1 / 50

Potential energy and conservation of energy

Potential energy and conservation of energy. Chapter 8. Potential energy  Energy of configuration Work and potential energy Conservative / Non-conservative forces Determining potential energy values: - Gravitational potential energy

jnormand
Download Presentation

Potential energy and conservation of energy

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Potential energy and conservation of energy Chapter 8

  2. Potential energy  Energy of configuration • Work and potential energy • Conservative / Non-conservative forces • Determining potential energy values: • - Gravitational potential energy • - Elastic potential energy • V. Conservation of mechanical energy • External work and thermal energy • External forces and internal energy changes

  3. I.Potential energy Energy associated with the arrangement of a system of objects that exert forces on one another. Units: 1J Examples: • Gravitational potential energy:associated with the state of separation between objects which can attract one another via the gravitational force. • Elastic potential energy:associated with the state of • compression/extension of an elastic object.

  4. II. Work and potential energy If tomato rises gravitational force transfers energy “from” tomato’s kinetic energy “to” the gravitational potential energy of the tomato-Earth system. If tomato falls down gravitational force transfers energy “from” the gravitational potential energy “to” the tomato’s kinetic energy.

  5. System Example • This system consists of Earth and a book • Do work on the system by lifting the book through dy • The work done is mgyb - mgya

  6. Potential Energy • The energy storage mechanism is called potential energy • A potential energy can only be associated with specific types of forces • Potential energy is always associated with a system of two or more interacting objects

  7. Gravitational Potential Energy • Gravitational Potential Energy is associated with an object at a given distance above Earth’s surface • Assume the object is in equilibrium and moving at constant velocity • The work done on the object is done by Fappand the upward displacement is

  8. Gravitational Potential Energy • The quantity mgyis identified as the gravitational potential energy, Ug Ug = mgy • Units are joules (J)

  9. Gravitational Potential Energy • The gravitational potential energy depends only on the vertical height of the object above Earth’s surface • In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero • The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration

  10. Conservation of Mechanical Energy • The mechanical energy of a system is the algebraic sum of the kinetic and potential energies in the system Emech = K + Ug • The statement of Conservation of Mechanical Energy for an isolated system is Kf + Uf= Ki+ Ui • An isolated system is one for which there are no energy transfers across the boundary

  11. Conservation of Mechanical Energy • Look at the work done by the book as it falls from some height to a lower height Won book = ΔKbook • Also, W = mgyb – mgya • So, ΔK = -ΔUg

  12. Elastic Potential Energy • Elastic Potential Energy is associated with a spring • The force the spring exerts (on a block, for example) is Fs = - kx • The work done by an external applied force on a spring-block system is W = ½ kxf2 – ½ kxi2 • The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

  13. Elastic Potential Energy • This expression is the elastic potential energy: Us = ½ kx2 • The elastic potential energy can be thought of as the energy stored in the deformed spring • The stored potential energy can be converted into kinetic energy

  14. Elastic Potential Energy • The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0) • The energy is stored in the spring only when the spring is stretched or compressed • The elastic potential energy is a maximum when the spring has reached its maximum extension or compression • The elastic potential energy is always positive • x2 will always be positive

  15. 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 (W1) transferring energy between KE of the object and some other form of energy of the system. • When the configuration change is reversed  force reverses the energy transfer, doing W2.

  16. Problem Solving Strategy – Conservation of Mechanical Energy • Define the isolated system and the initial and final configuration of the system • The system may include two or more interacting particles • The system may also include springs or other structures in which elastic potential energy can be stored • Also include all components of the system that exert forces on each other

  17. Problem-Solving Strategy • Identify the configuration for zero potential energy • Include both gravitational and elastic potential energies • If more than one force is acting within the system, write an expression for the potential energy associated with each force

  18. Problem-Solving Strategy • If friction or air resistance is present, mechanical energy of the system is not conserved • Use energy with non-conservative forces instead

  19. Problem-Solving Strategy • If the mechanical energy of the system is conserved, write the total energy as Ei = Ki + Ui for the initial configuration Ef = Kf + Uffor the final configuration • Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity

