Page 1 of 5 Queen s University at Kingston Faculty of Arts and Science Department of Physics PHYSICS 106 Final Examination April 16th, 2009 Professor: A. B. McLean Time allowed: 3 HOURS Instructions This examination is three hours in length. Part I: Put your student number on the scantron sheet in two places. Attempt all ten multiple choice questions and record your answers on the scantron sheet. Each question is worth one mark. The boxes should be penciled so that the letter is not visible. Select test form A. Part II: Put your student number on the front of the worksheets. Answer two out of three worksheet questions. Indicate clearly which questions you want marked or the first two answers in the worksheets will be marked. You may do rough working in the answer booklet. Each question is worth five marks. When finished put your scantron sheet and worksheet inside your rough working booklet and write your student number on the front. Please note: Proctors are unable to respond to queries about the interpretation of exam questions. Do your best to answer exam questions as written. 1
Part I: Attempt all 10 questions Page 2 of 5 Qu 1 Seven identical charges of magnitude Q = +1.0 nc are located at the vertices of a cube of side a = 1.0 cm. The potential at point P is? A 2400 V B 3200 V C 4600 V D 5100 V E 6200 V a P Q Qu 2 While spinning down from 500.0 r.p.m. to rest, a solid uniform flywheel does 5.1 kj of work. If the radius of the disk is 1.2 m, what is its mass? You may use the fact that the moment of inertia of a uniform disk about the center is I c = 1 2 mr2. A 4.1 kg B 5.2 kg C 4.4 kg D 6.0 kg E 6.8 kg Answer: B Qu 3 Three blocks of length L are arranged as shown. The block on top has mass 2m. The block in the middle has mass m and the block on the bottom has mass 3m. What is the maximum value for x for which the blocks are stable and do not collapse? A L/4 B L/3 C L/5 D 2L/3 E 5L/6 L 3m m 2m x 2
Part I: Attempt all 10 questions Page 3 of 5 Qu 4 I 1 2 V 4Ω 2Ω 4 V 4 V I 1 +I 2 I 2 2 V 2Ω Which of the following is correct: A I 1 = 3/5 A B I 1 = 7/5 A C I 2 = 6/5 A D I 1 + I 2 = 4/5 A E I 1 + I 2 = 7/5 A Answer: D Qu 5 An insulating solid sphere, with radius a = 1.0 m, has a charge of Q = +1.0 10 9 C distributed uniformly throughout its volume. Use Gauss s law to find the magnitude of the electric field a distance of 0.5 m from the center of the sphere. A 2.3 N/C B 4.5 N/C C 7.2 N/C D 7.2 N/C E 9.0 N/C Answer: B 3
Part I: Attempt all 10 questions Page 4 of 5 Qu 6 Three charges, Q 1 = Q 2 = 20 10 6 C and Q 3 = 40 10 6 C, are located in an equilateral triangle of side a = 1.0 m. The net force (F x, F y ) acting on Q 1 is? Q 3 y a a Q 1 x a Q 2 A (3.6, -1.4) N B (7.2, -2.4) N C (7.2, -3.2) N D (9.4, -1.8) N E (9.4, -2.4) N Qu 7 Positive charge Q is placed on a conducting spherical shell with inner radius a 1 and outer radius a 2. A point charge q is placed at the center of the cavity. The magnitude of the electric field at a point outside the shell, a distance r from the center, is: A. κ(q + Q)/r 2 B. κq/(a 2 1 r 2 ) C. κq/r 2 D. κq/a 2 1 E. κ(q + Q)/(a 2 1 r 2 ) 4
