# FACULTY OF SCIENCE SAMPLE FINAL EXAMINATION PHYSICS 198-101A (2000) MECHANICS AND WAVES

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FACULTY OF SCIENCE SAMPLE FINAL EXAMINATION PHYSICS 198-101A (2000) MECHANICS AND WAVES

FACULTY OF SCIENCE SAMPLE FINAL EXAMINATION PHYSICS 198-101A (2000) MECHANICS AND WAVES Examiners: INSTRUCTIONS: Professor R.B. Moore Date: Thursday, Dec. 14, 1999 Professor Z. Altounian Time: 9:00 - 12:00 Answer all 8 questions. All questions are worth 10 marks. The marks for the parts of each question are indicated. (80 marks are full marks for this exam.) Note that marks are given separately for indications of the correct method. Also, in some questions, marks are awarded for a diagram. Just stating the correct answer will not be enough for full marks. Formulae and other necessary data are at the back of the examination paper. Dictionaries are permitted Any type of calculator may be used. The examination questions are on two sheets. T h e complete examination package comprises 5 s h e e t s , including this cover sheet. You may keep the examination sheets if you wish. Physics 198-101A Sample Final Examination (2000) 1. page 1 of 2 A football player runs a pass-pattern which requires him to accelerate at a point 12 meters downfield, in a direction 30 degrees right of downfield, in 4 seconds, then, without changing speed, to cut back in a direction 45 degrees left of downfield for another 2.5 seconds at constant speed, whereupon he is to catch the ball in flight at that point. Draw a diagram of his path. (Include the distances.) (4 marks) Assuming he accelerated at a constant rate, draw a graph of his x-velocity vs time. (x is the direction to his right. Include numbers.) (4 marks) Draw a graph of his y-velocity vs time. (y is the down field direction.) (2 marks) 2. A student walks in a circle, taking steps of 80 cm. If at each step she changes direction by 1.5 degrees, what is the diameter of her circle? (3 marks for diagram, 1 mark for method 1 mark for answer) If her pace is 80 steps per minute, what is her angular velocity? (2 marks for method 1 mark for answer) What is her centripetal acceleration? (1 mark for method 1 mark for answer) 3. The temperature of a chamber is found to vary with time according to the equation T = 16.0 + 25.4 sin(0.00246t − 0.425) 0C What is the amplitude of the temperature oscillation? (1 marks) What is its period? (3 marks) What is its phase? (1 mark) Sketch this function as a graph. (5 marks) 4. A polarizing filter is placed in the light beam illuminating a microscope slide. Another polarizing filter in the body of the microscope intercepting the light after it has passed through the slide is oriented so that there is no light passing through to the eye-piece. A specimen that is optically active is then placed on the slide. EYEPIECE ANALYZING POLARIZER SPECIMEN LIGHT POLARIZER LIGHT SOURCE If the optical rotation of this specimen allows an electric field vector at the eye that is 5% of that passed through the first filter, as measured by a light meter, by what angle has the specimen rotated the plane of polarization? . (3 marks for drawing of electric field vectors, 3 marks for method and 4 marks for answer.) 