Lecture 08 Conservation of Energy

Transcription

1 Lecture 8 Conservation o Enery 8. The Nonisolated System: Conservation o Enery wor, mechanical waves, heat (drivin by a temperature dierence), matter transer (convection), electrical transmission (by means o electric currents), electromanetic radiation (photons) Conservation o Enery Equation: K U E W Q T T int MW MT T ET T ER What is the enery o a visible photon? (ev =? Joule) We can neither create nor destroy enery enery is always conserved. wor-inetic enery valid only when the object can be modeled as a particle i a rictional orce ests, how to express the wor with it? What s the dierence between inetic riction and resistive orce? I no external wor applied to the system, the loss o inetic enery will transer to wor done by the inetic riction orce. i K x dx I external orce est, the system may ain enery rom the external wor and loss enery to the inetic riction orce. K W x vi v vi dx v W = Fdx We said in previous section that no enery can be created or destroyed. the wor done by the inetic riction orce o? -> Internal Enery Where does E system K Eint E int d Nonisolated system dx isolated system The result o a riction orce is to transer inetic enery into internal enery.

2 Frictional orce -> Resistive orce -> Dra orce?? Example: A car travelin at a speed v slides a distance d to a halt ater its brae loc. Assumin that the car s initial speed is instead v at the moment the braes loc, estimate the distance it slides. 4d Example: A bloc o mass.6 is attached to a horizontal sprint that has a orce constant o. X 3 N/m. rom rest. The sprin is compressed. cm and is then released Calculate the speed o the bloc as it passes throuh the equilibrium position i a constant riction orce o 4. N retards its motion rom the moment it is released. W V x d 8. The Isolated System the wor done by the ravitational orce: W m zy yi zˆ myi my ˆ, y yi v yi y transorm to mechanical enery o the object W K my i my notice the minus sin W U wor is only intermediate substitutable quantity to express the enery transer W K -> K U inetic enery and potential enery o an object U in the system Deine Mechanical Enery: E, Emech K U mech I K U -> K K U U -> Ki U i K U i i -> i myi my system -> conservation o mechanical enery or an isolated

3 This result is called the principle o conservation o mechanical enery. (Now you can see where conservative orces ot their name.) We can write this principle in one more orm, as E K U mech Example: A child o mass m is released rom rest at the top o a water slide, at heiht h = 8.5 m above the bottom o the slide. Assumin that the slide is rictionless because o the water on it, ind the child's speed at the bottom o the slide. v h m/s Example: Ball in Free Fall m h y Example: The Pendulum ml B ml cos A Example: Two blocs are connected by a massless cord that passes over two rictionless pulleys, as in Fiure. One end o the cord is attached to an object o mass m=3. that is a distance R=.m rom the pulley on the let. The other end o the cord is connected to a bloc o mass m=6 restin on a table. From what anle must the 3. mass be released in order to just lit the 6. bloc o the table? Enery conservation: R( v cos ), mh, v Force balance: m m m, R 3

4 m m m ( cos ), 3m m cos m Elastic Potential Enery, U W x F x Fapp dx x E mech U K x, i the enery is conserved in the isolated system, E mech U K -> i x i x Example: A bead slides without riction around a loopthe-loop. The bead is released rom a heiht h = 3.5R. (a) What is its speed at point A? (b) How lare is the normal orce on it i its mass is 5.? Hint: (a) Transer potential enery to inetic enery, (b) calculate the centripetal orce that the wall exerted on the particle, the orce that the particle exert on the wall may need to subtract the ravitational term (a) v 3R (b) m normal orce is rom the plane in downward direction See AF68 & AF86. Gravitational potential enery: W x F x dx U U U U x y yi m F x dx y mdy m yi dy y y my y my yi i 4

5 elastic potential enery: U U x F x xdx xdx xdx x x x x i x The total enery E o a system can chane only by amounts o enery that are transerred to or rom the system. The Wor-Enery Theorem (compared with the wor-inetic enery theorem) W E E E mech mech K U E th E int isolated system -> W E 8.3 Situations Involvin Kinetic Friction 8.4 Chanes in Mechanical Enery or Nonconservative Forces No Friction Involved: W Emech K U Friction Involved:. Consider the inetic enery only: K d. Consider the mechanical enery: E K U d Derived rom the orce law: F ma, v v ad, Fd d By doin experiment, we ound that the bloc and the portion o the loor alon which it slides become warmer as the bloc slides E d th W Fd E E th Example: A ood shipper pushes a wood crate o cabbae heads (total mass m = 4 5

