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March 2, 2010

Chapter 7 Conservation of Energy Conservative force Non-conservative force potential energy & potential function. March 2, 2010. Kinetic Energy  Work (work-kinetic energy theorem). Mechanical energy is of two forms: kinetic (by motion) potential (by relative positions)

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March 2, 2010

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  1. Chapter 7Conservation of EnergyConservative forceNon-conservative forcepotential energy & potential function March 2, 2010

  2. Kinetic Energy  Work (work-kinetic energy theorem) Mechanical energy is of two forms:kinetic (by motion) potential (by relative positions) Both of the ability to do work (work-energy theorem); and kinetic  potential transformable. Energies in Nature come with several (known) forms:Mechanical; Electromagnetic & Chemical (heat); Atomic & nuclear …

  3. Conservative and Nonconservative Forces A force is conservative if the work done by the force does not depend on the path taken as the particle moves from one position to another. (only depends on the initial and final points.) In other words, A force is conservative if the work it does on a particle is zero when the particle moves around any closed path, returning to its initial position.

  4. Conservative and Nonconservative Forces y y x (1) x (2) work done around closed path W = Conservative Force Nonconservative Force

  5. Conservative and Nonconservative Forces y y x (1) x (2) work done around closed path W = Conservative Force Nonconservative Force

  6. Example of a nonconservative force: friction Friction always retards the motion, so the work it does around a closed path is nonzero. so the longer the path is, the more work done.

  7. Examples of conservative forces: Spring force Gravity Static electric force

  8. Conservative Forces: • In general, if the work done does not depend on the path taken, the force involved is said to be conservative. Performing integration: • Gravity is a conservativeforce: • Gravity near the Earth’s surface: • A spring produces a conservative force:

  9. Potential Energy U2 r2 r1 U1 The work done by a conservative force is equal and opposite to the change in the potential energy function. This can be written as: For any conservative force F we can define a potential energy function U in the following way:

  10. Potential Energy & Equilibrium: • For a given potential function U(x), one can find the force: • Its derivative w.r.t. x is Fx . At equilibrium: dU(x)/dx = 0. • its second derivative tells its stability • dF(x)/dx >0 or <0 • F physical; U auxiliary, • up to an arbitrary const.

  11. Conservative Forces and Potential Energies Change in P.E U = U2 - U1 Work W(1 to 2) P.E. function U Force F mg(y2-y1) -mg(y2-y1) mgy + C (R is the center-to-center distance, x is the spring stretch)

  12. Question You lift a 10-kg bag of flour from the floor to a shelf 2m above the floor. Which of the following statements are true? You increased the gravitational potential energy and performed negative work on the bag. You increased the gravitational potential energy and the earth performed positive work on the bag. You increased the gravitational potential energy and the earth performed negative work on the bag. You decreased the gravitational potential energy and the earth performed positive work on the bag.

  13. Question You did positive work on the bag (displacement along F). Gravity did negative work on the bag (opposite to displacement). The gravitational potential energy of the bag increased. You lift a 10-kg bag of flour from the floor to a shelf 2m above the floor. Which of the following statements are true? You increased the gravitational potential energy and performed negative work on the bag. You increased the gravitational potential energy and the earth performed positive work on the bag. You increased the gravitational potential energy and the earth performed negative work on the bag. You decreased the gravitational potential energy and the earth performed positive work on the bag.

  14. A woman runs up a flight of stairs. The gain in her gravitational potential energy is U. If she runs up the same stairs with twice the speed, what is her gain in potential energy? • U • 2U • U/2 • 4U • U/4

  15. A woman runs up a flight of stairs. The gain in her gravitational potential energy is U. If she runs up the same stairs with twice the speed, what is her gain in potential energy? • U • 2U • U/2 • 4U • U/4

  16. Question: Falling Objects v=0 v=0 v=0 H vf vi vp Free Fall Frictionless incline Pendulum (a) Vf > Vi > Vp(b) Vf > Vp > ViVf = Vp = Vi Three objects of mass m begin at height h with velocity zero. One falls straight down, one slides down a frictionless inclined plane, and one swings at the end of a pendulum. What is the relationship between their speeds when they have fallen to height zero?

  17. Question: Falling Objects v=0 v=0 v=0 H vf vi vp Free Fall Frictionless incline Pendulum (a) Vf > Vi > Vp(b) Vf > Vp > ViVf = Vp = Vi The only work is done by gravity, which is a conservative force. Three objects of mass m begin at height h with velocity zero. One falls straight down, one slides down a frictionless inclined plane, and one swings at the end of a pendulum. What is the relationship between their velocities when they have fallen to height zero?

  18. Question A ball, initially at rest, is dropped from a place 1.5 m above the floor and is observed to have a velocity V just before it hits the floor. If instead, the ball is dropped from a place that is 0.5 m above the floor, the velocity just before it hits the floor is: • 33% of V • 50% of V • 58% of V • 66% of V • 71% of V

  19. Question A ball, initially at rest, is dropped from a place 1.5 m above the floor and is observed to have a velocity V just before it hits the floor. If instead, the ball is dropped from a place that is 0.5 m above the floor, the velocity just before it hits the floor is: • 33% of V • 50% of V • 58% of V • 66% of V • 71% of V

  20. Loop-the-loop How high up from the ground must the cart stop to complete the loop-the-loop? To stay on the track, cart must have large enough speed at the top of the circle so that its centripetal acceleration is at least g: v2/R ≥ g So, at the top of the loop, Ek = ½mv2 ≥ ½ mgR. Therefore, the potential energy at the very start must be bigger than Ep(top) > Ep(2R) + Ek = mg(2R) + ½ mgR = 5mgR/2.

  21. Question h L q x A 1kg block slides 4 m down a frictionless plane inclined at 30 degrees to the horizontal. After reaching the bottom, it slides along a frictionless horizontal plane and strikes a spring with spring constant k=314 N/m. How far is the spring compressed when it stops the block?

  22. Question PE of block gets converted first into KE of block, and then into PE of spring. h L q x The decrease in the block’s height is h = L sinθ, so the magnitude of the change in potential energy when the block slides down the inclined plane is Ep = Ek = mgL sinθ. The spring compresses until its potential energy has that value, so ½kx2 = mgL sinθ x = (2mgL sinθ/k)1/2 = 35 cm. A 1kg block slides 4 m down a frictionless plane inclined at 30 degrees to the horizontal. After reaching the bottom, it slides along a frictionless horizontal plane and strikes a spring with spring constant k=314 N/m. How far is the spring compressed when it stops the block?

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