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Lecture D31 : Linear Harmonic OscillatorSpring-Mass System Spring Force F = −kx, k > 0 Newton’s Second Law (Define) Natural frequency (and period) Equation of a linear harmonic oscillator
Solution General solution or, Initial conditions Solution, or,
Graphical RepresentationDisplacement, Velocity and Acceleration
Energy Conservation No dissipation T + V = constant Potential EnergyEquilibrium Position No dissipation T + V = constant Potential Energy At Equilibrium −kδst +mg = 0,
Energy Conservation (cont’d)Kinetic Energy Conservation of energy Governing equation Above represents a very general way of de-riving equations of motion (Lagrangian Me-chanics)
Energy Conservation (cont’d)If V = 0 at the equilibrium position,
Examples Spring-mass systems Rotating machineryPendulums (small amplitude) Oscillating bodies (small amplitude) Aircraft motion (Phugoid) Waves (String, Surface, Volume, etc.) Circuits . . .
The Phugoid Idealized situationSmall perturbations (h′, v′) about steady level flight (h0, v0) L = W (≡ mg) for v = v0, but L ∼ v2,
The Phugoid (cont’d) Vertical momentum equationEnergy conservation T = D (to first order) Equations of motion
The Phugoid (cont’d) h′ and v′ satisfy a Harmonic Oscillator Equa-tion Natural frequency and Period Light aircraft v0 ∼ 150 ft/s → τ ∼ 20s Solution
The Phugoid (cont’d) Integrate v′ equation
Chapter 14 4 Oscillations. Section 14-1: Simple Harmonic Motion A simple harmonic oscillator is any system that oscillates. An example of this is a mass.
Pendulums Physics 202 Professor Lee Carkner Lecture 4 “The sweep of the pendulum had increased … As a natural consequence its velocity was also much greater.”
Department of Physics and Applied Physics , S2010, Lecture 23 Physics I LECTURE 23 5/10/10.
Math 5900 – Summer 2011 Lecture 1: Simple Harmonic Oscillations Gernot Laicher University of Utah - Department of Physics & Astronomy.
Chapter 13 VibrationsandWaves. Hooke’s Law F s = - k x F s = - k x F s is the spring force F s is the spring force k is the spring constant k is the spring.
Phy 212: General Physics II Chapter 15: Oscillations Lecture Notes.
Simple Harmonic Motion
Physics 111: Mechanics Lecture 14 Dale Gary NJIT Physics Department.
Copyright © 2009 Pearson Education, Inc. Lecture 1 – Waves & Sound a) Simple Harmonic Motion (SHM)
AP Physics B Topics for Test 3. AP Physics B Energy and Work (chapter 6) –Kinetic, Potential Energies –Spring energy –Work done by external force = Fxd.
PHYS 218 sec Review Chap. 13 Periodic Motion.
10/22/2012PHY 113 A Fall Lecture 211 PHY 113 A General Physics I 9-9:50 AM MWF Olin 101 Plan for Lecture 21: Chapter 15 – Simple harmonic motion.
Chapter 13 Oscillatory Motion.
Harmonic Motion AP Physics C.
Motion of a mass at the end of a spring Differential equation for simple harmonic oscillation Amplitude, period, frequency and angular frequency Energetics.
Chapter 13: Oscillatory Motions
NAZARIN B. NORDIN What you will learn: Load transfer, linear retardation/ acceleration Radius of gyration Moment of inertia Simple.
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