α(t) = ω 2 θ (t) κ I ω = g L L g T = 2π mgh rot com I rot
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3 α(t) = ω 2 θ (t) ω = κ I ω = g L T = 2π L g ω = mgh rot com I rot T = 2π I rot mgh rot com
4 Chapter 16: Waves
5 Mechanical Waves Waves and particles Vibration = waves - Sound - medium vibrates - Surface ocean waves - no net water is displaced - Mechanical waves - Newton s equations with medium - Electro-magnetic waves - NO MEDIUM light (photons) WAVES Pulse - single wave Periodic wave - sinusoidal waves Particle - Displacement - Velocity - Acceleration
6 Transverse vs Longitudinal Waves Transverse: Displacement of particle is perpendicular to the direction of wave propagation Longitudinal: Displacement (vibration) of particles is along same direction as motion of wave -Sound (fluids ) -Ocean currents - top vs bottom Traveling Waves - they travel from one point to another - Nodes move Standing Waves - they look like they re standing still - Nodes do not move
7 Transverse Waves In a transverse wave the motion of the particles of the medium is perpendicular to the direction of the wave s travel
8 Longitudinal Waves A longitudinal pulse travels along the medium but does not involve the transport of matter just energy Here are periodic longitudinal waves pick a single particle and follow its motion as the wave goes by
9 A Wave on the Water A water wave is a combination of a longitudinal and a transverse wave notice how the blue dots make a circular motion: How do we describe the such waves?
10 Descrip<on of transverse traveling wave Displacement (y) versus position (x) y(x,0) = y max sin( kx) Spatially Periodic ( repeats ) : kλ = 2π t = 0 t = δ Wave number k = 2π λ Wavelength t = 2δ Temporally Periodic ( repeats ) : ωτ = 2π t = 3δ t = 4δ y(0,t) = y max sin( ωt) ω = 2π T Displacement versus time does not show shape
11 Descrip<on of traveling wave: mathema<cal k = 2π λ ω = 2π T phase : kx ± ωt kx ωt kx + ωt Wave traveling in + x direction Wave traveling in - x direction What is the velocity at which the wave crests move?
12 Wave speed Velocity at which crests move = wave velocity or phase velocity phase : kx ωt = const. k dx dt ω = 0 v wave = dx dt = ω k = λ T = λf A wave crest travels a distance of one wavelength, λ, in one period, T
13 Descrip<on of transverse wave - To describe a wave (particle) on a string, the transverse displacement (y) depends on both the position (x) along the string and the time (t) Displacement Y versus position X Displacement Y versus time t Spatially Periodic ( it repeats ) Temporally Periodic ( it repeats ) vt = λ v = λ T = fλ f = 1 T
14 Example: Wavelengths of Radio Sta<ons Waves like radio, light, x-rays etc. are part of the electromagnetic spectrum. They travel with a velocity: v = c = speed of light = m /s What is the wavelength of talk radio WJBO am 1150? What is the wavelength of KLSU fm 91.1? f =1150 khz = Hz = Hz λ = c f = m /s /s = 261 m f = 91.1 MHz = Hz = Hz λ = c f = m /s /s = 3.29 m
15 Sample problem 16-1 & -2 ( ) y(x,t) = sin 72.1x 2.72t A wave traveling along a string is described by: in which the numerical constants are in SI units ( m, 72.1 rad/m, and 2.72 rad/s). a) Which direction are the waves traveling? b) What is the amplitude of the waves? Positive x-direction y max = m = 3.27 mm c) What is the wavelength? d) What is the period? e) What is the frequency? k = 72.1 rad/m λ = m = 87.1 mm ω= 2.72 rad/s Τ = 2.31 s Τ = 2.31 s f = Hz f) What is the velocity of the wave (v w )? v w = ω / k = λf =38 mm/s g) What is the displacement y at x = 22.5 cm and t=18.9 s? y = 1.92 mm make sure your calculator is in radians h) What is u, (or v t ) the transverse velocity, at x = 22.5 cm and t=18.9 s? i) What is a t, the transverse acceleration, at x = 22.5 cm and t=18.9 s? u=-ωy m cos(kx-ωt) a t = -ω 2 y(x,t)
16 Wave speed on stretched string v wave = λ T = ω k = λf Wavelength and period are NOT independent: Wave speed depends on physical properties: on properties of the medium in which it travels If particles of medium move, then velocity must be a function of elasticity and mass (inertia) {potential energy and kinetic energy} τδl R = (µδl) v 2 R
17 Velocity in String v wave = τ µ = λf For transverse wave in physical medium Conceptually 1) the speed which the waves moves to right depends on how quickly one particle is accelerated upward in response to net pulling force by neighbors. 2) Newton s 2nd law - stronger force greater upward acceleration (a = F/m) -> leads to a faster moving wave 3) Ability to pull on neighbors depends on how tight the string is -> TENSION greater tension greater force greater speed 4) Inertia F = ma: travel faster with small mass ( or linear mass density - mass/length)
18 Problem 16 21: wave speed in stretched string shows a plot of the displacement as a function of position at time t=0. The scale Find (a) amplitude (b) wavelength (c) wave speed (d) period of the wave
19 Problem 16 21: wave speed in stretched string shows a plot of the displacement as a function of position at time t=0. The scale Find (e) maximum speed of a particle in the string (f) If the wave form is determine all variables (g) The correct choice of sign in front of ω
1 f. result from periodic disturbance same period (frequency) as source Longitudinal or Transverse Waves Characterized by
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