5. A noisy piece of machinery is causing a 400 Hz tone at your ears, with a pressure -4 amplitude of 2.5 × 10 Pa. There is a so-called "active noise suppression system in place, which is essentially a sound amplifier which picks up the sound from the machine and plays it back through a loudspeaker to your ears but with a phase delay so that it tends to cancel the machine sound. If this loudspeaker gain is set so -4 that it does produce 2.5 × 10 Pa at your ears but at a phase of 160 degrees relative to the sound of the machine itself, what will be the sound pressure at your ears? (3 marks for phasor diagram, 4 marks for method and 3 marks for correct answer.) Physics 198-101A Sample Final Examination (2000) 6. page 2 of 2 You are sitting on a bathroom scale in an airplane. (Don't ask why you might be doing this, perhaps you are a bathroom-scale fetiscist.) As the plane enters an "airpocket" you notice that the scale reads 45 kg. If you know that you weigh 80 kg, what is the vertical acceleration of the plane ? (3 marks for diagram 2 marks for method, 2 marks for answer) If this acceleration persists, by how much has the plane risen (or fallen - specify which) in 5 seconds? (2 marks for method, 2 marks for answer) 7. At what altitude above the Earth's surface is the gravitational field 6.5 N/kg? (3 marks for method, 2 marks for answer) What is the gravitational potential at this height? ? (3 marks for method, 2 marks for answer). Take the field at the Earth's surface to be 9.8 N/kg and the potential to be zero at infinity. The mass of the Earth and the gravitational constant can be found on the symbol sheet at the back of the exam. 8. A torque of 1.2 N-m2 is applied to a 12 kg hoop of 1.6 m diameter that is rotating about its center. What is its angular acceleration? (2 marks for diagram, 2 marks for method and 1 mark for correct answer). What would be the kinetic energy of the hoop after 2 seconds of application of this torque? (1 mark for method, 1 mark for answer). At this 2 seconds the torque is released and the hoop left to spin. An electromagnetic clutch then locks the hoop to a disk on the same axis but with a moment of inertia of 8 kg-m2. What will be the rotational speed of the locked combination? (3 mark) Physics 198-101A Symbols used in Formula Sheet (Revised Nov. 2000) Quantity Area Base of triangle Height of triangle Radius Volume Horizontal coordinate Vertical coordinate Angle Arc length Angular speed Velocity, speed Angular acceleration Acceleration Time Displacement Displacement vector Initial velocity Final velocity Average velocity Initial angular velocity Final angular velocity Average angular velocity Velocity vector Acceleration vector Centripetal acceleration Density Mass Specific gravity Pressure Height, depth Acceleration of gravity Force Torque Force perpendicular. to r Distance of line from pivot Distance from center (pivot) Spring constant Stress Strain Young's Modulus Shear stress Shear strain Shear Modulus Gravitational constant Major mass Gravitational field Gravitational potential Distance from Earth center Distance from Sun center Radius of Sun Radius of Earth Mass of Sun Mass of Earth Quantity Work Symbol A b h r V x y θ sarc ω v α a t x, y, z, s s vi vf vave ωi ωf ω ave v a aC ρ m sg p h g F τ F r r k σ ε Y σs γ S G M g φG r r R R M M Symbol W Value, units m2 m m m m3 m m Radians (rad) m rad/s m/s m/s2 m/s2 s m m m/s m/s m/s rad/s rad/s rad/s m/s m/s2 m/s2 kg/m3 kg Pascal (Pa) m 9.8 m/s2 Newton (N) N-m N m m N/m Pa Pa Pa Pa 6.67×10-11 N-m2/kg2 kg N/kg Joule (J)/kg m m 6.97×108 m 6.37×106 m 2.0 ×1030 kg 5.98×1024 kg Value, units J Parallel comp. of l Distance, length Power Potential energy Potential energy of gravity Potential energy of a spring Kinetic energy Kinetic energy of rotation Moment of inertia Linear momentum Linear momentum vector Weight Tension Angular Momentum Wave distance Wave, oscillation amplitude Wave, oscillation frequency Wave, oscillation period Wavelength Wavenumber Angular frequency Phase angle Angle of incidence Angle of reflection Index of refraction Angle of refraction Speed of light Refractive