Physics 07, Lecture 18, Nov. 3 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand and use energy conservation in oscillatory systes. Understand the basic ideas of daping and resonance. Phase Contrast Microscopy Epithelial cell in brightfield (BF) using a 40x lens (NA 0.75) (left) and with phase contrast using a DL Plan Achroat 40x (NA 0.65) (right). A green interference filter is used for both iages. Physics 07: Lecture 18, Pg 1 Physics 07, Lecture 18, Nov. 3 Assignent HW8, Due Wednesday, Nov. 1 th Wednesday: Read through Chapter 15.4 Physics 07: Lecture 18, Pg Page 1
Periodic Motion is everywhere Exaples of periodic otion Earth around the sun Elastic ball bouncing up an down Quartz crystal in your watch, coputer cloc, ipod cloc, etc. Physics 07: Lecture 18, Pg 3 Periodic Motion is everywhere Exaples of periodic otion Heart beat In taing your pulse, you count 70.0 heartbeats in 1 in. What is the period, in seconds, of your heart's oscillations? Period is the tie for one oscillation T= 60 sec/ 70.0 = 0.86 s What is the frequency? f = 1 / T = 1.17 Hz Physics 07: Lecture 18, Pg 4 Page
A special ind of period oscillator: Haronic oscillator What do all haronic oscillators have in coon? 1. A position of equilibriu. A restoring force, which ust be linear [Hooe s law spring F = -x (In a pendulu the behavior only linear for sall angles: sin where = s / L) ] In this liit we have: F = -s with = g/l) 3. Inertia 4. The drag forces are reasonably sall Physics 07: Lecture 18, Pg 5 Siple Haronic Motion (SHM) In Siple Haronic Motion the restoring force on the ass is linear, that is, exactly proportional to the displaceent of the ass fro rest position Hooe s Law : F = -x If >> rapid oscillations <=> large frequency If << slow oscillations <=> low frequency Physics 07: Lecture 18, Pg 6 Page 3
Siple Haronic Motion (SHM) We now that if we stretch a spring with a ass on the end and let it go the ass will, if there is no friction,.do soething 1. Pull bloc to the right until x = A. After the bloc is released fro x = A, it will A: reain at rest B: ove to the left until it reaches equilibriu and stop there C: ove to the left until it reaches x = -A and stop there D: ove to the left until it reaches x = -A and then begin to ove to the right -A 0( X eq ) A Physics 07: Lecture 18, Pg 7 Siple Haronic Motion (SHM) We now that if we stretch a spring with a ass on the end and let it go the ass will. 1. Pull bloc to the right until x = A. After the bloc is released fro x = A, it will A: reain at rest B: ove to the left until it reaches equilibriu and stop there C: ove to the left until it reaches x = -A and stop there D: ove to the left until it reaches x = -A and then begin to ove to the right This oscillation is called Siple Haronic Motion -A 0( X eq ) A Physics 07: Lecture 18, Pg 8 Page 4
Siple Haronic Motion (SHM) The tie it taes the bloc to coplete one cycle is called the period. Usually, the period is denoted T and is easured in seconds. The frequency, denoted f, is the nuber of cycles that are copleted per unit of tie: f = 1 / T. In SI units, f is easured in inverse seconds, or hertz (Hz). If the period is doubled, the frequency is A. unchanged B. doubled C. halved Physics 07: Lecture 18, Pg 9 Siple Haronic Motion (SHM) An oscillating object taes 0.10 s to coplete one cycle; that is, its period is 0.10 s. What is its frequency f? Express your answer in hertz. f = 1/ T = 10 Hz Physics 07: Lecture 18, Pg 10 Page 5