6 ) across a concrete loor with a constant horizontal orce F o manitude 4 N. In a straiht-line displacement o manitude d =.5 m, the speed o the crate decreases rom v =.6 m/s to v =. m/s. (a) How much wor is done by orce F, and on what system does it do the wor? W 4.5 J (b) What is the increase Eth in the thermal enery o the crate and loor? E th W K J J Example: A child o mass m rides on an irreularly curved slide o heiht h =. m. The child starts at rest on the top. (a) Determine his speed at the bottom, assumin no riction is present. v h (b) I a orce o inetic riction acts on the child, how much mechanical enery does the system lose? Assume that v = 3. m/s and m =.. E mh mech mh.3. 3J Example: Two blocs are connected by a liht strin. The bloc o mass m is connected to a sprin o orce constant. The system is released rom rest when the sprin is unstreched. I the bloc o mass m alls a distance o h beore comin to rest, calculate the coeicient o inetic riction between the bloc o mass m and the surace. Hint: Potential enery is provided to stretch the sprin and to the loss o inetic riction. Overdamped system? mh h m h 6

7 Example: A. pacae o tamale slides alon a loor with speed v = 4. m/s. It then runs into and compresses a sprin, until the pacae momentarily stops. Its path to the initially relaxed sprin is rictionless, but as it compresses the sprin, a inetic rictional orce rom the loor, o manitude 5 N, acts on it. The sprin constant is, N/m. By what distance d is the sprin compressed when the pacae stops? 5 d 5 d 6, 5d 5d 6, 5 d Systems With Chemical Enery Example: You are drivin a - asoline-powered car at a constant speed o m/h (7.8 m/s) up a -percent rade. (a) I the eiciency is 5 percent, what is the rate at which the chemical enery o the car-earth-atmosphere system chanes? tan %. t E chem h.5 m mvsin ~ mv tan t de 9.8 chem W dt.5 Mass and Enery In 95, Albert Einstein published his special theory o relativity, a result o which is the amous equation E mc. A particle or system o mass m has rest enery mc. The enery is intrinsic to the particle. Enery in atomic and nuclear physics are usually expressed in units o electron volts (ev). A convenient unit or the masses o atomic particles is ev/c. Particle Symbol Rest Enery (MeV) Electron e -.5 Positron e +.5 Proton p Neutron n

8 Deuteron d Triton t 88.4 Helium-4 (alpha particle) 4 He The increase in mass with an increase in enery o 4 J is E 4 7 m c 3 Nuclear Enery Example: A typical nuclear usion reaction is written H+ 3 H-> 4 He+n. enery is released in this usion reaction? ~ 7. MeV How much Newtonian Mechanics and Relativity K v c E v c mc v K E : rest mass enery c E Newtonian mechanics is valid o the speed o the particle is much less than the speed o liht. Quantization o Enery 34 Plan s constant: h 6.66 J s The quantized enery o an oscillator: E n n h round state enery E En h h The quantized enery o a photon: electromanetic radiation. E photon h, is the requency o the For macroscopic bound system, the oscillation requencies or a sprin system are about to Hz. I, the spacin between allowed levels is 33 h ~ 6 J. The enery o a macroscopic system is in the order o J. The enery level spacin is too small to be observable. Example: For a diatomic molecule, a typical requency o vibration is 4, and a typical enery o -9 J. The spacin between allowed levels is then E 6 E ev.6 n En h 6 J, ~

9 Since the enery o molecule is in the order o ev, the enery spacin is not neliible. At let is a hydroen spectral tube excited by a 5 volt transormer. The three prominent hydroen lines are shown at the riht o the imae throuh a 6 lines/mm diraction ratin. An appromate classiication o spectral colors: Violet (38-435nm) Blue(435-5 nm) Cyan (5-5 nm) Green (5-565 nm) Yellow ( nm) Orane (59-65 nm) Red (65-74 nm) 8.5 Power de P dt P dw dt d dt F r F v 3 _ Watt _ J / s _ m / s _ hp 746 _W _ Wh 5 _ W _ h _ W 36 _ s 3.6 _ J 9