Physics 207 Lecture 24

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1 Physics 7 Lecture 4 Physics 7, Lecture 4, Nov. 7 gena: Mi-Ter 3 Review Elastic Properties of Matter, Mouli Pressure, Wor, rchiees Principle, Flui flow, Bernoulli Oscillatory otion, Linear oscillator, Penulus Energy, Daping, Resonance Transverse Waves, Pulses, Reflection, Transission, Power Longituinal Waves (Soun), Plane waves, Spherical waves Louness, Doppler effect ssignents: Proble Set 9 ue Tuesay, Dec. 5, :59 PM Ch. 8: 9, 7,, 39, 53a, Ch. 9:,, 5, 3, 43, 57 Mi-ter 3, Tuesay, Nov. 8, Chapters 4-7, 7, 9 inutes, 7:5-8:45 PM in roos 5 an 3 Psychology. McBurney stuents will go to roo 53 Chaberlin (Graes on Monay) Wenesay, Chapter 9 (Teperature, then Heat & Theroynaics) Physics 7: Lecture 4, Pg Soe efinitions Elastic properties of solis : Young s oulus: easures the resistance of a soli to a change in its length. F L L Shear oulus: easures the resistance to otion of the planes of a soli sliing past each other. Bul oulus: easures the resistance of solis or liquis to changes in their volue. Y tensile tensile strain stress F F V P B V / V F / = L / L F / S shear shear strain stress = / h F V - V Physics 7: Lecture 4, Pg Eaple: Statics with Young s Moulus sall person is riing a unicycle an is halfway between two posts apart. The guie wire was originally long, weighs. g an has cross sectional area of c. Uner the weight of the unicycle it sags own. at the center an there is a tension of 5 Newtons along the wire. Eaple: Statics with Young s Moulus sall person is riing a unicycle an is halfway between two posts apart. The guie wire was originally long, weighs. g an has cross sectional area of c. Uner the weight of the unicycle it sags own. at the center an there is a tension of 5 Newtons along the wire. (Values change fro class). (a) What is the Young's Moulus of the wire (to two significant figures)? (b) How long oes it tae a transverse wave (a pulse) to propagate fro the support to the unicycle (Treat wire as a siple string)? (c) f the pulse is now sai to be a perfectly sinusoial wave an has a frequency of Hz, what is the angular frequency? () t what transverse aplitue of the wave, in the vertical irection, will the wire's aiu acceleration just reach /s (a) What is the Young's Moulus of the wire (to two significant figures)? L = [( +. ) ½ - ] 5-3 F / 5 N/ Y = = = 5 L / L 5. N/ Physics 7: Lecture 4, Pg 3 Physics 7: Lecture 4, Pg 4 Eaple: Statics with Young s Moulus sall person is riing a unicycle an is halfway between two posts apart. The guie wire was originally long, weighs. g an has cross sectional area of c. Uner the weight of the unicycle it sags own. at the center an there is a tension of 5 Newtons along the wire. (b) How long oes it tae a transverse wave to propagate fro the support to the unicycle (Note: treat wire as a siple string)? tie = istance / velocity = / (T/ µ ) ½ = / (5 / (./)) ½ s t = / ( 6 ) ½ sec = / 3 sec =. secons Notice T/ µ= 5 = 6 /s & Y/ρ = 5. ( -6 )= 8 /s Math Suary The forula y(, t) = cos( ω t + φ ) y escribes a haronic wave of aplitue oving in the + irection. Each point on the wave oscillates in the y irection with siple haronic otion of angular frequency ω. π = The wavelength of the wave is v = ω The spee of the wave is The quantity is often calle wave nuber. Physics 7: Lecture 4, Pg 5 Physics 7: Lecture 4, Pg 6 Page

