Chapter 13 Lecture Essential University Physics Richard Wolfson 2nd Edition Oscillatory Motion © 2012 Pearson Education, Inc. Slide 13-1 In this lecture you’ll learn • To describe the conditions under which oscillatory motion occurs • To describe oscillatory motion quantitatively in terms of frequency, period, and amplitude • To explain simple harmonic motion and why it occurs universally in both natural and technological systems • To explain damped harmonic motion and resonance © 2012 Pearson Education, Inc. Slide 13-2 Oscillatory Motion • A system in oscillatory motion undergoes repeating, periodic motion about a point of stable equilibrium. • Oscillatory motion is characterized by – Its frequency f or period T=1/f. – Its amplitude A, or maximum excursion from equilibrium. • Shown here are positionversus-time graphs for two different oscillatory motions with the same period and amplitude. © 2012 Pearson Education, Inc. Slide 13-3 Simple Harmonic Motion • Simple harmonic motion (SHM) results when the force or torque that tends to restore equilibrium is directly proportional to the displacement from equilibrium. – The paradigm example is a mass m on a spring of spring constant k. d 2x m 2 kx dt • The angular frequency = 2p for this system is w= k m – In SHM, displacement is a sinusoidal function of time: x(t ) A cos t – Any amplitude A is possible. • In SHM, frequency doesn’t depend on amplitude. © 2012 Pearson Education, Inc. Slide 13-4 Quantities in Simple Harmonic Motion • Angular frequency, frequency, period: k w= m • Phase w 1 f = = 2p 2p k m 1 m T = = 2p f k – Describes the starting time of the displacement-versustime curve in oscillatory motion: x(t) = Acos(t + ) © 2012 Pearson Education, Inc. Slide 13-5 Velocity and Acceleration in SHM • Velocity is the rate of change of position: dx v(t ) A sin t dt • Acceleration is the rate of change of velocity: a(t ) dv 2 A cos t dt © 2012 Pearson Education, Inc. Slide 13-6 Other Simple Harmonic Oscillators • The torsional oscillator – A fiber with torsional constant provides a restoring torque. – Frequency depends on and rotational inertia: w= k I • Simple pendulum – Point mass on massless cord of length L: d 2 I 2 mgL sin mgL dt I mL2 for point mass w= g L © 2012 Pearson Education, Inc. Slide 13-7 SHM and Circular Motion • Simple harmonic motion can be viewed as one component of uniform circular motion. – Angular frequency in SHM is the same as angular velocity in circular motion. As the position vector r traces out a circle, its x– and y– components are sinusoidal functions of time. © 2012 Pearson Education, Inc. Slide 13-8 Energy in Simple Harmonic Motion • In the absence of nonconservative forces, the energy of a simple harmonic oscillator does not change. – But energy is transfered back and forth between kinetic and potential forms. © 2012 Pearson Education, Inc. Slide 13-9 Simple Harmonic Motion is Ubiquitous! • That’s because most systems near stable equilibrium have potential-energy curves that are approximately parabolic. – Ideal spring: U = 12 kx 2 = 12 mw 2 x 2 – Typical potential-energy curve of an arbitrary system: © 2012 Pearson Education, Inc. Slide 13-10 Damped Harmonic Motion • With nonconservative forces present, SHM gradually damps d 2x dx out: m kx b dt 2 dt – Amplitude declines exponentially toward zero: x (t ) Ae bt 2 m cos(t ) – For weak damping b, oscillations still occur at approximately the undamped frequency – With stronger damping, oscillations cease. • Critical damping brings the system to equilibrium most quickly. w= k m. (a) Underdamped, (b) critically damped, and (c) overdamped oscillations. © 2012 Pearson Education, Inc. Slide 13-11 Driven Oscillations • When an external force acts on an oscillatory system, we say that the system is undergoing driven oscillation. • Suppose the driving force is F0cosdt, where d is the driving frequency, then Newton’s law is d 2x dx m 2 kx b F0 cos d t dt dt • The solution is x (t ) A cos(d t ) where A( ) 0 k m F0 m (d2 02 ) 2 b 2d2 / m2 : natural frequency © 2012 Pearson Education, Inc. Slide 13-12 Resonance • When a system is driven by an external force at near its natural frequency, it responds with large-amplitude oscillations. – This is the phenomenon of resonance. – The size of the resonant response increases as damping decreases. – The width of the resonance curve (amplitude versus driving frequency) also narrows with lower damping. Resonance curves for several damping strengths; 0 is the undamped natural frequency k/m. © 2012 Pearson Education, Inc. Slide 13-13 Summary • Oscillatory motion is periodic motion that results from a force or torque that tends to restore a system to equilibrium. • In simple harmonic motion (SHM), the restoring force or torque is directly proportional to displacement. – The mass-spring system is the paradigm simple harmonic oscillator. x(t ) A cos t w= k m – Damped harmonic motion occurs when nonconservative forces act on the oscillating system. – Resonance is a high-amplitude oscillatory response of a system driven at near its natural oscillation frequency. © 2012 Pearson Education, Inc. Slide 13-14

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