Phsics 1 ecture 9 Goals ecture 9 v Describe oscillator otion in a siple pendulu v Describe oscillator otion with torques v Introduce daping in SHM v Discuss resonance v Final Ea Details l Sunda, Ma 13th 1:5a-1:5p in 15 Ag Hall & quiet roo l Forat: v Closed boo v Up to 4 8½1 sheets, hand written onl v Approiatel 5% fro Chapters 13-15 and 5% 1-1 v Bring a calculator l Special needs/ conflicts: All requests for alternative test arrangeents should be ade b hursda Ma1th (ecept for edical eergenc) Phsics 1: ecture 9, Pg 1 Phsics 1: ecture 9, Pg Mechanical Energ of the Spring-Mass Sste (t) = A cos( ωt + φ ) v(t) = -ωa sin( ωt + φ ) a(t) = -ω A cos( ωt + φ ) Kinetic energ is alwas K = ½ v = ½ (ωa) sin (ωt+φ) Potential energ of a spring is, U = ½ = ½ A cos (ωt + φ) And ω = / or = ω K + U = constant Phsics 1: ecture 9, Pg 3 SHM is a close as ae Mendota So can ou estiate the characteristic frequenc for a bobbing in the water? If ou have equilibriu and there is a linear restoring force, then es with ω = ( / ) ½ At equlibriu F = = F g F = g = ρ A g B w FB g B Phsics 1: ecture 9, Pg 4 SHM is a close as ae Mendota Deeper than eans B = ρ and a net force of F wag F = ρw A ) g A linear restoring force with = ρ w Ag and boat ass = A ρ w so ω = (g / ) ½ ighter boats bob ore quicl than heav ones (if the sae sie) Net ( Eaple of phase = A cos(ωt + φ) l You have identical vertical springs with identical asses. Both are undergoing siple haronic otion with frequenc f = 1/π (/) ½ l he 1 st ass alwas oves up when the nd ass is oves down. Vertical displaceent π π 1 st ass / tie =ωt nd ass F B g What is the phase difference between the two asses? A: B: π/ C: π D: 3π/ E: π Phsics 1: ecture 9, Pg 5 Phsics 1: ecture 9, Pg 6 Page 1
Phsics 1 ecture 9 he shaer cart l You stand inside a sall cart attached to a heav-dut spring, the spring is copressed and released, and ou shae bac and forth, attepting to aintain our balance. Note that there is also a sandbag in the cart with ou. l At the instant ou pass through the equilibriu position of the spring, ou drop the sandbag out of the cart onto the ground. l What effect does jettisoning the sandbag at the equilibriu position have on the aplitude of our oscillation? A. It increases the aplitude. B. It decreases the aplitude. C. It has no effect on the aplitude. Hint: At equilibriu, both the cart and the bag are oving at he shaer cart l Instead of dropping the sandbag as ou pass through equilibriu, ou decide to drop the sandbag when the cart is at its aiu distance fro equilibriu. l What effect does jettisoning the sandbag at the cart s aiu distance fro equilibriu have on the aplitude of our oscillation? A. It increases the aplitude. B. It decreases the aplitude. C. It has no effect on the aplitude. Hint: At aiu displaceent there is no inetic energ. their aiu speed. Phsics 1: ecture 9, Pg 7 Phsics 1: ecture 9, Pg 8 he shaer cart l What effect does jettisoning the sandbag at the cart s aiu displaceent fro equilibriu have on the aiu speed of the cart? A. It increases the aiu speed. B. It decreases the aiu speed. C. It has no effect on the aiu speed. Hint: At aiu displaceent there is no inetic energ. Phsics 1: ecture 9, Pg 9 he Pendulu (using torque) l A pendulu is ade b suspending a ass at the end of a string of length. Find the frequenc of oscillation for sall displaceents. sin Σ τ = Iα = -g sin() Σ τ α -g (d / ) = -g copare to a = - d / = (-g/) with (t)= cos( ωt + φ ) and ω =(g/) ½ g Phsics 1: ecture 9, Pg 1 he Pendulu l A pendulu is ade b suspending a ass at the end of a string of length. Find the frequenc of oscillation for sall displaceents. he Siple Pendulu l A pendulu is ade b suspending a ass at the end of a string of length. Find the frequenc of oscillation for sall displaceents. If sall then sin() tan. = sin. =. 5 tan.9 = sin.9 =.9 1 tan.17 = sin.17 =.17 15 tan.6 =.7 sin.6 =.6 Σ F = a = g cos() = a c = v / Σ F = a = -g sin() where = tan If sall then and sin() d/ = d/ a = d / = d / so a = -g = d / d / - g = g Phsics 1: ecture 9, Pg 11 and = cos(ωt + φ) or = sin(ωt + φ) with ω = (g/) ½ g Phsics 1: ecture 9, Pg 1 Page
