1. Lecture #4 Cassandra Paul Physics 7A Summer Session II 2008

2. Graphing On Monday we didn’t get to finish talking about graphing, let’s do that now.

3. Diving: Potential Energy 0m or 3m 2m or 5m -2m or 1m -3m or 0m At highest point, Tricia Woo is 2 meters above the board and 5 meters above the water, how should we calculate her PE? Where should we measure the height from? From board From floor

4. How can it not Matter!? KE trans Speed PE gravity Height ΔKE +ΔPE= 0 System: Diver Initial: Highest point Final: Just before hitting water ½ m(v f 2 -v i 2 ) + mg(h f -h i )= 0 (0.5)(50kg)(v f 2 -0) + (50kg)(10m/s 2 )(h f -h i )= 0 0m or 3m 2m or 5 m -2m or 1m -3m or 0m From board From floor (0 - 5) (-3 - 2) Δh is the same! Δh=-5 so v f = 10m/s +-

5. Instantaneous PE and KE ΔKE +ΔPE= 0 (KE f – KE i ) + (PE f - PE i ) = 0 KE f + PE f - KE i - PE i = 0 KE f + PE f = KE i + PE i = Etot KE anytime + PE anytime = Etot The sum total of all of the energies at one point in time is equal to the total energy of the system. In a closed system that value is constant throughout the process.

6. Let’s graph our instantaneous energy…. 0m 2m -2m -3m hPE(J)KE(J)Etot(J) height(mgh)Etot-PEsum 210000 00 -2-100020001000 -3-150025001000 Mass=50kg Say g = 10m/s 2

7. Practice this!!!! hPE(J)KE(J)Etot(J) height(mgh)Etot-PEsum 210000 00 -2-100020001000 -3-150025001000 X axis is height in meters Energy in Joules

8. Where have I set h=0? X axis is height in meters Energy in Joules A B C D E 5m The answer is D! Potential Energy is 1 meter above the water. Why? Look at the Graph we can tell that PE is zero At letter D, and KE is zero at A. From there we should Be able to work backwards to interpret the graph.

9. X axis is height in meters Energy in Joules X axis is height in meters Energy in Joules h=0 at board (3 meters above water) h=0 (2 meters above water) Same and different? PE can be negative KE can’t be negative. (½ mv 2) Etot is NOT the same for both cases KE line remains the same! (2500J just before hitting the ground)

10. Springs! Intro to Spring-Mass Oscillator Model (another model in blue pages in course notes)

11. Potential Energy: Springs Springs contain energy when you stretch or compress them. We will use them a lot in Physics 7. The indicator is how much the spring is stretched or compressed,  x, from its equilibrium position. k is a measure of the “stiffness” of the spring, with units [k] = kg/s 2.  x: Much easier to stretch a spring a little bit than a lot! ΔPEspring = (1/2) kΔx 2 x

12. Mass-Spring Systems k is a property of the spring only PE mass-spring does not depend on mass PE = 0 arbitrary ΔPE mass- spring = (1/2) kΔx 2 x x x

13. Mass-Spring Systems PE mass- spring ∆y KE Speed Δx = -2cm initial final

14. Mass-Spring Systems PE mass- spring ∆x KE Speed System: mass-spring Initial: mass at rest at 2cm Final: mass at x=0 X = 2cm (x initial final

15. Potential Energy and Forces: Springs, Gravitational The indicator is how much the spring is stretched or compressed,  x, from its equilibrium position. ΔPE spring = (1/2) kΔx 2 x ∆PE grav = h The indicator is the change in vertical distance that the object moved (I.e. change in the distance between the center of the Earth and the object)

16. PE vs displacement: Force Displacement from equilibrium y[+][-]

17. PE vs displacement: Force Displacement from equilibrium y[+][-] direction of force

18. PE vs displacement: Force direction of force Displacement from equilibrium y[+][-]

19. PE vs displacement: Force On this side force pushes up On this side force pushes down Equilibrium Forces from potentials point in direction that (locally) lowers PE Displacement from equilibrium y[+][-]

20. Graphing Energies What are the x-axis, y axis? Units? x axis (independent variable: height) y axis (dependent variable: PEgrav) Which quantity (energy) is the easiest to graph? Etot ? PEgrav? What about KE? Where should the origin (0) be placed? Where does it most make sense? Should the floor be 0m?

