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phy140_final_may06 - UNIVERSITY OF TORONTO Faculty of Arts...

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Unformatted text preview: UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL/MAY EXAMINATIONS 2006 PHY 140Y' Duration - 3 hours Write your name, student number, and TA’s name on ALL examination booklets used. The only allowed aids are hand-held calculators and an attached appendix of formulae. Answer BOTH sections A and B. Section A is worth 60%, and Section B is worth 40% of the total. Show all derivations, and justify each significant step in your answer. This exam has 8 pages. PART A (60%) Answer SIX of the following seven questions. Each question is of equal value. If you answer all seven questions», CLEARLY INDICATE which six you would like marked. FIG. 1: Question 1. 1. A person is standing against the inner wall of a rotating cylinder of radius R as shown in Figure 1. The rotation proceeds at a uniform angular velocity w. The floor of the cylinder suddenly drops down a distance L. The maximum coefficient of static friction is ,us and the coefficient of kinetic friction is pk. Express your answers to the following questions in terms of the valiables given in the question and g (the gravitational acceleration) only. PHYI40Y - Foundations of Physics 2005-2006 Final Exam, page 1 of 8 (a) For what values of u) will the person remain stationary against the wall when the floor drops out? (b) If instead, the conditions are such that the person begins sliding down the wall (i.e. accelerating from rest), how long does it take to slide down the distance L? (c) Does your solution for the time in part (b) ever give you an unphysical answer? Why? (1-2 sentences max). 2. A mass is connected to 2 rubber bands of length L as shown in Figure 2. Each rubber band has a constant tension T. You can neglect the force of gravity in this problem. The mass is displaced horizontally by a very small distance :1: and then released (so 2: < L always). The mass will exhibit simple harmonic motion. Express all your answers in terms of the variables given in the question. (Note: the picture is not to scale, a: is actually much smaller than portrayed). (a) Find a simple expression for the small angle 6 in terms of the displacement at of the mass using the fact that m < L. (b) What is the angular frequencey w of this motion? (0) If att = 0, m(0) = 3:0 and 11(0) = 0, find :r;(t) for any 15. FIG. 2: Question 2. PHY140Y - Foundations of Physics 2005-2006 Final Exam, page 2 of 8 FIG. 3: Question 3. 3. A small ball of mass ml — 1 kg is suspended at the end of a string of length L = 1 m. The ball is released from rest at an angle 6 = 60° as shown in Figure 3. Use 9 = 10ms‘2 throughout this problem. (a) What is the ball’s velocity when it reaches the vertical Ge. 9 = 0)? (b) If a second ball of mass m2 = 0.5 kg is at the base of the circle swept out by the first ball, and the 2 balls collide perfectly elastically, what maximum height does the ball on the string reach after the collision? (Assume the collision is instantaneous, occurs directly at 0 = 0 and that the balls do not interact at all after the collision). for part c with the bug: FIG. 4: Question 4. 4. A cylindrical hollow hoop of radius R and mass M is rolling without slipping along a horizontal floor at 11cm 2 120 when it comes to an incline of 9 degreesas shown in Figure 4. Express your answers in terms of the variables given in the problem and the gravitational acceleration 9 only. (a) Derive the moment of inertia of the hoop about its center starting from the equation I = f r2dm. (b) What distance does the hoop travel along the incline before reversing direction? PHYI40Y — Foundations of Physics 2005-2006 Final Exam, page 3 of 8 (c) What is the gravitational torque acting on a bug of mass mb stuck on the hoop as shown in Figure 4? (I promise the bug will jump off the hoop before it gets squished underneath it!) 5. A point charge q = —1.0 x 10-5 C is placed at the centre of a thick spherical conducting shell of inner radius 2.5 m and outer radius 4.0 m and total charge Q = 2.0 X 10‘5 C, as shown in Figure 5. Find 0 (3) FIG. 5: Question 5. (a) the surface charge density on the outer wall of the shell, _ (b) the surface charge density on the inner wall of the shell, and (c) the magnitude and direction of the electric field at the points (1), (2) and (3), situated 1.0 m, 3.0 m and 5.0 m from the centre of the sphere, respectively. 