2 Physics 7 Lecture 4 Velocity an cceleration Position: (t) = cos(ωt + φ) Velocity: v(t) = -ω sin(ωt + φ) cceleration: a(t) = -ω cos(ωt + φ) a = v a = ω a a = ω by taing erivatives, since: ( t ) v( t ) = v( t ) a( t ) = Physics 7: Lecture 4, Pg 7 Eaple: Statics with Young s Moulus sall person is riing a unicycle an is halfway between two posts apart. The guie wire was originally long, weighs. g an has cross sectional area of. Uner the weight of the unicycle it sags own. at the center an there is a tension of 5 Newtons along the wire. (c) f the pulse is sai to be a perfectly sinusonial wave an has a frequencyof Hz, what is the angular frequecy? 68 ra/s () t what transverse aplitue of the wave, in the vertical irection, will the wire's aiu acceleration just reach /s a(t) = -ω cos(ωt + φ) a a = ω = ( π) = /s Physics 7: Lecture 4, Pg 8 Pascal s Principle Consier the syste shown: ownwar force F is applie to the piston of area. F F Fluis: Pascal s Principle Pressure epens on epth: p = ρ g y Pascal s Principle aresses how a change in pressure is transitte through a flui. This force is transitte through the liqui to create an upwar force F. ny change in the pressure applie to an enclose flui is transitte to every portion of the flui an to the walls of the containing vessel. Pascal s Principle says that increase pressure fro F (F / ) is transitte throughout the liqui. F > F : s there conservation of energy? W = F Here W = F/ ( ) or W = P V F / = P = W F / = P = W so = F F Physics 7: Lecture 4, Pg 9 Physics 7: Lecture 4, Pg rchiees Principle The buoyant force is equal to the weight of the liqui that is isplace. f the buoyant force is larger than the weight of the object, it will float; otherwise it will sin. eal Fluis Strealines o not eet or cross Velocity vector is tangent to strealine Volue of flui follows a tube of flow boune by strealines strealine v Since the pressure at the botto of the object is greater than that at the top of the object, the water eerts a net upwar force, the buoyant force, on the object. W W? Strealine ensity is proportional to velocity Flow obeys continuity equation Volue flow rate Q = v is constant along flow tube. v v = v Follows fro ass conservation if flow is incopressible. Physics 7: Lecture 4, Pg Physics 7: Lecture 4, Pg Page

3 Physics 7 Lecture 4 Conservation of Energy for eal Flui Recall the stanar wor-energy relation W = K = K f - K i pply the principle to a section of flowing flui with volue V an ass = ρ V (here W is wor one on flui) Net wor by pressure ifference over ( = v t) y y v p Bernoulli Equation P + ½ ρ v + ρ g y = constant V v p Physics 7: Lecture 4, Pg 3 Reviewing Siple Haronic Oscillators Spring-ass syste Penula = ω where θ = ω θ (t) = cos( ωt + φ) θ = θ cos( ωt + φ) General physical penulu Torsion penulu κ MgR a z-ais F = - R θ CM Mg τ wire θ Physics 7: Lecture 4, Pg 4 Lecture 4, Eercise Physical Penulu penulu is ae by hanging a thin hoola-hoop of iaeter D on a sall nail. What is the angular frequency of oscillation of the hoop for sall isplaceents? ( CM = R for a hoop) θ = ω θ where MgR = τ pivot (nail) Saple Proble (nother physical penulu) PROBLEM: 3 g chil is sitting with his center of ass fro the frictionless pivot of a assless see-saw as shown. The see-saw is initially horizontal an, at 3 on the other sie, there is a assless Hooe's Law spring (constant N/) attache so that it sits perfectly vertical (but slightly stretche). Gravity acts in the ownwar irection with g = /s. F chil = b g Pivot F= () (B) (C) g D g D g D D Physics 7: Lecture 4, Pg 5 ssuing everything is static an in perfect equilibriu. (a) What force, F, is provie by the spring? Στ = = b g () - F (3) F = 6 N / 3 = N (b) Now the chil briefly bounces the see-saw (with a sall aplitue oscillation) an then oves with the seesaw. What is the angular frequency of the chil? Physics 7: Lecture 4, Pg 6 Saple Proble (nother physical penulu) PROBLEM: 3 g chil is sitting with his center of ass fro the frictionless pivot of a assless see-saw as shown. The see-saw is initially horizontal an, at 3 on the other sie, there is a assless Hooe's Law spring (constant N/) attache so that it sits perfectly vertical (but slightly stretche). F chil = b g Pivot F= = r θ Στ = = b g () - F (3) F = 6 N / 3 = N (b) Now the chil briefly bounces the see-saw (with a sall aplitue oscillation) an then oves with the see-saw. What is the angular frequency of the chil? α = θ / = -r -r r θ (τ/) ½ = (r / r b ) ½ (9 / 3 4 ) ½ = (4 / 4) ½ = 3 ra / s Physics 7: Lecture 4, Pg 7 SHM: Velocity an cceleration Position: (t) = cos(ωt + φ) Velocity: v(t) = -ω sin(ωt + φ) cceleration: a(t) = -ω cos(ωt + φ) a = v a = ω a a = ω by taing erivatives, since: ( t ) v( t ) = v( t ) a( t ) = Physics 7: Lecture 4, Pg 8 Page 3

