Chapter 13 Vibrations and Waves Hooke s Law Reviewed F kx When x is positive F is negative When at equilibrium x 0 F 0 When x is negative F is positive Sinusoidal Oscillation Pen traces a sine wave Graphing x vs t A T A amplitude length m time s T period Some Vocabulary x Acos t Acos 2 ft 2 t Acos T 1 f T 2 2 f T f Frequency Angular Frequency T Period A Amplitude phase Phases Phase is related to starting time 2 t x Acos T 2 t0 2 t t0 Acos if T T 90 degrees changes cosine to sine cos t sin t 2 Velocity and Acceleration vs time Velocity is 90 out of phase with x When x is at max v is at min Acceleration is 180 out of phase with x a F m k m x x T v T a T Find vmax with E conservation v and a vs t x Acos t v vmax sin t a amax cos t 1 2 1 2 kA mvmax 2 2 k vmax A m Find amax using F ma kx ma kA cos t mamax cos t amax k A m What is Requires calculus Since d A cos t Asin t dt k vmax A A m k m Formula Summary 1 f T 2 2 f T x Acos t v Asin t a 2A cos t k m Example13 1 An block spring system oscillates with an amplitude of 3 5 cm If the spring constant is 250 N m and the block has a mass of 0 50 kg determine a the mechanical energy of the system b the maximum speed of the block c the maximum acceleration a 0 153 J b 0 783 m s c 17 5 m s2 Example 13 2 A 36 kg block is attached to a spring of constant k 600 N m The block is pulled 3 5 cm away from its equilibrium positions and released from rest at t 0 At t 0 75 seconds a 3 489 cm a what is the position of the block b 1 138 cm s b what is the velocity of the block Example 13 3 A 36 kg block is attached to a spring of constant k 600 N m The block is pulled 3 5 cm away from its equilibrium position and is pushed so that is has an initial velocity of 5 0 cm s at t 0 a What is the position of the block at t 0 75 seconds a 3 39 cm Example 13 4a An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 Where x will be in cm if t is in seconds The amplitude of the motion is a 1 cm b 2 cm c 3 cm d 4 cm e 4 cm Example 13 4b An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 Here x will be in cm if t is in seconds The period of the motion is a 1 3 s b 1 2 s c 1 s d 2 s e 2 s Example 13 4c An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 Here x will be in cm if t is in seconds The frequency of the motion is a 1 3 Hz b 1 2 Hz c 1 Hz d 2 Hz e Hz Example 13 4d An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 Here x will be in cm if t is in seconds The angular frequency of the motion is a 1 3 rad s b 1 2 rad s c 1 rad s d 2 rad s e rad s Example 13 4e An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 Here x will be in cm if t is in seconds he object will pass through the equilibrium position the times t seconds 2 1 0 1 2 1 5 0 5 0 5 1 5 2 5 1 5 1 0 5 0 0 5 1 0 1 5 4 2 0 2 4 2 5 0 5 1 5 3 5 Simple Pendulum F mgsin x x sin x2 L2 L mg F x L Looks like Hooke s law k mg L Simple Pendulum F mgsin x x sin x2 L2 L mg F x L g L max cos t Simple pendulum g L Frequency independent of mass and amplitude for small amplitudes Pendulum Demo Example 13 5 A man enters a tall tower needing to know its height h He notes that a long pendulum extends from the roof almost to the ground and that its period is 15 5 s a How tall is the tower a 59 7 m b If this pendulum is taken to the Moon where the free fall acceleration is 1 67 m s2 what is the period of the pendulum there b 37 6 s Damped Oscillations In real systems friction slows motion TRAVELING WAVES Sound Surface of a liquid Vibration of strings Electromagnetic Radio waves Microwaves Infrared Visible Ultraviolet X rays Gamma rays Gravity Longitudinal Compression Waves Sound waves are longitudinal waves Compression and Transverse Waves Demo Transverse Waves Elements move perpendicular to wave motion Elements move parallel to wave motion Snapshot of a Transverse Wave x y Acos 2 wavelength x Snapshot of Longitudinal Wave x y Acos 2 y could refer to pressure or density Replace x with x vt Moving Wave if wave moves to the righ x vt Replace with x vt if y Acos 2 wave should move to left moves to right with velocity v Fixing x 0 v y Acos 2 t v f v f Moving Wave Formula Summary x y Acos 2 mft v f Example 13 6a A wave traveling in the positive x direction has a frequency of f 25 0 Hz as shown in the figure The wavelength is a 5 cm b 9 cm c 10 cm d 18 cm e 20 cm Example 13 6b A wave traveling in the positive x direction has a frequency of f 25 0 Hz as shown in the figure The amplitude is a 5 cm b 9 cm c 10 cm d 18 cm e 20 cm Example 13 6c A wave traveling in the positive x direction has a frequency of f 25 0 Hz as shown in the figure The speed of the wave is a 25 cm s b 50 cm s c 100 cm s d 250 cm s e 500 cm s Example 13 7a Consider the following expression for a pressure wav P 60 cos 2x 3t where it is assumed that x is in cm t is in seconds and P will be given in N m2 What is the amplitude a 1 5 N m2 b 3 N m2 c 30 N m2 d 60 N m2 e …
View Full Document
Unlocking...