6. A spaceship is approaching an asteroid directly in its path at a speed 2) 2 4c/ 5. When the spaceship is 1010 m from the asteroid (in the asteroid’s frame of reference), it fires a missile (call this Event A). The missile hits the asteroid 36.0 3 later, in the reference frame of the asteroid, and destroys it (call this Event B). (a) What is the spacetime interval A32 between Event A and Event B? (b) How long does it take the missile to reach the asteroid, in the reference frame of i. the spaceship, and ii. the missile? (c) How far is the spaceship from the asteroid when the missile hits it, in the reference frame of the spaceship? (d) What is the speed of the missile in the rest frame of the spaceship, as a fraction of c? PHY140Y — Foundations of Physics 2005—2006 Final Exam, page 4 of 8 (a) What is the frequency of a photon whose energy is equal to the total energy of an electron moving at v = 0.500? (b) Light of wavelength A incident on a metal surface causes electrons to be immediately ejected with a maximum kinetic energy of 6.00 eV. When the wavelength of the light is , doubled, the maximum energy of the electrons decreases to 1.00 eV (recall, 1 eV=1.6 x 10"19 J). i. Briefly explain why this observation is inconsistent with the classical picture of light. ii. What is the work function of the metal? iii. What is the initial wavelength A of the light? END OF PART A PHY140Y — Foundations of Physics 2005-2006 Final Exam, page 5 0f 8 PART B (40%) Answer ALL of the following three questions. Each question is of equal value. 8. Consider the pulley system shown in Figure 6. Assume that initially I am holding mg in place (so the system is at rest), that the walls and pulley are frictionless and that m1 2 m2. Also assume 0 S 6 _<_ 7r. Express all your answers in terms of the variables 9, 0, m1, m2 and L only. (a) Once I release 7712 what is the acceleration of 7m? (b) How long will it take for mg to reach the top of the hill? (c) Looking at your answer for part (a), under what 2 conditions is the acceleration zero? Explain how these make sense physically (1-2 sentences max). (d) What is the net work done on moving block m2 from its initial position to the top of the triangle? FIG. 6: Question 8. 9. (a) A wave on a string is described by y(:c,t) = (2.0 cm) x sin[(12.57 rad/m)x — (638 rad/s)t)], where a: is in m and t is in s. i. What is the wavelength of the wave? ii. What is the speed of the wave? iii. At t = 1.0 3, what is the velocity of the point of the string at x = 0.5 m? iv. At t = 1.0 3, what is the slope of the string at x = 0.5 m? (b) A 2.0 m length of wire with a mass of 75 g is vibrated at 30 Hz, producing a standing Wave with three antinodes. What is the tension in the wire? PHYI40Y — Foundations of Physics 2005-2006 Final Exam, page 6 0f 8 10. (a) Consider the electron wave function ca: [ml S 1.0 nm W93) = { c (1) HI E IV l—1 E: :3 B where z is in nm. i. Determine c. (b) The wavefunction of a particle in a potential U (:12) is 712(1/ m. Determine the potential energy U (as) of the particle. (c) Consider a particle whose potential energy U (m) is shown in Figure 7 (the arrows indicate the potential going to infinity at that point). The n = 0 level (grOund state) U (W) 3.0 2.0 1.0 FIG. 7: Question 7. THIS IS THE END OF THE EXAM. Have a nice day. PHYI40Y — Foundations of Physics 2005—2006 Final Exam, page 7 0f 8 Possibly Useful Facts k = 9.0 X 109 Nm2/C2, 60 = 8.85 x 10—12C2/Nm2, c = 3.00 x 108 m/s G = 6.67 x 10-11 Nm2/kg2, h = 6.63 x 10-34 J s, h = 1.06 x 10-34 J s melectron = 9.11 x 10—31 kg = 511 keV/c2 = 0.511 MeV/c2, 1 eV = 1.6 x 10-19 J (1 Fnet=ma= 71%: P:mV, Fk:lu'klnlw OSFsS/J'slnl) K: émvz Emech. : K+U; AK 2 Wnet: W : fF'dI‘, AU : _Wc: AEmech = Wnc A - B = AB 0080 = AIB, + AyBy + AZBZ, F3 = _fl d5, Ug = my F9 = ”Law, > U9 = ”"2“ , > F» = —kAs, Usp = yams? s=R6, v=rw~r§—Z, I |—— lac|='w27", a9=r0z=rQ dt xcm=fi2mixi=fif$dm, |A><B|=ABsin0, AXB=—BXA x 3:1}, ini‘I2’E, kxz=3, T=I‘XF, 2D:’7’=7‘i<}, Tnet=IOz = Emir? : fTZ dm) I = Icm +Md21 Krot 22211012, Ktot : Kcm+ Krot r x , é}; anet, L = [wit Am’ — 7(Ax — 'uAt), At' = 7(At —- vAx/cz), ”)1 = (1 — 122/02)—1/2 Ax: ’y’(Am +vAt',) At='y(At'+'uAm’ /c2) ,_ uz—v V , uy , uz um 1 — vuz/cz’ uy : 7(1m ~ fivum/cz)’ uz : 7(1 — yum/02) _ mc2 1/1—112/02, =‘/1——:2/c2 F12 = 19%;]2?’ E_ __ _’ (I): f E dA— _ genclosed 7” q A 60 kg kqlf a = -—A = : ri) E : E : __ E 7‘2 r, E ;E1 g“ 7"? 27mm" 260 _ , h '0: I, (41:27wa kzazr‘) 'U=_f)\, Ezhf) p:— Vu A A PHY140Y - Foundations of Physics 2005-2006 Final Exam, page 8 of 8 ...
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