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UW-Madison PHYSICS 207 - Physics 207 -Lecture 20

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Lecture 20SHM So FarSlide 3Slide 4The shaker cartSlide 6Slide 7What about Vertical Springs?Exercise Simple Harmonic MotionExercise Simple Harmonic MotionEnergy in SHMSHM and quadratic potentialsSlide 17What about Friction? A velocity dependent drag force (A model)What about Friction?Variations in the dampingDamped Simple Harmonic MotionDriven SHM with ResistanceResonance-based DNA detection with nanoparticle probesStick - Slip FrictionDramatic example of resonanceA short clipSlide 28Exercise Resonant MotionChapter 15, FluidsFluids (Ch. 15)FluidsSlide 33Slide 34Pressure vs. Depth Incompressible Fluids (liquids)Slide 36Physics 207: Lecture 20, Pg 1Lecture 20Goals:Goals:•Chapter 14Chapter 14 Understand and use energy conservation in oscillatory systems. Understand the basic ideas of damping and resonance.•Chapter 15Chapter 15 Understand pressure in liquids and gases Use Archimedes’ principle to understand buoyancy Understand the equation of continuity Use an ideal-fluid model to study fluid flow. Investigate the elastic deformation of solids and liquids•AssignmentAssignment HW8, Due Wednesday, Apr. 8th  Tuesday: Read all of Chapter 15Physics 207: Lecture 20, Pg 2SHM So FarThe most general solution is x(t) = A cos(t + )where A = amplitude  = (angular) frequency = 2 f = 2/T  = phase constant mkVelocity: v(t) = -A sin(t + )Acceleration: a(t) = -2A cos(t + ) LgSimple Pendulum:Hooke’s Law Spring:Spring constantInertiaPhysics 207: Lecture 20, Pg 3SHM So FarThe most general solution is x(t) = A cos(t + )where A = amplitude  = (angular) frequency = 2 f = 2/T  = phase constant Here  = 0Velocity: v(t) = -A sin(t + )Acceleration: a(t) = -2A cos(t + ) T = 2/ AAtimex(t)Physics 207: Lecture 20, Pg 4SHM So FarFor SHM without friction The frequency does not depend on the amplitude !The oscillation occurs around the equilibrium point where the force is zero! Mechanical Energy is constant, it transfers between potential and kinetic energies.Physics 207: Lecture 20, Pg 5The shaker cartYou stand inside a small cart attached to a heavy-duty spring, the spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you.At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground.What effect does jettisoning the sandbag at the equilibrium position have on the amplitude of your oscillation?A. It increases the amplitude.B. It decreases the amplitude.C. It has no effect on the amplitude.Hint: At equilibrium, both the cart and the bag are moving at their maximum speed. By dropping the bag at this point, energy (specifically the kinetic energy of the bag) is lost from the spring-cart system. Thus, both the elastic potential energy at maximum displacement and the kinetic energy at equilibrium must decreasePhysics 207: Lecture 20, Pg 6The shaker cartInstead of dropping the sandbag as you pass through equilibrium, you decide to drop the sandbag when the cart is at its maximum distance from equilibrium.What effect does jettisoning the sandbag at the cart’s maximum distance from equilibrium have on the amplitude of your oscillation?A. It increases the amplitude.B. It decreases the amplitude.C. It has no effect on the amplitude.Hint: Dropping the bag at maximum distance from equilibrium, both the cart and the bag are at rest. By dropping the bag at this point, no energy is lost from the spring-cart system. Therefore, both theelastic potential energy at maximum displacementand the kinetic energy at equilibrium must remain constant.Physics 207: Lecture 20, Pg 7The shaker cartWhat effect does jettisoning the sandbag at the cart’s maximum distance from equilibrium have on the maximum speed of the cart?A. It increases the maximum speed.B. It decreases the maximum speed.C. It has no effect on the maximum speed.Hint: Dropping the bag at maximum distance from equilibrium, both the cart and the bag are at rest. By dropping the bag at this point, no energy is lost from the spring-cart system. Therefore, both the elastic potential energy at maximum displacement and the kinetic energy at equilibrium must remain constant.Physics 207: Lecture 20, Pg 8What about Vertical Springs?For a vertical spring, if y is measured from the equilibrium position Recall: force of the spring is the negative derivative of this function:This will be just like the horizontal case:-ky = ma = j j kmF= -kyy = 0U ky122kydydUF NET22dtydmWhich has solution y(t) = A cos( t + ) kmwherePhysics 207: Lecture 20, Pg 9 Exercise Simple Harmonic MotionA mass oscillates up & down on a spring. It’s position as a function of time is shown below. At which of the points shown does the mass have positive velocity and negative acceleration ? Remember: velocity is slope and acceleration is the curvaturet y(t)(a)(b)(c)y(t) = A cos( t + ) v(t) = -A  sin( t + ) a(t) = -A  cos( t + )Physics 207: Lecture 20, Pg 13Exercise Simple Harmonic MotionYou are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T1.Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T2. Which of the following is true recalling that  = (g / L)½ (A) T1 = T2 (B) T1 > T2 (C) T1 < T2 T1T2Physics 207: Lecture 20, Pg 15Energy in SHMFor both the spring and the pendulum, we can derive the SHM solution using energy conservation. The total energy (K + U) of a system undergoing SMH will always be constant!This is not surprising since there are only conservative forces present, hence energy is conserved.-A A0xUUKEPhysics 207: Lecture 20, Pg 16SHM and quadratic potentialsSHM will occur whenever the potential is quadratic.For small oscillations this will be true:For example, the potential betweenH atoms in an H2 molecule lookssomething like this:-A A0xUUKEUxPhysics 207: Lecture 20, Pg 17See: http://hansmalab.physics.ucsb.eduSHM and quadratic potentialsCurvature reflects the spring constantor modulus (i.e., stress vs. strain orforce vs. displacement)Measuring modular proteins with an AFMUxPhysics 207: Lecture 20, Pg 18What about


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UW-Madison PHYSICS 207 - Physics 207 -Lecture 20

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