Springs in parallel. Two springs in parallel. Springs in series. Springs in series. Resonance. Forced vibrations and resonance. 2 C. 2 1/2 m.

Transcription

1 Springs in parallel w springs in parallel Suppse yu had tw identical springs each with frce cnstant frm which an bject f mass m was suspended. he scillatin perid fr ne spring is. What wuld the scillatin perid be if the tw springs were cnnected in parallel? A. B. / C. 1/ D. / 1/ p = = π p = m stiffer y = y p = y p = One spring parallel Springs in series Springs in series Suppse yu had tw identical springs each with frce cnstant frm which an bject f mass m was suspended. he scillatin perid fr ne spring is. = y One spring What wuld the scillatin perid be if the tw springs were cnnected in series? A. = Less stiff s B. / C. 1/ m = π D. / 1/ s = y y Net displacement =y = y s = s series rced vibratins and resnance he peridic frce puts energy int the system Resnance When the driving scillatins has a frequency that matches the scillatin frequency f the standing waves in the system then a large amunt f energy can be put int the system. he push frequency must be at the same frequency as the frequency f the swing. he driving frce is in resnance with the natural frequency. 1

2 Cupled Oscillatins When tw scillatrs are cupled by an interactin, energy can be transferred frm ne scillatr t anther. he rate f energy transfer is faster when the tw scillatrs are in resnance. ast transfer Slw transfer 1. Waves Wave prperties speed wavelength Superpsitin f waves Reflectin f waves at an interface Wave n a string Speed f wave n a string Sund waves Speed f Sund Intensity f Sund Waves A disturbance that carries energy Mechanical Waves- water wave, sund must prpagate thrugh matter. Electrmagnetic Waves radi, x-ray, light can prpagate thrugh a vacuum. increasing time Wave n a string ransverse and Lngitudinal Waves ransverse Wave - he displacement is perpendicular t the directin f prpagatin Lngitudinal Wave- he displacement is parallel t the directin f prpagatin Examples ransverse waves ransverse wave n a string Electrmagnetic waves (speed = 3.00x10 8 m/s) Lngitudinal waves Sund waves in air (speed = 340 m/s)

3 Seismic waves are transverse and lngitudinal Seismgraph recrd after an earthquae. P waves- lngitudinal faster v~ 5000 m/s (granite) S waves transverse slwer v~ 3000 m/s (granite) P S ime Simple Harmnic Waves Wavelength - Spatial Perid Harmnic scillatins Peridic displacement vs distance Wave travels distance λ during ne perid Wave velcity Units λ(meters) (secnds) meters/secnd v = λ(meters)f(1/secnds) meters/secnd 1 v(meters/secnd) f = = λ(meters) 1/secnds λ =λf he wave travel at a velcity f ne wavelength in ne perid. 3

4 Example ransverse wave n a string A radi statin transmits at a frequency f 100 MHz. ind the wavelength f the electrmagnetic waves. (speed f light =3.0x10 8 m/s) λf v λ= f 8 3.0x10 = = 3.0m 6 100x10 ransverse wave simulatin transverse wave simulatin ave01aapplet.html r a transverse wave each segment underges simple harmnic mtin. Speed f the transverse wave n a string. V -> m µ = mass density L L m µ speed f transverse wave n a string depends n the tensin n the string and the mass density Example A transverse wave with a speed f 50 m/s is t be prduced n a stretched spring. If the string has a length f 5.0 m and a mass f g, what tensin n the string is required. Superpsitin Principle When tw waves verlap in space the displacement f the wave is the sum f the individual displacements. m/l vm = L (50m / s) (0.060g) = = 30N 5.0m 4

5 Interference Superpsitin f harmnic waves depends n the relative phase f the tw waves Can lead t Cnstructive Interference Destructive Interference Cnstructive Interference Wave 1 Wave Superpsitin distance he tw waves have the same phase Destructive Interference Wave 1 Wave Superpsitin Distance -> he tw waves are ut f phase (by 180, r π) Reflectin and ransmissin. When a wave reaches a bundary, part f the wave is reflected and part f the wave is transmitted. he amunt reflected and transmitted depends n hw well the media is matched at the bundary. he sign f the reflected wave depends n the resistance at the bundary. Mis-match at the bundary part f the wave will be reflected at the bundary ixed End- Inversin Reflectin ree End- N Inversin match wea resistanc mis-match strng resistance mis-match Bundary 5

6 A wave n a string ges frm a thin string t a thic string. What picture best represents the wave sme time after hitting the bundary? Befre questin Displacement- parallel Sund waves A B C D Density Pressure Prduced by cmpressin and rarefactin f media (air) Sund waves are lngitudinal resulting in displacement in the directin f prpagatin. he displacements result in scillatins in density and pressure. requencies f sund wave Speed f sund infra-snic Audible Sund ultra- snic 10 0,000 requency (Hz) Wavelength (m) in air Speed f sund in a fluid B ρ P B = V/V Bul mdulus m ρ= V Density Similarity t speed f a transverse wave n a string elastic _prperty int ertial _prperty Speed f sund in air B ρ Why is the speed f sund higher in Helium than in air? Why is the speed f sund higher in water than in air? γp ρ γ is a cnstant that depends n the nature f the gas γ =7/5 fr air. P - Pressure ρ - Density Since P is prprtinal t the abslute temperature by the ideal gas law. PV=nR v is dependent n 331 (m/s) 73 6

7 ind the speed f sund in air at 0 C = 343m / s 73 Example Yu are standing in a canyn and shut. Yu hear yur ech 3.0 s later. Hw wide is the canyn? (v sund =340 m/s) d d = vt vt (340 m / s)(3.0 s) d = = = 510m Example he maximum sensitivity f the human ear is fr a frequency f abut 3 Hz. What is the wavelength f the sund at this frequency? v λ = f 340m / s = = 0.11m = 11cm 3 3x10 Hz Energy and Intensity f sund waves pwer energy P = time area A Intensity pwer P I = = area A (units W/m ) Sund intensity level I β= 10lg decibels (db) I he ear is capable f distinguishing a wide range f sund intensities. I = 10-1 W/m the threshld f hearing decibel is a lgarithmic unit. It cvers a wide range f intensities. 7

8 Questin Questin he sund intensity f an ipd earphne can be as much as 10 db. Hw is this pssible? What is the intensity f sund at a rc cncert? (W/m ) I β= 10lg = 10 I I 10 lg = = 1 I0 10 I I 0 1 = 10 A) he ipd is very pwerful B) he area f the earphne is very small C) he ipd is a digital device D) Rc music can be very lud 1 1 I = 10 I 0 = = 1 W/m 1 he sund intensity f an ipd earphne can be as much as 10 db. Hw is this pssible? Spherical and plane waves A = 4πr area f sphere he earphne is placed directly in the ear. he intensity at the earphne is the pwer divided by a small area. Say the area is abut 1cm. 4 4 P IA 1w /m (10 = = m ) = 10 W A small amunt f pwer prduces a high intensity. r a pint surce the intensity decreases as 1/r P I = 4 π r P = pwer f surce Suppse yu are standing near a ludspeaer that can is blasting away with 100 W f audi pwer. Hw far away frm the speaer shuld yu stand if yu want t hear a sund level f 10 db. ( assume that the sund is emitted unifrmly in all directins.) P P I = A = 4 π r r = P 4πI 100W = =.8m 4 π(1w/m ) 8