4 Physics 7 Lecture 4 Lecture 4, Eercise Siple Haronic Motion ass oscillates up & own on a spring. t s position as a function of tie is shown below. t which of the points shown oes the ass have positive velocity an negative acceleration? Reeber: velocity is slope an acceleration is the curvature (a) y(t) (b) (c) t Eaple ass = g on a spring oscillates with aplitue = c. t t = its spee is at a aiu, an is v=+ /s What is the angular frequency of oscillation ω? What is the spring constant? General relationships E = K + U = constant, (/) ½ So at aiu spee U= an ½ v = E = ½ thus = v / = () /(.) = 8 N/, ra/sec Physics 7: Lecture 4, Pg 9 Physics 7: Lecture 4, Pg Lecture 4, Eaple nitial Conitions ass hanging fro a vertical spring is lifte a istance above equilibriu an release at t =. Which of the following escribe its velocity an acceleration as a function of tie (upwars is positive y irection): () v(t) = - v a sin( ωt ) a(t) = -a a cos( ωt ) Lecture 4, Eercise nitial Conitions ass hanging fro a vertical spring is lifte a istance above equilibriu an release at t =. Which of the following escribe its velocity an acceleration as a function of tie (upwars is positive y irection): () v(t) = - v a sin( ωt ) a(t) = -a a cos( ωt ) (B) v(t) = v a sin( ωt ) a(t) = a a cos( ωt ) y (B) v(t) = v a sin( ωt ) a(t) = a a cos( ωt ) y t = (C) v(t) = v a cos( ωt ) a(t) = -a a cos(ωt ) t = (C) v(t) = v a cos( ωt ) a(t) = -a a cos(ωt ) (both v a an a a are positive nubers) (both v a an a a are positive nubers) Physics 7: Lecture 4, Pg Physics 7: Lecture 4, Pg Energy of the Spring-Mass Syste We now enough to iscuss the echanical energy of the oscillating ass on a spring. Reeber, (t) = cos( ωt + φ ) v(t) = -ω sin( ωt + φ ) a(t) = -ω cos(ωt + φ) What about Friction? Friction causes the oscillations to get saller over tie This is nown as DMPNG. s a oel, we assue that the force ue to friction is proportional to the velocity, F friction = - b v Kinetic energy is always K = ½ v.4. K = ½ [ -ω sin( ωt + φ )] n the potential energy of a spring is, U = ½ U = ½ [ cos (ωt + φ) ] ωt Physics 7: Lecture 4, Pg 3 Physics 7: Lecture 4, Pg 4 Page 4

5 Physics 7 Lecture 4 b = What about Friction? We can guess at a new solution. = ep( bt ) cos( ω t + φ ) With, b = b + + = ω o an now ω b / What about Friction? ( t) = ep( bt ) cos( ω t + φ) if ωo > b / What oes this function loo lie? -.8 Physics 7: Lecture 4, Pg 5 - ωt Physics 7: Lecture 4, Pg 6 Lecture 4, Eercise Resonant Motion Consier the following set of penulus all attache to the sae string n then traveling waves on a string y D C f start bob D swinging which of the others will have the largest swing aplitue? () (B) (C) B General haronic waves y (,t ) = cos ( ωt ) = π π πf = T ω v = f = Waves on a string F v = µ P = µ vω E = µω tension ass / length Physics 7: Lecture 4, Pg 7 Physics 7: Lecture 4, Pg 8 Soun Wave Properties Displaceent: The aiu relative isplaceent s of a point on the wave. Displaceent is longituinal. Maiu isplaceent has iniu velocity s(, t) = s a s / = ωs cos[( π / ) ωt)] a sin[( π / ) ωt)] Molecules pile up where the relative velocity is aiu (i.e., s/ = ω s a ) Wavelength s s a P a =ρvωs a Eaple, Energy transferre by a string Two strings are hel at the sae tension an riven with the sae aplitue an frequency. The only ifference is that one is thicer an has a ass per unit length that is four ties larger than the thinner one. Which string (an by how uch) transfers the ost power? (Circle the correct answer.) () the thicer string by a factor of 4. (B) the thicer string by a factor of. (C) they transfer an equivalent aount of energy. (D) the thinner string by a factor of. (E) the thinner string by a factor of 4. T T P = µ ω µ ω µ 4 µ µ thic 4 v = 4 = P thin Physics 7: Lecture 4, Pg 9 Physics 7: Lecture 4, Pg 3 Page 5

Physics 7 Lecture 4 Eaple, pulses on a string transverse pulse is initially traveling to the right on a string that is joine, on the right, to a thicer string of higher ass per unit length.

