Welcome back to Physics 215

1. Welcome back to Physics 215 Today’s agenda: • Standing waves and normal modes • Sound waves • Brief review for final • Course evaluations

2. Current homework assignment • HW12: • Knight Textbook Ch. 14: 48, 58, 76 • Knight Textbook Ch. 20: 62 • exam-style problem – print out from website • due Friday, Dec. 10th in recitation

3. Final Exam: Thursday (Dec .16), 12:45PM • Exam will be 2 hours long • In Room 208 (here!) • Cumulative! • Material covered: • Textbook chapters 1-15, 20 • Lectures -- all (slides online) • Workshop Activities -- all • Homework assignments -- all (select solutions online) • Exams 1, 2 & 3 (solutions online) • As with Exams 1, 2 & 3, Final is closed book, but you may bring calculator and onehandwritten 8.5” x 11” sheet of notes -- this may be a different sheet from Exams 1, 2 & 3. • Practice versions of Final Exam posted online

4. Sinusoidal Waves [Can also use sine in the definition since cos(q) = sin(q-p/2)] Angular frequency Amplitude Angular wave number Initial phase Speed of sinusoidal wave:

5. Power in a wave • Consider energy of SHM E = (1/2)kA2 = (1/2)mw2A2 • What is m? • Total mass of excited oscillators in time T is Tvm • Also need power = energy per unit time delivered by wave E/T = P = (1/2)vmw2A2

6. Reflection of waves Fixed end Free end • Reflection – reversal of wave velocity Pulse not inverted f(x - vt) a+ f(x + vt) Pulse inverted f(x - vt) a– f(x + vt)

7. Interference • When two waves propagate through same region – combine to give some new wave motion • interfere • Resultant wave motion is simply sum of individual wave motions (superposition)

8. Resultant wave Two different waves Superposition of waves • Waves add up algebraically y(x,t) = y1(x,t) + y2(x,t) As a consequence, waves can pass through each other without being altered

9. Standing Waves • Consider wave on rubber hose (demo) • If I drive system with just right frequency • hose exhibits standing wave pattern • some parts of hose never move, others oscillate always maximally. Motion of different parts of medium in phase • no energy transport

10. Mathematics of standing waves y(x, t)= Acos(wt - kx) - Acos(wt + kx) = 2Asin(wt)sin(kx) Possible values of k ? If string of length L is clamped at both ends, need yyL . Therefore, need kL = p,2p,3p,... 14243 Amplitude oscillating with time 123 Wave that does not move (v=0)

11. antinodes Normal Modes y(x, t)= 2Asin(wt)sin(kx) /2 = L nodes  = L /2 = L 0 L

12. A string is clamped at both ends and plucked so that it vibrates in a standing wave mode between positions a and b. Take upward motion of the string to correspond to positive velocities. When the string is in position c, the instantaneous velocity of points along the string: is zero everywhere is positive everywhere is negative everywhere depends on location

13. A string is clamped at both ends and plucked so that it vibrates in a standing wave mode between positions a and b. Take upward motion of the string to correspond to positive velocities. When the string is in position b, the instantaneous velocity of points along the string: is zero everywhere is positive everywhere is negative everywhere depends on location

14. Molecule Direction of velocity of the sound wave Sound Waves • Longitudinal oscillation of gas (or liquid) molecules • As a consequence, gas pressure oscillates • Velocity of sound depends on the medium vsound in air ≈ 330 m/s • We hear frequency of the wave (f) as a pitch of the sound -- high pitch = high frequency (short wavelength)

15. Intensity • Intensity -- average power transported per unit area • Decibel scale I0 = threshold of hearing • threshold of pain 120 dB • elevated train 90 dB • quiet automobile 50 dB • average whisper 20 dB • threshold of hearing 0 dB

16. Kinematics -1D • Know: definitions/meaning instantaneous velocity and acceleration • Graphical interpretation: slope of x(t) and v(t) curves • Area under v(t) curve  x, etc. • Constant acceleration formulae: v = v0 + at x = x0 + v0t + (1/2)at2 etc.

17. Kinematics – 2D • Velocity tangent to path • Acceleration – 2 components – radial v2/r (centripetal) and tangential – rate of change of speed • Know about adding/subtracting vectors graphically and in component form

18. Forces • Free body diagrams, labeling - FAB • Types: normal N, tension T (massless string), friction Fk = mk N, weight W,... • Newton’s laws • Fnet = ma, F12 = -F21 • Internal/external forces

19. Work, momentum, etc. • F.Dx – scalar product of vectors • W-KE theorem: net work done on body equals change in K = (1/2)mv2 • Impulse I = FDt  net impulse on body gives change in momentum p = mv • Conservation of momentum (collisions) and conservation (or not) of mechanical energy -- elastic vs. inelastic

20. Rotational Dynamics • Center of mass • equilibrium: Fnet = 0, tnet = 0 (torque). • |t|=r||F|sin(q) magnitude. Direction along axis of rotation (fixed) (clockwise or anticlockwise) • Angular velocity w and acceleration a • Moment of inertia: Ia = t

21. More rotational dynamics • Angular momentum L = Iw • Angular (rotational) kinetic energy K = (1/2)Iw2 for body rotating about fixed axis • Rolling without slipping – relation between linear and rotational acceleration a = Ra

22. Gravitation Inverse square law: F = Gm1m2/r2 F12 m1 r m2

23. Fluid Mechanics • Pressure vs. depth: P = P0 + gh • Archimedes principle • Buoyant force = weight of displaced fluid

24. Periodic Motion, Waves • Defining equation for SHM a = -w2x with w2 = k/m for spring • Solutions: x = Acos(wt+f). Energy E = (1/2)kx2 + (1/2)mv2 conserved. • Amplitude, frequency, wavelength, angular frequency, wavenumber for traveling wave. Standing waves. Power in wave. Wave speed for stretched wire. Sound waves.

25. Wave formulae P = (1/2)v2A2

26. Reading assignment • Prepare for final exam • Have a great semester break!