forces between the rim and thetool grind small pieces of the tool. The wheel has a radius of 20cm and rotates at2.5 rev/s. The coefficient of kinetic friction between the wheel and the tool is 0.32.At what rate is energy being transferred from the motor driving the wheel and thetool to the kinetic energy of the material thrown from the tool?vF 180N rev 2π (0.2m) P f v ( 57.6 N )(3.14m / s ) 181Wv 2.5 3.14m / s s 1rev Pmotor 181Wf k µ k N µ k F (0.32)(180 N ) 57.6 NPower dissipated by friction Power sup plied motor82. A block with a kinetic energy of 30J is about to collide with a spring at itsrelaxed length. As the block compresses the spring, a frictional force betweenthe block and floor acts on the block. The figure below gives the kinetic energy ofthe block (K(x)) and the potential energy of the spring (U(x)) as a function ofthe position x of the block, as the spring is compressed. What is the increase inthermal energy of the block and the floor when (a) the block reaches position 0.1m and (b) the spring reaches its maximum compression?NIsolated system E 0 0 Emec Eth Eth Emec(a) x 0.1mfmgGraph : K f 20 J , U f 3JEmec,i K i 30 JEmec, f K f U f 23J Emec 23J 30 J 7 J Eth 7 J(b) xmax v 0 K 0 x 0.21mEmec,i K i 30 JEmec, f U f 14 J Emec 14 J 30 J 16 J Eth 16 J11

B1. A 2kg block is pushed against a spring with spring constant k 500 N/mcompressing it 20 cm. After the block is released, it travels along a frictionlesshorizontal surface and a 45º incline plane. What is the maximum height reached bythis block?12

changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system cannot change . Emec K U 0 - When the mechanical energy of a system is conserved , we can relate the sum of kinetic energy and

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