6 Physics 7 Lecture 4 Eaple, pulses on a string transverse pulse is initially traveling to the right on a string that is joine, on the right, to a thicer string of higher ass per unit length. The tension reains constant T throughout. Part of the pulse is reflecte an part transitte. The rawing to the right shows the before (at top) an after (botto) the pulse traverses the interface. There are however a few istaes in the botto rawing. entify two things wrong in the botto setch assuing the top setch is correct. Original pulse (before interface) Soun Wave, longituinal wave Displaceent: The aiu relative isplaceent s of a point on the wave. Displaceent is longituinal. Maiu isplaceent has iniu velocity s(, t) = s a s / = ωs Wavelength cos[( π / ) ωt)] sin[( π / ) ωt)] a Molecules pile up where the relative velocity is aiu (i.e., s/ = ω s a ) s P = ρv ω s a s a P a =ρvωs a Physics 7: Lecture 4, Pg 3 Physics 7: Lecture 4, Pg 3 Waves, Wavefronts,, an Rays f the power output of a source is constant, the total power of any wave front is constant. The ntensity at any point epens on the type of wave. Pav P = = 4πR Pav P = = av const av ntensity of souns The aplitue of pressure wave epens on Frequency ω of haronic soun wave Spee of soun v an ensity of eiu ρ of eiu Displaceent aplitue s a of eleent of eiu ntensity of a soun wave is Pa = ωvρs a Pa = ρ v Proportional to (aplitue) This is a general result (not only for soun) Threshol of huan hearing: = - W/ Physics 7: Lecture 4, Pg 33 Physics 7: Lecture 4, Pg 34 Soun Level, Eaple β = log What is the soun level that correspons to an intensity of. -7 W/? β = log (. -7 W/ /. - W/ ) = log. 5 = 53 B Rule of thub: n apparent oubling in the louness is approiately equivalent to an increase of B. This factor is not linear with intensity Physics 7: Lecture 4, Pg 35 Doppler effect, oving sources/receivers f the source of soun is oving Towar the observer sees saller way fro observer sees larger v fobserver = fsource v ± vs f the observer is oving Towar the source sees saller way fro source sees larger v ± vo fobserver = fsource v f both are oving v ± v o fobserver = fsource v vs Doppler Eaple uio Eaples: police car, train, etc. (Recall: v is vector) Doppler Eaple Visual Physics 7: Lecture 4, Pg 36 Page 6

7 Physics 7 Lecture 4 Physics 7, Lecture 4, Nov. 7 gena: Mi-Ter 3 Review Elastic Properties of Matter, Mouli Pressure, Wor, rchiees Principle, Flui flow, Bernoulli Oscillatory otion, Linear oscillator, Penulus Energy, Daping, Resonance Transverse Waves, Pulses, Reflection, Transission, Power Longituinal Waves (Soun), Plane waves, Spherical waves Louness, Doppler effect ssignents: Proble Set 9 ue Tuesay, Dec. 5, :59 PM Ch. 8: 3, 8, 3, 4, 58 Ch. 9:,, 5, 3, 43, 57 Mi-ter 3, Tuesay, Nov. 8, Chapters 4-7, 7, 9 inutes, 7:5-8:45 PM in roos 5 an 3 Psychology. (Graes on Monay) Wenesay, Chapter 9 (Teperature, Heat an Theroynaics) Physics 7: Lecture 4, Pg 37 Page 7

m A 9. The length of a simple pendulum with a period on Earth of one second is most nearly (A) 0.12 m (B) 0.25 m (C) 0.50 m (D) 1.0 m (E) 10.

m A 9. The length of a simple pendulum with a period on Earth of one second is most nearly (A) 0.12 m (B) 0.25 m (C) 0.50 m (D) 1.0 m (E) 10. P Physics Multiple Choice Practice Oscillations. ass, attache to a horizontal assless spring with spring constant, is set into siple haronic otion. Its axiu isplaceent fro its equilibriu position is. What