Download Simple Harmonic Motion: Lecture 21 from Introductory Physics I and more Study notes Physics in PDF only on Docsity! PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 21 Simple Harmonic Motion !, f, T determined by mass and spring constant A, " determined by initial conditions: x(0), v(0) Last Lecture f = 1 T ! = 2" f = 2" T x = Acos(!t "#) v = "!Asin(!t "#) a = "! 2A(cos!t "#) ! = k m Example 13.4b An object undergoing simple harmonic motion follows the expression, x(t) = 4 + 2cos[! (t " 3)] The period of the motion is: a) 1/3 s b) 1/2 s c) 1 s d) 2 s e) 2/! s Here, x will be in cm if t is in seconds Example 13.4c An object undergoing simple harmonic motion follows the expression, x(t) = 4 + 2cos[! (t " 3)] The frequency of the motion is: a) 1/3 Hz b) 1/2 Hz c) 1 Hz d) 2 Hz e) ! Hz Here, x will be in cm if t is in seconds Example 13.4d An object undergoing simple harmonic motion follows the expression, x(t) = 4 + 2cos[! (t " 3)] 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 Here, x will be in cm if t is in seconds Simple Pendulum F = !mgsin" sin" = x x 2 + L 2 # x L F # ! mg L x ! = g L " = "max cos(!t #$) Simple pendulum Frequency independent of mass and amplitude! (for small amplitudes) ! = g L Pendulum Demo TRAVELING WAVES •Sound •Surface of a liquid •Vibration of strings •Electromagnetic •Radio waves •Microwaves •Infrared •Visible •Ultraviolet •X-rays •Gamma-rays •Gravity Longitudinal (Compression) Waves Elements move parallel to wave motion. Example - Sound waves Transverse Waves Elements move perpendicular to wave motion. Examples - strings, light waves Snapshot of Longitudinal Wave # y could refer to pressure or density y = Acos 2! x " #$ % &' ( )* Moving Wave moves to right with velocity v Fixing x=0, Replace x with x-vt if wave moves to the right. Replace with x+vt if wave should move to left. y = Acos 2! x " vt # "$ % &' ( )* y = Acos !2" v # t !$ % &' ( )* f = v ! , v = f! Moving Wave: Formula Summary y = Acos 2! x " m ft # $% & '( )* + , - . / 0 v = f! - moving to right + moving to left 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 wave, 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) 120 N/m2 P = 60 !cos 2x " 3t( ) Example 13.7b Consider the following expression for a pressure wave, where it is assumed that x is in cm,t is in seconds and P 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 P = 60 !cos 2x " 3t( ) Example 13.8 Which of these waves move in the positive x direction? a) 5 and 6 b) 1 and 4 c) 5,6,7 and 8 d) 1,4,5 and 8 e) 2,3,6 and 7 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) Speed of a Wave in a Vibrating String For other kinds of waves: (e.g. sound) • Always a square root • Numerator related to restoring force • Denominator is some sort of mass density v = T µ where µ = m L Example 13.9 A string is tied tightly between points A and B as a communication device. If one wants to double the wave speed, one could: a) Double the tension b) Quadruple the tension c) Use a string with half the mass d) Use a string with double the mass e) Use a string with quadruple the mass
