PHY 113 C General Physics I 11 AM 12:15 PM TR Olin 101

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1 PHY 113 C General Phsics I 11 AM 1:15 PM R Olin 101 Plan for Lectre 16: Chapter 16 Phsics of wave motion 1. Review of SHM. Eamples of wave motion 3. What determines the wave velocit 4. Properties of periodic waves 10//013 PHY 113 C Fall Lectre 16 1

2 10//013 PHY 113 C Fall Lectre 16

3 Some comments on Simple Harmonic Motion Sppose that o now: (in standard nits) (= cos(pt+3p) What is? = m(p) m=1 g d F ma m dt d differential eqation : dt m general soltion : ( Acos(ωt φ) provided that ω m 10//013 PHY 113 C Fall Lectre 16 3

4 Some comments on Simple Harmonic Motion Sppose that o now: (in standard nits) (= cos(pt+3p) m=1 g d F ma m dt d differential eqation : dt m general soltion : ( Acos(ωt φ) provided What is the freqenc of oscillations? w=pf=p f=1 Hz that ω m 10//013 PHY 113 C Fall Lectre 16 4

5 Some comments on Simple Harmonic Motion Sppose that o now: (in standard nits) (= cos(pt+3p) m=1 g d F ma m dt d differential eqation : dt m general soltion : ( Acos(ωt φ) provided What is amplitde of the displacement? ma =m that ω m 10//013 PHY 113 C Fall Lectre 16 5

6 Some comments on Simple Harmonic Motion Sppose that o now: (in standard nits) (= cos(pt+3p) m=1 g d F ma m dt d differential eqation : dt m general soltion : ( Acos(ωt φ) provided What is the maimm velocit? v(=-(p) cos(pt+3p) v ma =4p that ω m 10//013 PHY 113 C Fall Lectre 16 6

7 Some comments on Simple Harmonic Motion Sppose that o now: (in standard nits) (= cos(pt+3p) m=1 g d F ma m dt d differential eqation : dt m general soltion : ( Acos(ωt φ) provided What is the maimm acceleration? a(=-(p) cos(pt+3p) a ma =8p that ω m 10//013 PHY 113 C Fall Lectre 16 7

8 Some comments on Simple Harmonic Motion Sppose that o now: (in standard nits) (= cos(pt+3p) m=1 g d F ma m dt d differential eqation : dt m general soltion : ( Acos(ωt φ) provided What is the displacement at t=0.3 s? (0.3)= cos(p(0.3)+3p) =(0.309)=0.618 m that ω m 10//013 PHY 113 C Fall Lectre 16 8

9 Some comments on driven Simple Harmonic Motion F(=F 0 sin( F where F ω 0 sin differential eqation : general soltion : m ( d ma m dt d F0sin dt m F0 / m Acos(ωt φ) w 10//013 PHY 113 C Fall Lectre 16 9 sin

10 Webassign qestion (Assignment 14) Damping is negligible for a g object hanging from a light, 6.30-N/m spring. A sinsoidal force with an amplitde of 1.70 N drives the sstem. At what freqenc will the force mae the object vibrate with an amplitde of m? ( Acos(ωt φ) F / m w 0 sin ω m 0 ( X 0sin amplitde X X 0 F0 / m / m m F0 mx 0 10//013 PHY 113 C Fall Lectre 16 10

11 he wave eqation (, position time Wave variable What does the wave eqation mean? Eamples Mathematical soltions of wave eqation and descriptions of waves t c 10//013 PHY 113 C Fall Lectre 16 11

12 Eample: Water waves Sorce: Needs more sophistocated analsis: 10//013 PHY 113 C Fall Lectre 16 1

13 Mechanical waves occr in continos media. he are described b a vale () which changes in both time ( and position () and are characterized b a wave velocit c: =f(-c or =f(+c 10//013 PHY 113 C Fall Lectre 16 13

Waves on a string: pical vales for c: 310 8 m/s light waves ~1000 m/s wave

14 Waves on a string: pical vales for c: m/s light waves ~1000 m/s wave on a string 343 m/s sond in air 10//013 PHY 113 C Fall Lectre 16 14

15 ransverse wave: 10//013 PHY 113 C Fall Lectre 16 15

16 Longitdinal wave: 10//013 PHY 113 C Fall Lectre 16 16

17 General traveling wave t = 0 t > 0 10//013 PHY 113 C Fall Lectre 16 17

18 iclicer qestion: t=0 t=1 s t=s 10//013 PHY 113 C Fall Lectre 16 18

19 10//013 PHY 113 C Fall Lectre Basic phsics behind wave motion -- eample: transverse wave on a string with tension and mass per nit length m A B dt d m dt d m μ μ sin θ sin θ A B q B B B B θ tan sin θ 0 1 Lim A B μ t

20 10//013 PHY 113 C Fall Lectre 16 0 he wave eqation: Soltions: (, = f ( c c t (for a string) μ where c fnction of an shape Let Note: c f t f t f f f f f f ct

21 iclicer qestion Is it significant to write the wave eqation with the special smbols? t c A. Yes B. No 10//013 PHY 113 C Fall Lectre 16 1

22 Eamples of soltions to the wave eqation: Moving plse : Periodic wave: (, 0 e ct (, 0 sin ct φ phase factor π λ π c πf ω wave vector not spring constant!!!, ( 0 sinπ λ t φ λ c 10//013 PHY 113 C Fall Lectre 16

23 Periodic traveling waves:, ( 0 sinπ λ Amplitde t φ λ wave length (m) phase (radians) c period (s); = 1/f velocit (m/s) 10//013 PHY 113 C Fall Lectre 16 3

24 Snapshot of periodic wave at t=t 0 l f l = c ime plot of periodic wave at = 0 1/f 10//013 PHY 113 C Fall Lectre 16 4

25 Combinations of waves ( sperposition ) Note that : sin A sin B sin 1 1 A B cos A B right t (, 0 sinπ φ left (, 0 λ sinπ λ t φ Standing wave: right, (, ( left 0 sin π λ φ cos πt 10//013 PHY 113 C Fall Lectre 16 5

26 Smmar of wave properties: Wave speed c depends on the process and/or medim in which wave occrs. Eample : Wave on a string with tension and mass/length : c Eample : Sond wave in air with pressre p and densit : c p Eample : Light wave de to copled electric and magnetic fields : c m / s (fndamental constan 10//013 PHY 113 C Fall Lectre 16 6

27 Eample from webassign: 10//013 PHY 113 C Fall Lectre 16 7

28 Periodic traveling waves:, ( 0 sinπ λ Amplitde t φ λ wave length (m) phase (radians) c period (s); = 1/f velocit (m/s) 10//013 PHY 113 C Fall Lectre 16 8

29 10//013 PHY 113 C Fall Lectre 16 9 Eample from webassign: maimm transverse speed : φ λ π cos ), ( λ φ λ π sin ), ( t t t c t t

EXERCISES WAVE EQUATION. In Problems 1 and 2 solve the heat equation (1) subject to the given conditions. Assume a rod of length L.

EXERCISES WAVE EQUATION. In Problems 1 and 2 solve the heat equation (1) subject to the given conditions. Assume a rod of length L. .4 WAVE EQUATION 445 EXERCISES.3 In Problems and solve the heat eqation () sbject to the given conditions. Assme a rod of length.. (, t), (, t) (, ),, > >. (, t), (, t) (, ) ( ) 3. Find the temperatre