Siple Haronic Motion Note in the (x,t) graph that the vertical axis represents the x coordinate of the oscillating object, and the horizontal axis represents tie. Which points on the x axis are located a displaceent A fro the equilibriu position? A. R only B. Q only C. both R and Q Position tie Physics 07: Lecture 18, Pg 11 Suppose that the period is T. Siple Haronic Motion Which of the following points on the t axis are separated by the tie interval T? A. K and L B. K and M C. K and P D. L and N E. M and P tie Physics 07: Lecture 18, Pg 1 Page 6
Siple Haronic Motion Now assue that the t coordinate of point K is 0.0050 s. What is the period T, in seconds? T = 0.0 s How uch tie t does the bloc tae to travel fro the point of axiu displaceent to the opposite point of axiu displaceent t = 0.01 s tie Physics 07: Lecture 18, Pg 13 Siple Haronic Motion Now assue that the x coordinate of point R is 0.1. What distance d does the object cover during one period of oscillation? d = 0.48 What distance d does the object cover between the oents labeled K and N on the graph? d = 0.36 tie Physics 07: Lecture 18, Pg 14 Page 7
SHM Dynaics At any given instant we now that F = a ust be true. But in this case F = - x and a = d x dt So: - x = a = d x dt a x F = - x d x dt x = a differential equation for x(t)! Siple approach, guess a solution and see if it wors! Physics 07: Lecture 18, Pg 15 SHM Solution... Try either cos ( ω t ) or sin ( ω t ) Below is a drawing of A cos ( ω t ) where A = aplitude of oscillation T = π/ω A π π π A [with ω = (/) ½ and ω = π f = π /T ] Both sin and cosine wor so need to include both Physics 07: Lecture 18, Pg 16 Page 8
x(t) = Cobining sin and cosine solutions B cos ωt + C sin ωt = A cos ( ωt + φ) = A (cos ωt cos φ sin ωt sin φ ) =A cos φ cos ωt A sin φ sin ωt) Notice that B = A cos φ C = -A sin φ tan φ= -C/B π θ π cos sin 0 x Use initial conditions to deterine phase φ! Physics 07: Lecture 18, Pg 17 Energy of the Spring-Mass Syste We now enough to discuss the echanical energy of the oscillating ass on a spring. x(t) = A cos ( ωt + φ) If x(t) is displaceent fro equilibriu, then potential energy is v(t) = dx/dt U(t) = ½ x(t) = A cos ( ωt + φ) v(t) = A ω (-sin ( ωt + φ)) And so the inetic energy is just ½ v(t) Finally, a(t) = dv/dt = -ω A cos(ωt + φ) K(t) = ½ v(t) = (Aω) sin ( ωt + φ) Physics 07: Lecture 18, Pg 18 Page 9
Energy of the Spring-Mass Syste x(t) = A cos( ωt + φ ) v(t) = -ωa sin( ωt + φ ) a(t) = -ω A cos(ωt + φ) Kinetic energy is always K = ½ v = ½ (ωa) sin (ωt+φ) Potential energy of a spring is, And ω = / or = ω U = ½ x = ½ A cos (ωt + φ) U = ½ ω A cos (ωt + φ) Physics 07: Lecture 18, Pg 19 Energy of the Spring-Mass Syste x(t) = A cos( ωt + φ ) v(t) = -ωa sin( ωt + φ ) a(t) = -ω A cos(ωt + φ) And the echanical energy is K + U =½ ω A cos (ωt + φ) + ½ ω A sin (ωt + φ) K + U = ½ ω A [cos (ωt + φ) + sin (ωt + φ)] K + U = ½ ω A = ½ A which is constant Physics 07: Lecture 18, Pg 0 Page 10
Energy of the Spring-Mass Syste So E = K + U = constant =½ A ω = ω = At axiu displaceent K = 0 and U = ½ A and acceleration has it axiu (or iniu) At the equilibriu position K = ½ A = ½ v and U = 0 E = ½ A U~cos K~sin π π θ Physics 07: Lecture 18, Pg 1 SHM So Far The ost general solution is x = A cos(ωt + φ) where A = aplitude ω = (angular) frequency φ = phase constant ω = For SHM without friction, The frequency does not depend on the aplitude! This is true of all siple haronic otion! The oscillation occurs around the equilibriu point where the force is zero! Energy is a constant, it transfers between potential and inetic Physics 07: Lecture 18, Pg Page 11
