Physics 114C - Mechanics Lecture 19 (Walker: Ch. 8.1-2) Potential Energy February 10, 2014

1. Physics 114C - MechanicsLecture 19 (Walker: Ch. 8.1-2)Potential EnergyFebruary 10, 2014 John G. Cramer Professor Emeritus, Department of Physics B451 PAB jcramer@uw.edu

2. Announcements • HW#5 is due at 11:59 PM on Thursday, February 13.HW#6 is due at 11:59 PM on Thursday, February 21. • Clicker scores as of today will be posted on Catalyst soon. • We will have Exam 2 on Friday, February 14. (Happy Valentine’s Day!) It will cover Chapters 5-8 and will be similar to Exam 1 in its structure. Bring a Scantron sheet, a straight-edge, and an calculator with good batteries. • Exam 2 will again have assigned seating. If you have not already doneso and would like to request a left-handed seat, right-handed aisle seat, or front row seat, E-mail your request to me ASAP. Physics 114A - Lecture 18

3. Lecture Schedule (Part 2) We are here. Physics 114A - Lecture 18

4. Conservative andNonconservative Forces Conservative force: the work it does is stored in the form of energy that can be released at a later time Examples of a conservative force: gravity, spring. Example of a nonconservative force: friction. Also: the work done by a conservative force in moving an object around a closed path is zero;this is not true for a nonconservative force. Physics 114A - Lecture 18

5. Conservative Force: Gravity Work done by gravity on a closed path is zero: Physics 114A - Lecture 18

6. Nonconservative Force: Friction Work done by friction on a closed path is not zero: Physics 114A - Lecture 18

7. Heat: Calories vs. joules Heat energy is often produced by nonconservative forces. Energy, particularly heat energy, is sometimes quantified in units of calories (cal), defined as the heat energy required to raise 1 g (= 1 ml = 1 cm3) of water by 1 degree Celsius. 1 cal = 4.186 J. The equivalent SI unit is the kilocalorie, denoted by Cal or kcal, which is the heat energy required to raise 1 kg (or 1 liter) of water by 1 degree Celsius. 1 Cal = 1 kcal = 4.186 x 103 J. This is the unit often used to indicate the energy content of food, and is measured by burning the food in a calorimeter and then measuring the heat output. Physics 114A - Lecture 18

8. Path Dependence of Gravitational Energy General Observation: the work done by gravity is independent of the details of the path followed by the object and depends only on Dy, the change in height. Physics 114A - Lecture 18

9. Conservative Forces and Paths The work done by a conservative force is zero on any closed path: Physics 114A - Lecture 18

10. Conservative Forces& Potential Energy Consider a more general case in which a force F (similar to gravity) depends only on the position along any path taken by a particle moving from point A to point B. The potential energy U also depends only on location [U=U(r )]. • Potential energy is an energy of position. The system has a unique value of potential energy when the object is at A, a different value when the object is at B, etc. Thus, the net change in potential energy is DU=UB-UA, and is the same whether the object moves form A to B along Path 1 or along Path 2. • Potential energy is transformed to kinetic energy with DK=-DU. If DU is independent of path, then DK must also be independent of path. In moving from A to B, the particle must arrive at B with the same kinetic energy, no matter which path is taken. • The change in the particle’s kinetic energy is related to the amount of work done on the particle by force F. According to the work-kinetic-energy theorem, DK=W. Because DK is independent of path, the work W done by the force in moving from A to B must also be independent of path. A potential energy can be associated with any conservative force. Physics 114A - Lecture 18

11. Work Done byConservative Forces: Gravity If we pick up a ball and put it on the shelf, we have done work on the ball. We can get that energy back if the ball falls back off the shelf; in the meantime, we say the energy is stored as potential energy. (8-1) Physics 114A - Lecture 18

12. Work Done byConservative Forces: Gravity Gravitational potential energy: Physics 114A - Lecture 18

13. Gravitational Potential Energy Consider lifting a barbell of mass m to a height h. The barbell starts at rest and ends at rest, so the net change in its kinetic energy is zero. Considering the barbell as a particle, the Work-Energy Theorem tells us that the net work done on the barbell is zero. There are two forces on it during the lift, gravity (-mg) and the force from your hands (+mg). The work done on the bar- bell by gravity is -mgh, and the work done by your hands is +mgh. Consider the barbell and Earth to be a system of twoparticles. You are not part of this system. You exert three forces in this Earth-barbell system: your hands push up on the barbell, your feet push downward, and you gravitationally attract the Earth. Since the amount by which you displace the Earth by standing on it and attracting it are negligibly small, the only work done on the system is by your hands on the barbell. Thus, the net work that you do on the Earth-barbell system is +mgh, the total work done by all three forces. The energy transferred to the Earth-barbell system is stored as gravit-ational potential energy associated with the position of the barbell relative to the Earth. Physics 114A - Lecture 18

