Waves and Fields (PHYS 195): Week 9 Spring 2018 v1.5

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1 Waves and Fields (PHYS 195): Week 9 Spring 2018 v1.5 Intro: This week we finish our discussion of electric potential and electrostatics before moving on to current and a bit of magnetostatics. Reminder: Mid-term II will be held in lab starting April 3. The topics include, predominantly, resonance, waves and electric fields. Reading: For Monday: HRW page 730 (on energy density of the electric field) HRW , Wednesday: HRW Friday: HRW 29.1 Physics Topics: Energy density of the electric field Moving charges - current Magnetic field Biot-Savart force Lorentz force qv B Magnetic moment The Il B force Math Topics: Cross products (review) Problems: All these problems are optional and are example mid-term questions. Instructions that you may expect: This is a closed book exam held under the auspices of the Hamilton Honor Code. All work must be completely your own. Useful bits of information are included in the problems and on the Handy Relations page. Please show all your work and do not hesitate to ask questions! Assume the speed of sound is 343 m/s throughout. (1) (10 pts.) Equations and Derivations - choose one of the following: (a) Derive the wave equation for transverse waves on a string. Assume the string has linear mass density µ and tension F T. Start with a careful diagram and include explanations of your steps. What is the wave speed? (b) Derive the wave equation for sound waves in a fluid. Assume a bulk modulus of B and a density ρ. Start with a careful diagram and include explanations of your steps. What is the wave speed? 1

2 2 (2) (10 pts.) In 2014 scientists announced the discovery of an underground sea of liquid water on the moon Enceladus, which orbits Saturn. They found the sea by carefully mapping the gravitational field of the moon. They tracked the Cassini spacecraft using radio waves. Kenneth Chang, writing in the New York Times, described the method, As the pull of Enceladuss gravity sped and then slowed the spacecraft, the frequency of the radio signal shifted, just as the pitch of a train whistle rises and falls as it passes by a listener. Radio signals are a form of light and travel at c = m/s. (a) What is the name of the physical effect used to find the sea? (b) They had astonishing precision and were able to measure a change of velocity as small as δv = 0.09 mm/s. What relative frequency difference (δf/f) must they been able to measure? [If you are curious how this is possible, the experimenters used atomic clocks, which keep time within a part in to ] (3) (10 pts.) In the lab room there are some funny plastic tubes. If you whirl them around your head (please try it!) they make sounds. (a) What is a possible explanation of this phenomenon? (b) What is the length of one of the tubes? (c) Estimate the lowest frequency produced by the whirling tube. Use v = 343 m/s for your phase velocity. (d) Derive the relation for all of the frequencies likely produced by the tube. (4) (10 pts.) A lyre, or simple harp, is made of a wood frame and a set of strings fixed on both ends. Suppose that one nylon string is tuned to D 4 at 294 Hz, has a length of m, and has a linear mass density of 7.2 g m 1. (a) Why is it that when you pluck a string that you hear the 294 Hz standing wave (and higher harmonics)? (b) Find the wave length of the fundamental mode. (c) What is the tension in the string? Assume that the fundamental mode gives the note D 4. (d) Find the distance between the nodes for the 3rd harmonic. (e) What is the phase velocity for this string? (f) In light of these results, comment on the assumption in part c. (5) Now dipoles will not be part of the Midterm II A dipole with Q = 7.40 nc, d = 0.60 cm is in a uniform electric field of strength 3200 V/m. The angle between the dipole moment and the electric field is θ o = 8.0 o 0.14 rad at t = 0. Assume the dipole has a moment of inertia I = kg m 2. (a) Find the torque on the dipole at t = 0.

3 3 (b) What will the motion be if the dipole starts at rest at angle θ o? A qualitative answer is fine. (6) (10 pts.) Organ sounds: Suppose that you have two organ pipes, each with one open and one closed end. One pipe has a length of 1.26 ± 0.02 m and the other has a length of 1.22 ± 0.02 m. (a) If you played them simultaneously in their second harmonic describe in words what you would hear. Include a sketch of the harmonic. (b) Find the frequency of the sound that you would hear. Please include the uncertainty. (c) Find the frequency of variation in the volume (the wha-wha s) that you would hear. Please include the uncertainty. For this question please assume v = 343 ± 2 m/s. (7) (10 pts.) The Electric Curling Championship has arrived! The goal is to move what I have called the puck to the circle on the right so that it ends there at rest. The particles (or stones) are all of equal mass m = 18 kg, have a high coefficient of static friction and so are essentially fixed, and have a radius of 0.13 m. Currently on the ice are one stone of charge +1.0 C at coordinates (x, y) = (0, 1.0) and another +1.0 C stone at (x, y) = (0, 1.0), in addition to the +1.0 C puck at ( 6.0, 0). The point P is at (4.0, 0). Assume all these distances are in meters. (a) Carefully sketch the electric field of the current configuration of charges. (b) Find the electric potential due to the two stones, which are on the y-axis, as a function of position on the x axis. (c) Once the puck is moving is has a coefficient of kinetic friction µ = Estimate the minimum amount of kinetic energy you must impart to the puck stone so that it arrives at point P. (8) (10 pts.) Two sinusoidal waves of the same frequency travel in the same direction along a string. If y m1 = 2.2 cm, y m2 = 2.2 cm, and the relative phase is φ = π/6, what is the resulting wave s amplitude and phase? (9) (10 pts.) Electrical circuits provide one of the best examples of resonant systems. The equation of motion is L d2 q dt 2 + R dq dt + 1 C q = V o cos(ωt)