What about Vertical Springs? For a vertical spring, if y is easured fro the equilibriu position U = 1 y Recall: force of the spring is the negative derivative of this function: du F = = y dy This will be just lie the horizontal case: -y = a = d y dt Which has solution y(t) = A cos( ωt + φ) where ω = j y = 0 F= -y Physics 07: Lecture 18, Pg 3 The Siple Pendulu (In class torques, τ = Iα, were used instead but the results are the sae) A pendulu is ade by suspending a ass at the end of a string of length L. Find the frequency of oscillation for sall displaceents. Σ F y = a c = T g cos(θ) = v /L Σ F x = a x = -g sin(θ) If θ sall then x L θ and sin(θ) θ dx/dt = L dθ/dt a x = d x/dt = L d θ/dt so a x = -g θ = L d θ / dt L d θ / dt - g θ = 0 and θ = θ 0 cos(ωt + φ) or θ = θ 0 sin(ωt + φ) with ω = (g/l) ½ z θ L y T g Physics 07: Lecture 18, Pg 4 x Page 1
Velocity and Acceleration Position: x(t) = A cos(ωt + φ) Velocity: v(t) = -ωa sin(ωt + φ) Acceleration: a(t) = -ω A cos(ωt + φ) x ax = A v ax = ωa a ax = ω A by taing derivatives, since: dx( t ) v( t ) = dt a( t ) = dv t dt ( ) 0 x Physics 07: Lecture 18, Pg 5 Physics 07, Lecture 18, Nov. 3 Assignent HW8, Due Wednesday, Nov. 1 th Wednesday: Read through Chapter 15.4 The rest are for Wednesday, plus daping, resonance and part of Chapter 15. Physics 07: Lecture 18, Pg 6 Page 13
The shaer cart You stand inside a sall cart attached to a heavy-duty spring, the spring is copressed and released, and you shae bac and forth, attepting to aintain your balance. Note that there is also a sandbag in the cart with you. At the instant you pass through the equilibriu position of the spring, you drop the sandbag out of the cart onto the ground. What effect does jettisoning the sandbag at the equilibriu position have on the aplitude of your oscillation? It increases the aplitude. It decreases the aplitude. It has no effect on the aplitude. Hint: At equilibriu, both the cart and the bag are oving at their axiu speed. By dropping the bag at this point, energy (specifically the inetic energy of the bag) is lost fro the spring-cart syste. Thus, both the elastic potential energy at axiu displaceent and the inetic energy at equilibriu ust decrease Physics 07: Lecture 18, Pg 7 The shaer cart Instead of dropping the sandbag as you pass through equilibriu, you decide to drop the sandbag when the cart is at its axiu distance fro equilibriu. What effect does jettisoning the sandbag at the cart s axiu distance fro equilibriu have on the aplitude of your oscillation? It increases the aplitude. It decreases the aplitude. It has no effect on the aplitude. Hint: Dropping the bag at axiu distance fro equilibriu, both the cart and the bag are at rest. By dropping the bag at this point, no energy is lost fro the spring-cart syste. Therefore, both the elastic potential energy at axiu displaceent and the inetic energy at equilibriu ust reain constant. Physics 07: Lecture 18, Pg 8 Page 14
The shaer cart What effect does jettisoning the sandbag at the cart s axiu distance fro equilibriu have on the axiu speed of the cart? It increases the axiu speed. It decreases the axiu speed. It has no effect on the axiu speed. Hint: Dropping the bag at axiu distance fro equilibriu, both the cart and the bag are at rest. By dropping the bag at this point, no energy is lost fro the spring-cart syste. Therefore, both the elastic potential energy at axiu displaceent and the inetic energy at equilibriu ust reain constant. Physics 07: Lecture 18, Pg 9 Exercise Siple Haronic Motion 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 velocity and negative acceleration? Reeber: velocity is slope and acceleration is the curvature (a) y(t) (b) (c) t Physics 07: Lecture 18, Pg 30 Page 15