Part I: Attempt all 10 questions Page 5 of 5 Qu 8 The figure shows a current entering a truncated solid cone made of a conducting metal. The electron drift speed at the 3.0 mm diameter end of the cone is 4.0 10 4 m/s. What is the electron drift speed at the 1.0 mm diameter end of the wire? A 3.6 10 3 m/s B 1.2 10 3 m/s C 1.3 10 4 m/s D 4.4 10 5 m/s E 5.2 10 5 m/s 3mm 1mm I = 1.0 A Answer: A Qu 9 A projectile is launched over horizontal ground as shown. At which of the points A-E is v perpendicular to a? y C B D v i 45 o 0 A x E Qu 10 Two boys, with masses of 40 kg and 60 kg, respectively, stand on a horizontal frictionless surface holding the ends of a light 10 m long rod. The boys pull themselves together along the rod. What distance will the 40 kg boy have moved when they meet? A 4 m B 5 m C 6 m D 10 m E need to know the forces they exert Answer: C 5
Part II: Long Question Worksheets (attempt 2 out of 3) Student Number: Worksheet One (a) A marble rolls down a track and around a loop-the-loop of radius R without slipping. The marble has mass m and radius r. Calculate a formula for the minimum height h the track must have for the marble to make it around the loop-the-loop without falling off. For the marble I c = 2 5 mr2. 2r h R continued over... 1
(b) A cube slides down a frictionless track and around a loop-the-loop of radius R. The cube has mass m and the length of each side is 2r. Calculate a formula for the minimum height h the track must have for the cube to make it around the loop-the-loop without falling off. 2r h R continued over... 2
(c) If the answers to parts (a) and (b) are the same, explain why they are the same. If the answers to parts (a) and (b) are different, explain why they are different. 3
Part II: Long Question Worksheets (attempt 2 out of 3) Student Number: Worksheet Two a The figure shows a thin rod with charge Q that has been bent into a semi-circle of radius R. Find an expression for the electric potential at the center. Q R center b A disk with a hole has inner radius a and outer radius b. The disk is uniformly charged with total charge Q and it lies in the xy plane. Show that the on-axis electric potential at a distance z from the center of the disk is V (z) = κ2πσ [ b2 + z 2 ] a 2 + z 2 You may use: b a xdx x2 + z 2 = [ x2 + z 2 ] b a continued over... 4
Put your answer to part b on this page. continued over... 5
c Using the result from part b, and E z = dv/dz, calculate the electric field at distance z from the center of the disk. d Use the result from part c to calculate the electric field above an infinite uniformly charged plane. 6
Part II: Long Question Worksheets (attempt 2 out of 3) Student Number: Worksheet Three A rectangular current loop, with sides of length a and b, is placed in a magnetic field that is uniform and parallel to the x-axis. The loop is pivoted about points P 1 and P 2, and its axis of rotation is the line that joins these two points. Conventional current flows in the direction shown. θ is the angle between the normal to the current loop (ˆn) and the B field. a Draw the forces on each side of the loop. Indicate in what direction the loop will rotate? P 1 b 3 I pivot 1 z a n θ B 2 I 4 P 2 pivot b pivot 0 z I θ x n B y 0 x perspective view side view b Add a vector to both diagrams, showing the magnetic moment µ. c How is the magnitude of the magnetic moment related to the size of the loop and the current I? 7