  20. Conservation of Energy Example 1 (Drop a Ball) • Initial conditions: Ei= Ki + Ui = mgh • The ball is dropped, so Ki = 0 • The configuration for zero potential energy is the ground • Conservation rules applied at some point y above the ground gives ½ mvf2 + mgy = mgh

  21. Conservation of Energy Example 2 (Pendulum) • As the pendulum swings, there is a continuous change between potential and kinetic energies • At A, the energy is potential • At B, all of the potential energy at A is transformed into kinetic energy • Let zero potential energy be at B • At C, the kinetic energy has been transformed back into potential energy

  22. Conservation of Energy Example 3 (Spring Gun) • Choose point A as the initial point and C as the final point EA = EC KA + UgA + UsA = KC + UgC + UsB ½ kx2 = mgh

  23. Also valid for elastic potential energy Spring compression Spring forcedoes –W on block energy transfer from kinetic energy of the block to potential elastic energy of the spring. fs Spring extension Spring forcedoes +W on block  energy transfer from potential energy of the spring to kinetic energy of the block. fs

  24. III. Conservative / Nonconservative forces • If W1=W2always  conservative force. Examples:Gravitational force and spring force  associated potential energies. • If W1≠W2 nonconservative force. Examples:Drag force, frictional force  KE transferred into thermal energy. Non-reversible process. - Thermal energy:Energy associated with the random movement of atoms and molecules. This is not a potential energy.

  25. Conservative force:The net work it does on a particle moving around every 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 not depend on the particle’s path. Conservative forceWab,1=Wab,2 Proof: Wab,1+ Wba,2=0  Wab,1= -Wba,2 Wab,2= - Wba,2 Wab,2= Wab,1

  26. IV. Determining potential energy values Force F is conservative Gravitational potential energy: Change in the gravitational potential energy of the particle-Earth system.

  27. Reference configuration The gravitational potential energy associated with particle-Earth system depends only on particle’s vertical position “y” relative to the reference position y=0, not on the horizontal position. Elastic potential energy: Change in the elastic potential energy of the spring-block system.

  28. Reference configuration  when the spring is at its relaxed length and the block is at xi=0. Remember!Potential energy is always associated with a system.

  29. V. Conservation of mechanical energy Mechanical energyof a system: Sum of the its potential (U) and kinetic (K) energies. Emec= U + K Only conservative forces cause energy transfer within the system. Assumptions: The system is isolated from its environment  No external force from an object outside the system causes energy changes inside the system. ΔEmec= ΔK + ΔU = 0

  30. In an isolated system where only conservative forces cause energy changes, the kinetic energy and potential energy can change, but their sum, the mechanical energy of the system cannot change. • When the mechanical energy of a system is conserved, we can relate the sum of kinetic energy and potential energy at one instant to that at another instant without considering the intermediate motion and without finding the work done by the forces involved.

  31. A bead slides without friction around a loop-the-loop. The bead is released from a height h = 3.50R. (a) What is its speed at point A? (b) How large is the normal force on it if its mass is 5.00 g?

  32. Emec= constant y x Potential energy curves

  33. Finding the force analytically: • 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)

  34. Turning point:a point x at which the particle reverses its motion (K=0). Kalways ≥0 (K=0.5mv2 ≥0) Examples: x= x1Emec= 5J=5J+K K=0 x<x1Emec= 5J= >5J+K K<0 impossible Equilibrium points: where the slope of theU(x)curve is zero F(x)=0 ΔU = -F(x) dx  ΔU/dx = -F(x)

  35. ΔU(x)/dx = -F(x) -F(x) Slope Emec,1 Emec,2 Equilibrium points Emec,3 Example:x ≥ x5Emec,1=4J=4J+KK=0 and alsoF=0 x5neutral equilibrium x2<x<x1, x5<x<x4Emec,2= 3J= 3J+KK=0Turning points x3 K=0, F=0 particle stationary Unstable equilibrium x4Emec,3=1J=1J+K  K=0, F=0, it cannot move to x>x4or x<x4, since thenK<0 Stable equilibrium

