Practice Final Exa PY 05 Monday 004 May 3 Nae There are THREE forula pages. Read all probles carefully before attepting to solve the. Your work ust be legible, and the organization ust be clear. Correct answers without adequate explanation will be counted wrong. Incorrect explanations ixed in with correct explanations will be counted wrong. Make explanations coplete but brief. Do not write a lot of prose. Write on the backs of pages if you need ore space, and write a note saying where we should look. Include diagras! Show what goes into a calculation, not just the final nuber: Give physical units with your results. 3 6 ( 8 0 )( 5 0 ) 4 ------------------------------------------ = 5 0 5 4 ( 0 )( 4 0 ) If you cannot do soe portion of a proble, invent a sybol for the quantity you can t calculate (explain that you re doing this), and do the rest of the proble.
Proble (5 pts) A ass of 0.7 kg connected to a spring of unknown stiffness oscillates with little friction, aking one round trip in. seconds. What is the spring stiffness k s? Proble (5 pts) Making the approxiation that the density of the Earth is roughly unifor, calculate the agnitude of the rotational angular oentu of the Earth around its own center. Proble 3 (0 pts) 6 The Moon has a ass of 7 0 kg and a radius of.75 0. There is no air, so no air resistance. Suppose you launch a rocket fro the surface of the Moon so that when it is very far fro the Moon its speed is 00 /s. What is the required launch speed?
Proble 4 (0 pts) A person jups out of an airplane above the surface of the Earth, and falls a distance h. (a) ( pts) The work done on the person by the Earth is: Positive, Negative, Zero, or Insufficient Inforation to Answer? (b) ( pts) The change in gravitational potential energy of the person+earth syste is: Positive, Negative, Zero, or Insufficient Inforation to Answer? (c) ( pts) After falling a distance h, the person s parachute opens and he proceeds to fall an additional distance h, albeit at a uch reduced speed. During this second phase of the fall, the work done on the person by the Earth is copared to that done in part (a). The work done is: More Than, Less Than, The Sae As, or Insufficient Inforation to Answer? Pluto orbits the Sun in an elliptical orbit with a period of 49 years. (d) ( pts) During one full period of Pluto, the work done on Pluto by the Sun is: Positive, Negative, Zero, or Insufficient Inforation to Answer? (e) ( pts) In its elliptical orbit, Pluto is currently oving away fro the Sun. The work done on Pluto by the Sun during the next 0 days is: Positive, Negative, Zero, or Insufficient Inforation to Answer? Proble 5 (5 pts) What is the total energy of an electron that is traveling at a speed of 0.97c, where c is the speed of light? Give a nuerical answer. Proble 6 (0 pts) An object had oentu 46,, 8 ( kg /s) when a net force 30, 0, N started to act on the object. After 0.0 seconds, what was the new oentu of the object? 3
Proble 7 (0 pts) Here is a portion of the orbit of an asteroid around the Sun in an elliptical orbit, oving fro A to B to C to D. B C r B r C A r A r D D (a) (0 pts) For the syste consisting of the Sun plus the asteroid, graph the gravitational potential energy U, the kinetic energy K, and the su K+U, as a function of the separation distance between Sun and asteroid. Label each curve. Along the r axis are shown the various distances between Sun and asteroid. (b) (0 pts) Did you reeber to label the curves? 0 +energy r A r B r C r D energy Proble 8 (0 pts) Here is a portion of a progra for a process siilar to Rutherford scattering. An alpha particle interacts with a silver nucleus. Coplete the issing pieces. (Not shown are the stateents that set the initial conditions before the while loop.)... deltat = e-3 while t < e-0: r = alpha.pos - silver.pos rag = sqrt(r.x** + r.y** + r.z**) rhat = Fag = 9e9*47**(.6e-9)**/rag** F = alpha.p = silver.p = alpha.pos = alpha.pos + (alpha.p/alpha.)*deltat silver.pos = silver.pos + (silver.p/silver.)*deltat t = t+deltat 4
Proble 9 (0 pts) A particular quantu syste has energy levels, K+U, shown in the diagra. A bea of high-energy electrons runs through a bottle that contains a large nuber of these systes, so that there are any systes in all of these energy states at all ties, and photons are continually eitted fro the bottle. (a) (5 pts) What are the energies of the eitted photons? Indicate the transitions on the diagra. 0.5 ev.0 ev.0 ev 4.5 ev (b) (5 pts) Next the electron bea is turned off, and the bottle is cold. A bea of light with a wide range of energies, fro 0. to 0 ev, shines through the bottle. On the other side of the bottle, what photon energies in the bea are soewhat depleted ( dark spectral lines )? Explain with the help of a diagra. Proble 0 (0 pts) A spring whose stiffness is 3500 N/ is used to launch a 4 kg block straight up in the classroo. The spring is initially copressed 0., and the block is initially at rest when it is released. When the block is.3 above its starting position, what is its speed? 5
Proble (0 pts) In outer space two rocks collide and stick together, without rotation. Here are the asses and initial velocities of the two rocks: Rock : ass = 5 kg, initial velocity = 0, 30, 0 /s Rock : ass = 3 kg, initial velocity = 5,, 0 /s (a) (0 pts) Use the oentu principle to find the (vector) velocity of the stuck-together rocks after the collision. (b) (0 pts) Use the energy principle, and your result fro part (a), to calculate the increase in theral energy of the rocks. 6
