Hooke s Law Reviewed F kx Chapter 13 Vibrations and Waves When x is positive F is negative When at equilibrium x 0 F 0 When x is negative F is positive 1 2 Sinusoidal Oscillation Graphing x vs t Pen traces a sine wave A T A amplitude length m T period time s 3 Phases Some Vocabulary x A cos t A cos 2 ft 2 t A cos T 4 Phase is related to starting time f 1 T 2 f 2 t x A cos T 2 T 2 t 0 2 t t 0 A cos if T T 90 degrees changes cosine to sine f Frequency Angular Frequency T Period A Amplitude phase cos t sin t 2 5 6 x T 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 Find vmax with E conservation 1 2 1 2 kA mvmax 2 2 k vmax A m v and a vs t x A cos t v vmax sin t a amax cos t v T Find amax using F ma kx ma kA cos t mamax cos t k amax A m a T 7 What is 8 Formula Summary Requires calculus Since d A cos t Asin t dt k vmax A A m f 1 T 2 f x A cos t v Asin t k m a 2 A cos t 2 T k m 9 Example13 1 10 Example 13 2 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 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 the mechanical energy of the system b the maximum speed of the block a what is the position of the block a 3 489 cm c the maximum acceleration b what is the velocity of the block b 1 138 cm s a 0 153 J b 0 783 m s c 17 5 m s2 11 12 Example 13 4a Example 13 3 An object undergoing simple harmonic motion follows the expression x t 4 2 cos t 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 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 a What is the position of the block at t 0 75 seconds a 3 39 cm 13 14 Example 13 4b Example 13 4c An object undergoing simple harmonic motion follows the expression An object undergoing simple harmonic motion follows the expression x t 4 2 cos t 3 x t 4 2 cos t 3 Here x will be in cm if t is in seconds 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 The frequency of the motion is a 1 3 Hz b 1 2 Hz c 1 Hz d 2 Hz e Hz 15 16 Example 13 4d Example 13 4e An object undergoing simple harmonic motion follows the expression An object undergoing simple harmonic motion follows the expression x t 4 2 cos t 3 x t 4 2 cos t 3 Here x will be in cm if t is in seconds 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 The object will pass through the equilibrium position at the times t seconds a b c d e 17 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 18 Simple Pendulum Simple Pendulum F mgsin x x sin 2 2 L x L mg F x L F mgsin x x sin 2 2 L x L mg F x L g L max cos t Looks like Hooke s law k mg L 19 Simple pendulum 20 Pendulum Demo g L Frequency independent of mass and amplitude for small amplitudes 21 Damped Oscillations 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 22 In real systems friction slows motion 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 23 24 TRAVELING WAVES Longitudinal Compression Waves Sound Surface of a liquid Vibration of strings Electromagnetic Radio waves Microwaves Infrared Visible Ultraviolet X rays Gamma rays Gravity Sound waves are longitudinal waves 25 Compression and Transverse Waves Demo 26 Transverse Waves Elements move perpendicular to wave motion Elements move parallel to wave motion 27 Snapshot of a Transverse Wave 28 Snapshot of Longitudinal Wave x y A cos 2 wavelength x y A cos 2 x y could refer to pressure or density 29 30 Moving Wave x vt y A cos 2 Replace x with x vt if wave moves to the right Replace with x vt if wave should move to left Moving Wave Formula Summary x y A cos 2 ft 0 moves to right with velocity v v f Fixing x 0 v y A cos 2 t f v v f 31 Example 13 6a 32 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 wavelength is 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 b c d e a b c d e 5 cm 9 cm 10 cm 18 cm 20 cm 5 cm 9 cm 10 cm 18 cm 20 cm 33 Example 13 7a Example 13 6c Consider the following expression for a pressure wave 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 b c d …
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