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String Waves

String Waves. Physics 202 Professor Lee Carkner Lecture 8. Exam #1 Friday, Dec 12. 10 multiple choice 4 problems/questions You get to bring a 3”X5” card of equations and/or notes Start making it now I get my inspiration from your assignments

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String Waves

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  1. String Waves Physics 202 Professor Lee Carkner Lecture 8

  2. Exam #1 Friday, Dec 12 • 10 multiple choice • 4 problems/questions • You get to bring a 3”X5” card of equations and/or notes • Start making it now • I get my inspiration from your assignments • Make sure you know how to do homework, PAL’s/Quizdom, discussion questions • Bring calculator – be sure it works for you

  3. Velocity and the Medium • The speed at which a wave travels depends on the medium • If you send a pulse down a string what properties of the string will affect the wave motion? • Tension (t) • The string tension provides restoring force • If you force the string up, tension brings it back down & vice versa • Linear density (m = m/l =mass/length) • The inertia of the string • Makes it hard to start moving, makes it keep moving through equilibrium

  4. Wave Tension in a String

  5. Force Balance on a String Element • Consider a small piece of string Dl of linear density m with a tension t pulling on each end moving in a very small arc a distance R from rest • There is a force balance between tension force: • F = (tDl)/R • and centripetal force: • F = (mDl) (v2/R) • Solving for v, • v = (t/m)½ • This is also equal to our previous expression for v v = lf

  6. String Properties • How do we affect wave speed? • v = (t/m)½ = lf • A string of a certain linear density and fixed tension has a fixed wave speed • Wave speed is solely a property of the medium • We set the frequency by how fast we shake the string up and down • The wavelength then comes from the equation above • The wavelength of a wave on a string depends on how fast you move it and the string properties

  7. Tension and Frequency

  8. Energy • A wave on a string has both kinetic and elastic potential energy • We input this energy when we start the wave by stretching the string • Every time we shake the string up and down we add a little more energy • This energy is transmitted down the string • This energy can be removed at the other end • The energy of a given piece of string changes with time as the string stretches and relaxes • The rate of energy transfer is this change of energy with time • Assuming no energy dissipation

  9. Power Dependency • The average power (energy per unit time) is thus: • P=½mvw2ym2 • If we want to move a lot of energy fast, we want to add a lot of energy to the string and then have it move on a high velocity wave • v and m depend on the string • ym and w depend on the wave generation process

  10. Equation of a Standing Wave • Equation of standing wave: • yr = [2ym sin kx] cos wt • The amplitude varies with position • e.g. at places where sin kx = 0 the amplitude is always 0 (a node)

  11. Nodes and Antinodes • Consider different values of x (where n is an integer) • For kx = np, sin kx = 0 and y = 0 • Node: • x=n (l/2) • Nodes occur every 1/2 wavelength • For kx=(n+½)p, sin kx = 1 and y=2ym • Antinode: • x=(n+½) (l/2) • Antinodes also occur every 1/2 wavelength, but at a spot 1/4 wavelength before and after the nodes

  12. Resonance? • Under what conditions will you have resonance? • Must satisfy l = 2L/n • n is the number of loops on a string • fractions of n don’t work • v = (t/m)½ = lf • Changing, m, t, or f will change l • Can find new l in terms of old l and see if it is an integer fraction or multiple

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