d Explicitly calculate the forces on each side of the loop ( F 1 derive a formula for the torque on the current loop. F 4 ) and e Graph the magnitude of the torque for angles in the range π θ π. continued over... 8
f Can you make a motor using the current loop shown above? If not, what could be changed to cause the loop to rotate continuously in one direction? 9
PHYS106 EQUATION SHEET Page 1 of 6 Vectors a = a xˆx + a y ŷ + a z ẑ and b = b xˆx + b y ŷ + b z ẑ a b = ab cos θ = a x b x + a y b y + a z b z cos θ = (a b)/ab a b = ab sin θê = ˆx(a y b z a z b y ) + ŷ(a z b x a x b z ) + ẑ(a x b y a y b x ) a = a 2 x + a 2 y + a 2 z â = a/a Math f x n e x 1 x ln x sin x cos x tan x f(g(x)) df dx nx n 1 e x 1 x 2 1 x cos x sin x sec 2 x df dg dg dx Roots of quadratic ax 2 + bx + c = 0 are x = 1 2a ( b ± b 2 4ac). Area of circle = πr 2, circumference of circle = 2πr, area of sphere = 4πr 2, volume of sphere = 4πr 3 /3 1
PHYS106 EQUATION SHEET Page 2 of 6 Kinematics r = xˆx + yŷ + zẑ v = dr dt a = dv dt Constant acceleration t t 12 Constant angular acceleration t t 12 x 12 = v 1 t + 1 2 at2 v 12 = at θ 12 = ω 1 t + 1 2 αt2 ω 12 = αt v 2 2 = v 2 1 + 2ax 12 ω 2 2 = ω 2 1 + 2αθ 12 Projectiles: R = v 2 o sin 2θ/g h = v 2 yo/2g Uniform circular motion (ω = dθ/dt = 2πf = 2π/T ) v = ωr v = ωrˆθ ˆθ is tangential a r = ω 2 r = v2 r a r = ω 2 rˆr = v2 r ˆr ˆr is radial Dynamics F net = i F i = ma Gravity F = GMm r 2 ˆr F = mg U = mgh G = 6.67 10 11 Nm 2 kg 2 g = 9.81 ms 2 2
PHYS106 EQUATION SHEET Page 3 of 6 Momentum, impulse and torque p = mv F = dp dt l = r p τ = dl dt Impulse p 12 = p 2 p 1 = J = t2 t 1 F(t)dt τ = r F τ = rf sin θ = r F = rf v 1 = m 1 m 2 m 1 + m 2 u 1 + 2m 2 m 1 + m 2 u 2 v 2 = 2m 1 m 1 + m 2 u 1 + m 2 m 1 m 1 + m 2 u 2 Constant Value Units e 1.6 10 19 C ɛ o 8.854 10 12 N 1 m 2 C 2 κ = 1/4πɛ o 9 10 9 Nm 2 C 2 m e 9.1 10 31 kg m n 1.7 10 27 kg m p 1.7 10 27 kg κ m = µ /4π 10 7 TmA 1 µ o 4π 10 7 TmA 1 3
PHYS106 EQUATION SHEET Page 4 of 6 Systems F ext i = Ṗ If F ext i = 0 then P = P i i Elastic collision : K 1 = K 2 Inelastic collision : K 1 K 2 Center of mass : x c = 1 m i x i M i y c = 1 m i y i M i z c = 1 m i z i M i I = i m i r 2 i I p = I c + Md 2 P = Mv c F = dp dt = Ma c L = I ω i τ i = dl dt = I α For rolling motion with no slipping, x c = θr, v c = ωr, a c = αr K = 1 2 I cω 2 + 1 2 Mv2 c K = 1 2 I cω 2 (fixed axis) W 12 = 2 1 F dl W = F d P = dw dt = F v W 12 = 2 1 τdθ P = τω F = kx U = 1 2 kx2 W 12 = K 12 = 1 2 mv2 2 1 2 mv2 1 W 12 +U 12 = 0 (conservative) 4
PHYS106 EQUATION SHEET Page 5 of 6 Statics F i = 0 i τ i = 0 i Electrostatics F = κqq tˆr E = F E = κqˆr r 2 q t r 2 U = κqq t r V = U q t V = κq r E = i E i E = κ dqˆr r 2 dq = λdl = σda = ρdν Φ E = A E da = Q enc ɛ o E = Q enc A ɛ p = QD (pointing from Q to +Q) τ = p E U = p E V p = κp cos θ r 2 Magnetostatics 2 V 12 = E dl 1 2 U 12 = F dl 1 E = dv dl F = qv B df = Idl B r c = mv qb τ = µ B µ = I A U = µ B db = k m Idl ˆr r 2 T = 2πm qb L B dl = µ I enc 5
PHYS106 EQUATION SHEET Page 6 of 6 Circuits Kirchhoff s Laws: The Loop law: i V i = 0. The Node law: i I i = 0. Equivalent resistances: Series r eq = i r i. Parallel r 1 eq = i r 1 i V = IR P = IV = I 2 R = V 2 R R = ρ L A ρ = 1 σ J = nev d J = I A 6