power (diopters) Focal length Distance of object from lens Distance of image from lens Height of image Height of object Magnification Electric field strength Two slit separation Width of slit Diameter of circular beam Resolution Numerical aperture Diameter of a lens (aperture) Decibels Harmonic number Fundamental frequency Electrical current Electrical potential Electrical resistance Number of kg-moles Universal gas constant Temperature Diameter of tube Viscosity Amplitude of Sound pressure Decibel relative Decibel absolute l l P PE PEG PEk KE KEω I p p W T L z A f T λ κ ω φ θi θr n θt c D f so si hi ho M E s w d R NA a db n fo I V R n R T d η pamp dB DBO m m Watt (W) J J J J J kg-m2 kg-m/s kg-m/s N N kg-m2/s m m Hz s m m-1 rad/s rad rad rad rad 3.00 × 108 m/s m-1 m m m m m V/m m m m m m Hz Ampere (A) Volt (V) Ohm ( Ω ) 8314.41 J/kgmole/K K m N-s/m2 Pa Physics 198-101A Formula sheet (revised Nov 2000) GEOMETRY Area of a triangle Area of a circle Surface area of a sphere Volume of a sphere 1 4 A = πr 2 A = 4πr 2 A = bh V = πr 3 2 3 y x y Right-angled triangle: r 2 = x 2 + y 2 ; sin θ = ; cosθ = ; tan θ = r r x sarc v aT θ2 Radian measures: θ = Small angle limit: sin θ = tan θ = θ ; cosθ = 1 − ; ω= ; α= r r r 2 KINEMATICS 1 Motion at constant velocity: s = vt; θ = ωt ( ω = 2πf ; f = ) T 1 1 vave = vi + v f ; v f = vi + at; s = vavet = vi t + at 2 ; v 2f = vi2 + 2 as 2 2 Motion at constant acceleration: 1 1 ω ave = ω i + ω f ; ω f = ω i + αt; θ = ω avet = ω i t + αt 2 ; ω 2f = ω i2 + 2αθ 2 2 v2 2 Vector addition of velocities: v AC = v AB + v BC Centripetal acceleration: ac = ω r = r STATICS F ρ ρ m = Density: ρ = Specific gravity: sg = ; Weight: W = mg : Pressure: p = ; ρwater 1000 V A ( ( ) ) Hydrostatic pressure: p = ρgh Balance: ΣF = 0; Στ = 0; τ = F r = Fr = Fr sin θ ; Spring formula: F = − kx F ∆l σ Fs σ ; Y = ; σs = εs = γ ;S = s Stress, strain: σ = ; ε = A l ε A εs GRAVITY: Law of gravitation mM FG = − G 2 r DYNAMICS Gravitational field g = −G Gravitational potential: M r2 φG = − G M = rg r 1 2 1 1 kx ; KE = mv 2 ; KEω = Iω 2 2 2 2 dp dL 1 2 2 Force, torque: F = ma = ; W = mg; p = mv; τ = Iα = ; L = Iω ; I = ∑ mi ri ; Idisk = mr i dt dt 2 1 x = A sin(ωt + φ ); ω = 2πf ; T = ; f Mass - spring Pendulum π Simple Harmonic Motion: v x = Aω cos(ωt + φ ) = Aω sin(ωt + φ + ); g k 2 ω= ω= l m 2 2 2 ax = − Aω sin(ωt + φ ) = Aω sin(ωt + φ + π ) = −ω x Work, energy: W = Fl = Fl cosθ ; P = Fv cosθ ; PEG = mgh; PEk = GEOMETRICAL OPTICS 1 vi 1 1 1 1 ; sin θ c = ; Lens refractive power: D = : Lens eqn: + = vt n so si f f h s 250 mm 1 1 1 Magnifying power of eyepiece = : Close combined thin lenses = + ; D = D1 + D2 Magnification i = − i ho so f f1 f2 f f Compound microscope magnification: M = Mobjective × Meyepiece Telescope magnification: M = objective feyepiece Reflection: θ r = θ i ; Refraction: sin θ i = n sin θ t ; n = WAVES 2π Wave speed: v = fλ ; λ For light in a vacuum v = c ; For sound in normal air (20C) v = 343 m/s; Polarization: Et = Ei cosθ Wave propagation y = A sin(ωt − kz ) ; k = Interference and diffraction (central maximum-small angle): λ λ λ λ Two-slit: θ = Single slit: θ = 2 Circular beam: θ = 2.44 Circle of confusion: θ = 1.22 w s d d λ λ a Resolution of a microscope: R = 1.22 ; NA = Resolution of a telescope: θ = 1.22 f NA d SOUND dBO p p2 amp 20 −5 p = 2 . 8 × 10 × 10 ; ; One decibel = 12.2% increase in pressure. dB = 20 log10 2 ; dB0 = 20 log10 amp −5 p 2 . 8 × 10 1 3 4 5 6 Harmonics: fn = nf0 ; Fifth = ; Fourth = ; Third = ; Minor Third = ; Semi-tone: 12 2 =1.0595 (5.95% increase) 2 3 4 5 MISCELLANEOUS V πd 4 p; Electrical current: I = ; Ideal Gas Law: pV = nRT ; Viscous flow: v flow = 128ηl R