  36. VI. Work done on a system by an external force Work is energy transfer“to”or “from”a system by means of an external force acting on that system. When more than one force acts on a system their net work is the energy transferredtoorfromthe system. W = ΔEmec= ΔK+ ΔU Ext. force No Friction:

  37. Remember! • ΔEmec= ΔK+ ΔU = 0only when: - System isolated. • No external forces act on a system. • All internal forces are conservative. Friction:

  38. General: Example:Block sliding up a ramp.

  39. A 15.7 kg block is dragged over a rough, horizontal surface by a72.2 Nforce acting at21°above the horizontal. The block is displaced4.5 m, and the coefficient of kinetic friction is0.177. Find the work done on the block by (a) the72.2 Nforce, (b) the normal force, and (c) the gravitational force. (d) What is the increase in internal energy of the block-surface system due to friction? (e) Find the total change in the block's kinetic energy. y F N F 21 Fy fk Fx Fg d

  40. Thermal energy: Friction due to cold welding between two surfaces. As the block slides over the floor, the sliding causes tearing and reforming of the welds between the block and the floor, which makes the block-floor warmer. Work done on a system by an external force, friction involved

  41. VI. Conservation of energy • Total energy of a system = Emechanical + Ethermal + Einternal • - The total energy of a system can only change by amounts of energy transferred“from”or “to”the system.  Experimental law • The total energy of an isolated system cannot change. (There cannot be energy transfers to or from it).

  42. Isolated system: • In an isolated system we can relate the total energy at one instant to the total energy at another instant without considering the energies at intermediate states.

  43. VII. External forces and internal energy changes Example:skater pushes herself away from a railing. There is a forceFon her from the railing that increases her kinetic energy. • One part of an object (skater’s arm) • does not move like the rest of body. • ii) Internal energy transfer (from one part • of the system to another) via the external forceF. Biochemical energy from muscles transferred to kinetic energy of the body. Change in system’s mechanical energy by an external force

  44. Change in system’s internal energy by a external force Proof:

  45. 129.A massless rigid rod of length Lhas a ball of mass m attached to one end. The other end is pivoted in such 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 Awith initial speed v0. The ball just barely reaches pointDand then stops. D y L C A x v0 Fc T B mg

  46. 7.A particle is attached between two identical springs on a horizontal frictionless table. Both springs have spring constant kand are initially unstressed. (a) If the particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs show that the force exerted by the springs on the particle is (b) Determine the amount of work done by this force in moving the particle from x = A to x = 0; (c) show that the potential energy of the system is:

  47. 61.In the figure below, a block slides along a path that is without friction until the block reaches the section of length L=0.75m, which begins at height h=2m. In that section, the coefficient of kinetic friction is 0.4. The block passes through point A with a speed of 8m/s. Does it reach point B (where the section of friction ends)? If so, what is the speed there and if not, what greatest height above point A does it reach? N f C mg

  48. 101.A 3kg sloth hangs 3m above the ground. (a) What is the gravitational potential energy of the sloth-Earth system if we take the reference point y=0 to be at the ground? If the sloth drops to the ground and air drag on it is assumed to be negligible, what are (b) the kinetic energy and (c) the speed of the sloth just before it reaches the ground?

  49. 130.A metal tool is sharpen by being held against the rim of a wheel on a grinding machine by a force of 180N. The frictional forces between the rim and the tool grind small pieces of the tool. The wheel has a radius of 20cm and rotates at 2.5 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 the tool to the kinetic energy of the material thrown from the tool? v F=180N

  50. 82.A block with a kinetic energy of 30J is about to collide with a spring at its relaxed length. As the block compresses the spring, a frictional force between the block and floor acts on the block. The figure below gives the kinetic energy of the block (K(x)) and the potential energy of the spring (U(x)) as a function of the position x of the block, as the spring is compressed. What is the increase in thermal energy of the block and the floor when (a) the block reaches position 0.1 m and (b) the spring reaches its maximum compression? N f mg

More Related