4 4 Comparing to our usual mass-on-a-spring case m d2 x dt 2 + bdx dt + kx = F o cos(ωt) shows that in these circuits charge q plays the role of position x; a coil of wire producing a magnetic field with parameter L is like mass; resistance R is like drag; and charged plates producing an electric field with parameter 1/C is like a spring. An alternating electric potential is the analog of the driving force. (a) What is the resonant angular frequency of this electrical circuit? (b) What is the quality factor Q for this circuit? (c) In cell phones, radios and many other devices an antenna, driven by a radio wave, provides the driving potential. In designing such a device would you want to have a low Q or high Q circuit? (d) Sketch the amplitude of the charge q as a function of the driving frequency at late times. (e) Sketch the phase of the charge q, compared to the driving potential, as a function of the driving frequency at late times. (10) HRW Lab: will instead be Mid-term II A look ahead... Next week we finish our brief study of magnetic fields. Waves in the form of light are coming up, see Chapter 33.

5 5 Handy Relations General: τ = Iα I = r 2 dm F = du dx P = F V = B A V F B = ρgv The Taylor series of a function f(x) around x = x o is f(x) = f(x o ) + df dx (x x o ) + 1 d 2 f x=xo 2 dx 2 (x x o ) d 3 f x=xo 6 dx 3 (x x o ) x=xo Handy approximation: Oscillations: (1 + x) n 1 + nx for small x F = kx and T = 2π and ω = 2πf ω k For spring-like SHM ω o = m. For a simple pendulum ω o = g l. Q = mω o /b E = 1 2 mω2 oa 2 A damped harmonic oscillator has the equation of motion d 2 x dx + 2β dt2 dt + ω2 ox = 0 with β = b 2m. The angular frequency is ω d = ωo 2 β 2 For a driven system d 2 x dx + 2β dt2 dt + ω2 ox = F o m cos(ωt) at late times x(t) = A(ω) cos[ωt δ(ω)] with ( ) 2βω A(ω) = ωo 2 ω 2 Waves: For waves on a string: F o /m and δ(ω) = arctan [(ωo 2 ω 2 ) 2 + 4β 2 ω 2 ] 1/2 ] k eff = d2 U dx 2 x=x eq y(x, t) = y m sin(kx ± ωt) k = 2π λ, v = ω k Standing wave condition for waves fixed at both ends L = nλ 2, n = 1, 2, 3,... v = λf

6 6 Phase: ϕ 2π = L λ Phasors: For two waves, with amplitudes y m1 and y m2 and relative phase ϕ, represented by phasors z 1 and z 2, the resulting length of the sum z = z 1 + z 2 = re iα is r = z = ym1 2 + y2 m2 + 2y m1y m2 cos ϕ. The phase is given by y m2 sin ϕ tan α = y m1 + y m2 cos ϕ. Sound: The phase speed of sound in air is about 343 m/s. Light: P = 2π 2 µvf 2 A 2 o, I = 2π 2 vρf 2 A 2, I = P o 1 4π r 2 β = 10 log I I o with I o = W/m 2 f b = f 1 f 2, d sin θ = mλ ( ) v ± f vo = f v v S sin θ = v v obj Fields: f = f c ± v c v µ o = 4π 10 7 Tm/A, ɛ o = C 2 /Nm 2 Coulomb force (magnitude) F = 1 qq 4πɛ o r 2 For a single charge Q E = 1 Q 4πɛ o r 2 ˆr, V = 1 Q 4πɛ o r U = qv V b V a = b a E dl, and E = V For a dipole τ = p E with p = qd Uncertainty: If you like you are welcome to add independent uncertainties in quadrature. For addition and subtraction, add the uncertainties in quadrature, z = x + y or z = x y then δz = δx 2 + δy 2 For multiplication and division then add the relative uncertainties in quadrature, (δx z = xy or z = x/y then δz ) 2 ( ) 2 δy z = + x y

7 7 For a power, multiply the relative uncertainty by the power, i.e. if z = x n then δz z = nδx x. In general for a calculated quantity q = q(x,..., z) then δq = ( q x δx ) ( ) 2 q x δz

8 8 Solution Hints: (1) See class notes and the text. (2) (a) Doppler effect for light (b) ( f f 1 + v ) But the reflection gives twice the shift δf c f = 2δv c = (3) (a) resonance (b) (c) 220 Hz (d) see notes (4) (a) resonance (b) 67.2 cm (c) 280 N (d) 11.2 cm (e) 198 m/s (5) (a) Nm into the page (b) SHM (6) (a) beats (b) (c) (7) (a) a sketch (b) L = 76 ± 1 cm 208 ± 3 Hz 7 ± 7 Hz V = 1 1 2πɛ o x2 + 1 (c) It is not possible. (This part is hard.) (8) 5.3 cm and 0.26 rad (9) (a) is 1/ LC. The sketches for (d) and e) are the same as before. (10) Use superposition and subtract a small disc from a big disk. On the midterm you would be given the result for a disk from class.

Waves and Fields (PHYS 195): Week 4 Spring 2019 v1.0

Waves and Fields (PHYS 195): Week 4 Spring 2019 v1.0 Intro: This week we finish our study of damped driven oscillators and resonance in these systems and start on our study of waves, beginning with transverse