14. Example: A Falling Bottle A 0.350 kg bottle falls from rest from a shelf that is 1.75 m above the floor. (a) Find the potential energy of the bottle-Earth system when the bottle is on the shelf. (b) Find the kinetic energy of the bottle-Earth system just before impact with the floor. Physics 114A - Lecture 18

15. Example: Pike’s Peak or Bust An 82.0 kg mountain climber is in the final stage of the ascent of Pike’s Peak, which 4,301 m above sea level. (a) What is the change in gravitational potential energy as the climber gains the last 100.0 m of altitude? Use U=0 at sea level. (b) Do the same calculation with U=0 at the top of the peak. Physics 114A - Lecture 18

16. Example: A Mountain Bar A candy bar called the Mountain Bar has an energy content when metabolized of 212 Cal = 212 kcal. This is equivalent to 8.87 x 105 J. If an 81.0 kg mountain climber eats a Mountain Bar and magically converts all of its energy content into gravitational potential energy, how much altitude Dy should he be able to gain? Physics 114A - Lecture 18

17. Work Done byConservative Forces: Springs (8-4) Springs: Physics 114A - Lecture 18

18. Elastic Potential Energy Another system that stores energy associated with its configuration is a spring. If you stretch or compress a spring, energy associated with the length of the spring is stored as elastic potential energy. In the figure you compress the spring, pushing it with equal and opposite forces F1and F2. These forces sum to zero, so the net force on the spring is zero and there is no change in the kinetic energy of the spring. The energy transfer associated with the work you do on the spring is stored, not as kinetic energy, but as elastic potential energy. The configuration of the spring has been changed, in that its length is decreased at the ends by Dℓ1 and Dℓ2. The net work is positive because F1,2║Dℓ1,2. Physics 114A - Lecture 18

19. Elastic Potential Energy Physics 114A - Lecture 18

20. Example: The Potential Energyof a Basketball Player A system consists of a 110 kg basketball player, the rim of a basketball hoop, and the Earth. Assume that the potential energy of the system is zero when the player is standing on the floor and the rim is horizontal. The CM of the player is 0.8 m above the floor when he is standing and 1.30 m when he is hanging. The force constant of the rim is 7.2 kN/m. Find the total potential energy of the system when the player is hanging on the front of the rim and bending it downward by 15 cm. Physics 114A - Lecture 18

21. Conservative Forcesand Potential Energy A potential energy can be associated with any conservative force. Both are location-dependent and reversible potential energies. Note that friction is nota conservative force and is irreversible. Physics 114A - Lecture 18

22. The Zero of Potential Energy Physics 114A - Lecture 18

23. Example:The Speed of a Falling Rock A rock is released from rest. Use both Betty’s and Bill’s perspectives to calculate its speed just before it hits the ground. Physics 114A - Lecture 18

24. Clicker Question 1 A small child slides down four frictionless sliding boards. Which relation below describes the relative magnitudes of her speeds at the bottom? c) vA=vB=vC=vD a) vC>vA>vB>vD e) vC<vA<vB<vD b) vA>vB=vC>vD d) vA<vB=vC<vD Physics 114A - Lecture 18

25. Example: The Speed of a Sled Christine runs forward with her sled at 2.0 m/s. She hops onto the sled at the top of a 5.0 m high, very slippery slope. What is her speed at the bottom? K1 + Ug1 = K0 + Ug0 ½mv12+mgy1 = ½mv02+mgy0 v1 = [v02 + 2gy1]½ = [(2.0 m/s)2+2(9.80 m/s2)(5.0 m)]½ = 10.1 m/s Notice that the steepness of the slope and/or whether it has bumps and dips does not matter in determining the answer. Only the change in height and the initial speed are relevant to the answer. Physics 114A - Lecture 18

26. End of Lecture 18 • HW#5 is due at 11:59 PM on Thursday, February 13.HW#6 is due at 11:59 PM on Thursday, February 20. • Clicker scores as of today will be posted on Catalyst soon. • We will have Exam 2 on Friday, February 14. It will cover Chapters 5-8 and will be similar to Exam 1 in its structure. Bring a Scantron sheet, a straight-edge, and an calculator with good batteries. • Exam 2 will again have assigned seating. If you have not already done so and would like to request a left-handed seat, right-handed aisle seat, or front row seat, E-mail your request to me ASAP. Physics 114A - Lecture 18