PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 21 Last Lecture Simple Harmonic Motion x Acos t v Asin t a 2A cos t k m 1 f T 2 2 f T f T determined by mass and spring constant A determined by initial conditions x 0 v 0 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 here x will be in cm if t is in seconds he amplitude of the motion is 1 cm 2 cm 3 cm 4 cm 4 cm Example 13 4b An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 ere 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 ere 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 ere x will be in cm if t is in seconds he angular frequency of the motion is 1 3 rad s 1 2 rad s 1 rad s 2 rad s rad s Example 13 4e An object undergoing simple harmonic motion follows the expression x t 4 2cos t 3 ere x will be in cm if t is in seconds 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 Elements move parallel to wave motion Example Sound waves Transverse Waves Elements move perpendicular to wave motion Examples strings light waves Compression and Transverse Waves Demo 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 Moving Wave Replace x with x vt x vt if wave moves to the right y Acos 2 Replace with x vt if 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 v f x y Acos 2 mft moving to right moving to left 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 b c d e 5 cm 9 cm 10 cm 18 cm 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 b c d e 5 cm 9 cm 10 cm 18 cm 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 b c d e 25 cm s 50 cm s 100 cm s 250 cm s 500 cm s Example 13 7a nsider the following expression for a pressure wave P 60 cos 2x 3t ere it is assumed that x is in cm t is in seconds a 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 120 N m2 Example 13 7b nsider the following expression for a pressure wave P 60 cos 2x 3t ere it is assumed that x is in cm t is in seconds a will be given in N m2 What is the wavelength a 0 5 cm b 1 cm c 1 5 cm d cm e 2 cm Example 13 7c nsider the following expression for a pressure wave P 60 cos 2x 3t ere it is assumed that x is in cm t is in seconds a will be given in N m2 What is the frequency a 1 5 Hz b 3 Hz c 3 Hz d 3 2 Hz e 3 Hz Example 13 7d nsider the following expression for a pressure wave P 60 cos 2x 3t ere it is assumed that x is in cm t is in seconds a will be given in N m2 What is the speed of the wave a 1 5 cm s b 6 cm s c 2 3 cm s d 3 2 cm s e 2 cm s Example 13 8 ich of these waves move in the positive x direction 1 y 21 3 cos 3 4x 2 5t 2 y 21 3 cos 3 4x 2 5t 3 y 21 3 cos 3 4x 2 5t 4 y 21 3 cos 3 4x 2 5t 5 y 21 3 cos 3 4x 2 5t 6 y 21 3 cos 3 4x 2 5t 7 y 21 3 cos 3 4x 2 5t 8 y 21 3 cos 3 4x 2 5t a b c d e 5 and 1 and 5 6 7 1 4 5 2 3 6 6 4 and 8 and 8 and 7 Speed of a Wave in a Vibrating String T m v where L For other kinds of waves e g sound Always a square root Numerator related to restoring force Denominator is some sort of mass density Example 13 9 …
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