Phsics 1 ecture 9 What about Vertical Springs? l For a vertical spring, if is easured fro the equilibriu position U = 1 l ecall: force of the spring is the negative derivative of this function: du F = = d l his will be just lie the horiontal case: d - = a = Which has solution (t) = A cos( ωt + φ) where ω = Phsics 1: ecture 9, Pg 13 j = F= - Eercise Siple Haronic Motion l A ass oscillates up & down on a spring. It s position as a function of tie is shown below. At which of the points shown does the ass have positive velocit and negative acceleration? eeber: velocit is slope and acceleration is the curvature (a) (t) (b) (c) t Phsics 1: ecture 9, Pg 14 Eaple l A ass = g on a spring oscillates with aplitude A = 1 c. At t = its speed is at a aiu, and is v=+ /s v What is the angular frequenc of oscillation ω? v What is the spring constant? General relationships E = K + U = constant, ω = (/) ½ So at aiu speed U= and ½ v = E = ½ A thus = v /A = () /(.1) = 8 N/, ω = rad/sec Phsics 1: ecture 9, Pg 15 Eaple Initial Conditions l A ass hanging fro a vertical spring is lifted a distance d above equilibriu and released at t =. Which of the following describe its velocit and acceleration as a function of tie (upwards is positive direction): (A) 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 ) t = (C) v(t) = v a cos( ωt ) a(t) = -a a cos(ωt ) (both v a and a a are positive nubers) d Phsics 1: ecture 9, Pg 16 Eercise Initial Conditions l A ass hanging fro a vertical spring is lifted a distance d above equilibriu and released at t =. Which of the following describe its velocit and acceleration as a function of tie (upwards is positive direction): (A) v(t) = - v a sin( ωt ) a(t) = -a a cos( ωt ) Eercise Siple Haronic Motion l You are sitting on a swing. A friend gives ou a sall push and ou start swinging bac & forth with period 1. l Suppose ou were standing on the swing rather than sitting. When given a sall push ou start swinging bac & forth with period. (B) v(t) = v a sin( ωt ) a(t) = a a cos( ωt ) t = (C) v(t) = v a cos( ωt ) a(t) = -a a cos(ωt ) (both v a and a a are positive nubers) d Which of the following is true recalling that ω = (g/) ½ (A) 1 = (B) 1 > (C) 1 < Phsics 1: ecture 9, Pg 17 Phsics 1: ecture 9, Pg 18 Page 3
Phsics 1 ecture 9 A od Pendulu l A pendulu is ade b suspending a thin rod of length and ass M at one end. Find the frequenc of oscillation for sall displaceents. Σ τ = I α = - r F = (/) g sin() I rod at end = /3 - /3 α / g -1/3 d / = ½ g CM ω = 3 g g General Phsical Pendulu l Suppose we have soe arbitraril shaped solid of ass M hung on a fied ais, that we now where the CM is located and what the oent of inertia I about the ais is. l he torque about the rotation () ais for sall is (sin ) d τ = Mg = I -Mg sin -Mg d = ω where ω = Mg I = cos(ωt + φ) τ α -ais CM Mg Phsics 1: ecture 9, Pg 19 Phsics 1: ecture 9, Pg orsion Pendulu l Consider an object suspended b a wire attached at its CM. he wire defines the rotation ais, and the oent of inertia I about this ais is nown. l he wire acts lie a rotational spring. v When the object is rotated, the wire is twisted. his produces a torque that opposes the rotation. v orque is proportional to the angular displaceent: τ = - κ where κ is the torsion constant v ω = (κ/i) ½ τ I wire Eercise Period l All of the following torsional pendulu bobs have the sae ass and radius with ω = (κ/i) ½ l Which pendulu rotates the slowest (i.e. has the longest period) if the wires are identical? (A) (B) (C) (D) Phsics 1: ecture 9, Pg 1 Phsics 1: ecture 9, Pg What about Friction? A velocit dependent drag force (A odel) d d b = d b d + + = We can guess at a new solution. ( t) = A ep( bt ) cos( ω t + φ ) and now ω Note / What about Friction? A daped eponential ( t) = A ep( b t) cos ( ω t + φ ) if A 1. 1.8.6.4. ωo > b / -. With, ω = b = ω o b -.4 -.6 -.8-1 ωt Phsics 1: ecture 9, Pg 3 Phsics 1: ecture 9, Pg 4 Page 4
Phsics 1 ecture 9 Variations in the daping Daped Siple Haronic Motion ω = ω o ( b / ) Sall daping tie constant (/b) ow friction coefficient, b << l A downward shift in the angular frequenc l here are three atheaticall distinct regies ωo > b / ωo = b / ωo < b / Moderate daping tie constant (/b) Moderate friction coefficient (b < ) Phsics 1: ecture 9, Pg 5 underdaped criticall daped overdaped Phsics 1: ecture 9, Pg 6 Eercise l Daped oscillations: A can of coe is attached to a spring and is displaced b hand ( =.5 g & = 5. N/) he coe can is released, and it starts oscillating with an aplitude of A =.3. How daped is the sste? A. Underdaped (ultiple oscillations with an eponential deca in aplitude) B. Criticall daped (siple decaing otion with at ost one overshoot of the sste's resting position) C. Overdaped (siple eponentiall decaing otion, without an oscillations) Phsics 1: ecture 9, Pg 7 Driven SHM with esistance l Appl a sinusoidal force, F cos (ωt), and now consider what A and b do, d b d F + + = cos ωt Not Zero!!! F / A = bω b/ sall ( ω ω ) + ( ) stead state aplitude b/ iddling b large ω ω ω Phsics 1: ecture 9, Pg 8 For hursda l eview for final! Phsics 1: ecture 9, Pg 9 Page 5