Exaple A ass = g on a spring oscillates with aplitude A = 10 c. At t = 0 its speed is at a axiu, and is v=+ /s What is the angular frequency of oscillation ω? What is the spring constant? General relationships E = K + U = constant, ω = (/) ½ So at axiu speed U=0 and ½ v = E = ½ A thus = v /A = x () /(0.1) = 800 N/, ω = 0 rad/sec x Physics 07: Lecture 18, Pg 31 Hoe Exercise Siple Haronic Motion You are sitting on a swing. A friend gives you a sall push and you start swinging bac & forth with period T 1. Suppose you were standing on the swing rather than sitting. When given a sall push you start swinging bac & forth with period T. Which of the following is true recalling that ω = (g/l) ½ (A) T 1 = T (B) T 1 > T (C) T 1 < T Physics 07: Lecture 18, Pg 3 Page 16
Hoe Exercise Siple Haronic Motion You are sitting on a swing. A friend gives you a sall push and you start swinging bac & forth with period T 1. Suppose you were standing on the swing rather than sitting. When given a sall push you start swinging bac & forth with period T. If you are standing, the center of ass oves towards the pivot point and so L is less, ω is bigger, T is saller (A) T 1 = T (B) T 1 > T (C) T 1 < T Physics 07: Lecture 18, Pg 33 BTW: The Rod Pendulu (not tested) A pendulu is ade by suspending a thin rod of length L and ass M at one end. Find the frequency of oscillation for sall displaceents. Σ τ z = I α = - r x F = (L/) g sin(θ) (no torque fro T) -[ L /1 + (L/) ] α L/ g θ -1/3 L d θ/dt = ½ g θ z θ T g x CM L Physics 07: Lecture 18, Pg 34 Page 17
BTW: General Physical Pendulu (not tested) Suppose we have soe arbitrarily shaped solid of ass M hung on a fixed axis, that we now where the CM is located and what the oent of inertia I about the axis is. The torque about the rotation (z) axis for sall θ is (sin θ θ ) τ = -MgR sinθ -MgRθ d θ MgR θ = I dt d dt θ = ω θ ω = MgR where θ = θ 0 cos(ωt + φ) τ I α z-axis θ R x CM Mg Physics 07: Lecture 18, Pg 35 Torsion Pendulu Consider an object suspended by a wire attached at its CM. The wire defines the rotation axis, and the oent of inertia I about this axis is nown. The wire acts lie a rotational spring. When the object is rotated, the wire is twisted. This produces a torque that opposes the rotation. In analogy with a spring, the torque produced is proportional to the displaceent: τ = - κ θ where κ is the torsional spring constant ω = (κ/i) ½ τ I wire θ Physics 07: Lecture 18, Pg 36 Page 18
Hoe Exercise Period All of the following torsional pendulu bobs have the sae ass and ω = (κ/i) ½ Which pendulu rotates the fastest, i.e. has the longest period? (The wires are identical) R R R R (A) (B) (C) (D) Physics 07: Lecture 18, Pg 37 Reviewing Siple Haronic Oscillators Spring-ass syste Pendula d x = ω x where ω = dt d θ = ω θ dt x(t) = A cos( ωt + φ) θ = θ 0 cos( ωt + φ) General physical pendulu ω = MgR I a x z-axis F = -x R θ xcm Mg wire Torsion pendulu ω = κ I τ I θ Physics 07: Lecture 18, Pg 38 Page 19
Energy in SHM For both the spring and the pendulu, we can derive the SHM solution using energy conservation. The total energy (K + U) of a syste undergoing SMH will always be constant! This is not surprising since U there are only conservative -A forces present, hence energy is conserved. 0 A U K E x Physics 07: Lecture 18, Pg 39 SHM and quadratic potentials SHM will occur whenever the potential is quadratic. For sall oscillations this will be true: For exaple, the potential between H atos in an H olecule loos soething lie this: U U x K E U -A 0 A x Physics 07: Lecture 18, Pg 40 Page 0