Proble (30 pts) Fro Young s odulus for aluinu we found that the effective interatoic spring stiffness was 6 N/. See forula sheet for additional data. (a) (0 pts) For a nanoparticle consisting of 3 aluinu atos (using the Einstein odel of 9 independent oscillators), calculate the teperature when there are 5 quanta of energy in the nanoparticle. Show your work clearly. (b) (0 pts) Calculate the heat capacity per aluinu ato at this teperature. Show your work clearly. 7
Proble 3 (0 pts) A string is wrapped around a unifor-density disk of ass M and radius R. Attached to the disk are four lowass rods, each with a sall ass at the end, a distance b fro the center of the disk. The apparatus is initially at rest on a nearly frictionless surface. Then you pull the string with a constant force F. At the instant when the center of the disk has oved a distance d, a length L of string has unwound off the disk, so your hand has oved a distance L+d. (a) (0 pts) Write out the energy principle for the point particle syste. Use this to find the final speed v of the center of ass of the syste. b M, R d F L + d F (b) (0 pts) Write out the energy principle for the real syste. Use this, and your result fro part (a), to find the angular final speed ω of the apparatus. 8
FUNDAMENTAL PHYSICAL LAWS Principle of relativity: Physical laws work in the sae way for observers in unifor otion as for observers at rest. The superposition principle: the effective force on an object is the net force, the vector su of all forces acting on the object, each force unaffected by the presence of other interactions. The oentu principle, and the definition of oentu. (These ust be eorized.) The energy principle, and the definitions of work and energy. (These ust be eorized.) The angular oentu principle. (This ust be eorized.) The relationship aong position, velocity, and tie. (This ust be eorized.) The second law of therodynaics or the entropy principle: If a closed syste is not in equilibriu, the ost probable consequence is that the entropy will increase. MULTIPARTICLE SYSTEMS K total = K trans + K rel to c = K trans + K rot + K vib K trans --M total v c = P ----------- total M nonrelativistically L P total = M total v c nonrelativistically K rot rotation = -------- = --Iω I I = r + r + 3 r 3 +... L A = r A p for a point particle τ A = r A F L A = L trans,a + L rot = ( R c,a P total ) + L rot for a ultiparticle syste L rot = Iω EVALUATING GENERAL PHYSICAL QUANTITIES E ( pc) = ( c ) ; E = Pc for assless photon ω = π T v = ωr Ω ( q + N )! = ---------------------------- S = klnω q! ( N )! -- T = ds ------ de EVALUATING SPECIFIC PHYSICAL QUANTITIES F grav G ------------- GM = ; can be approxiately g ; change can be approxiately (gy) r U grav = ---------------- r F electric ----------- Q Q = ---------------- 4πε 0 r U electric ----------- Q Q = ------------- 4πε 0 r F spring = k s s, opposite the stretch U spring = --k s s + U 0 U U U F = < ------, ------, ------ > Power = energy/tie (watts = joules/second) x y z E theral = C T air resistance: F = --CρAv sliding friction: vˆ f µf N oscillator: E N N h k s --- k = ; classically ω s 3.6 ev = ---- hydrogen ato: E (N =,,...) N = ------------------ N Y F A k ------------- s, interatoic k = = ---------------------------- v L L d sound = d s, interatoic atoic ---------------------------- atoic dp ω Circular otion at constant speed: ----- dp = ------------------------------r, ----- ω for << c dt v c dt r v dθ π where ω = ----- = -----, v dt T π r = ----------- = ω r T 9
x k = A cos ---- s t ω k = ---- s = πf = π ----- T sliding friction: f µf N oscillator: E N N h k s --- k = + E ; classically hydrogen ato: (N =,,...) 0 ω s 3.6 ev = ---- E N = ------------------ N C = E ------ T (high-teperature liit: --k per quadratic energy ter; solid is 3k/ato) A coponent of angular oentu is an integer or half-integer ultiple of h. The square agnitude of angular oentu has quantized values L = ll ( + )h, where l has integer (0,,,...) or half-integer values (/, 3/, 5/,...). Sphere Cylinder or disk Thin rod (about axis shown) Solid cylinder (about axis shown) L L I = --MR 5 I = R --MR I = -----ML I R = -----ML + 4 --MR CONSTANTS G = 6.7 0 N /kg g = 9.8 N/kg c = 8 3 0 /s 34 h = 6.6 0 J s h h 34 3 = ----- =.05 0 J s k =.4 0 J/K π 9 N /C 9 ----------- = 9 0 proton or positron charge: e =.6 0 coulob 4πε 0 9 ev =.6 0 J 3 Avogadro s nuber = 6 0 olecules/ole 3 proton neutron hydrogen ato = 0 kg/ole electron = kg 3 7 3 = 0 kg 9 0 6 0 atos/ole 4 M Earth = 6 0 kg 6 Radius of Earth = 6.4 0 M Moon = 7 0 kg 8 Distance fro Earth to Moon = 4 0 30 M Sun = 0 kg Distance fro Earth to Sun =.5 0 Typical atoic radius r 0 0 Proton radius r 0 5 5, R nucleus (.3 0 )A 3 Heat capacity of water = 4. (J/K)/gra. Energy of visible light is fro about.8 ev to about 3. ev. Effective stiffness of interatoic bond in aluinu is 6 N/, lead is 5 N/. 0
Mass of one ole of atos: carbon gras (0.0 kg) nitrogen 4 gras (0.04 kg) oxygen 6 gras (0.06 kg) aluinu 7 gras (0.07 kg) iron 56 gras (0.056 kg) lead 07 gras (0.07 kg) CONVERSION FACTORS kg near the Earth s surface weighs. pounds. inch =.5 c foot = 30 c