21. Cassandra, how can we do a practice problem with a pendulum? We’ve never learned anything about pendulums!!!!!!!! Practice: Pendulum Yes you have! Just use the Model, You’ll be surprised How much it will tell you.

22. Practice: Pendulum A 2kg pendulum swings to a maximum height of 3 meters. At it’s lowest point it is one meter above the floor. Find the maximum speed of the pendulum, and then graph Etot, PE and KE as a function of distance. A)6.32 m/s B)7.74 m/s C)60m/s D)None of these Initial Final (Still in motion) 2m 3m

23. Initial 2m 3m Final (Still in motion) Let’s use the instantaneous method instead of the Energy Diagram SUM OF ALL ENERGIES PRESENT = Etot PE anytime + KE anytime = Etot PE top + KE top = Etot mgh + 0 = Etot (2)(10)(3)J + 0 = Etot =60J Set h=0 at floor PE bottom + KE bottom = Etot mgh + ½ mv 2 = 60J (2)(10)(1) + ½ 2v 2 = 60J  v=6.32m/s h (m)PE (J)KE (J)Etot (J) 3600 1204060

24. h (m)PE (J)KE (J)Etot (J) 3600 1204060 Initial 2m 3m h (m)PE (J)KE (J)Etot (J) 3600 2402060 1204060 Height  Energy 

25. Back to Springs…

26. What is Ebond anyway?? We will model real atoms of liquids and solids as oscillating masses and springs Particle Model of Matter OK so springs are cool for physicists, but does understanding the spring help us understand anything else?? Three-phase model of matter Energy-interaction model Mass-spring oscillator Particle model of matter  Particle model of bond energy  Particle model of thermal energy Thermodynamics Ideal gas model Statistical model of thermodynamics r

27. Intro to Particle Model of Matter This model helps us understand how particles interact with each other at the molecular level. (another blue page in your course notes)

28. Particle Model of Matter We will model real atoms of liquids and solids as oscillating masses and springs r Goal : To understand macroscopic phenomena (e.g. melting, vaporizing) and macrocopic properties of matter such as phases, temperature, heat capacities, in terms of microscopic constituents and its behavior.

29. Atom 1 (anchored) Atom 2 (bonded or un-bonded) Model Bonded Atoms as Masses on Spring ~ two atomic size particles interacting via“pair-wise potential” a.k.a. Lennard-Jones Potential

30. Ro the distance that the particles are at equilibrium r0r0 1 2 We observe the system oscillating. At one instance we take a snapshot of the oscillation and see this: r > r 0 Which way is the force on particle #2? A)To the right because the particle is traveling to the right.  B)We can’t tell because we don’t know which way the particle is traveling C)To the left because the spring is pulling from the left.  D)We can’t tell which way it’s traveling but we know the force is to the right.  E)There is no force acting particle 2.

31. Lennard-Jones Potential (pair-wise potential) Not to scale (particles are just about touching at equilibrium) Distance  Energy  IT’S JUST A WEIRD SPRING! Equilibrium separation r o

32. Link to Applet: http://polymer.bu.edu/java/java/intermol/ind ex.html http://polymer.bu.edu/java/java/intermol/ind ex.html Don’t worry about this graph on the right.

33. Introduction to the Particle Model Potential Energy between two atoms separation Flattening: atoms have negligible forces at large separation. Repulsive: Atoms push apart as they get too close r PE Distance between the atoms

34. Equations to memorize, and more importantly know how to use for Monday’s Quiz ½ mΔ(v 2 )= ½ m (v f 2 -v i 2 ) = ΔKEtrans mgΔh = ΔPEgrav ½ k (Δx f 2 -Δx i 2 ) = ΔPEspring mcΔT = ΔEth ± l ΔmΔH l = ΔEb Also, we don’t use an equation for rotational energy, but know how to tell if it’s there.

35. DL sections Swapno: 11:00AM EversonSection 1 Amandeep: 11:00AM Roesller Section 2 Yi: 1:40PM Everson Section 3 Chun-Yen: 1:40PM Roesller Section 4