Chapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.

Transcription

1 Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1

2 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 Slide 13-

3 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. Slide 13-3

4 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. The angular frequency = p for this system is m d x dt kx w = k m In SHM, displacement is a sinusoidal function of time: x( t) Acos t Any amplitude A is possible. In SHM, frequency doesn t depend on amplitude. Slide 13-4

5 Quantities in Simple Harmonic Motion Angular frequency, frequency, period: w = Phase k m f = w p = 1 p Describes the starting time of the displacement-versustime curve in oscillatory motion: k m x(t) = Acos( t + ) T = 1 f = p m k Slide 13-5

6 Velocity and Acceleration in SHM Velocity is the rate of change of position: dx v( t) Asin t dt Acceleration is the rate of change of velocity: dv dt a( t) Acos t Slide 13-6

7 Other Simple Harmonic Oscillators The torsional oscillator A fiber with torsional constant provides a restoring torque. Frequency depends on and rotational inertia: w = Simple pendulum k I Point mass on massless cord of length L: I d I mgl sin mgl dt ml for point mass w = g L Slide 13-7

8 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. Slide 13-8

9 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. Slide 13-9

10 Simple Harmonic Motion is Ubiquitous! That s because most systems near stable equilibrium have potential-energy curves that are approximately parabolic. Ideal spring: U = 1 kx = 1 mw x Typical potential-energy curve of an arbitrary system: Slide 13-10

11 Damped Harmonic Motion With nonconservative forces present, SHM gradually damps out: d x dx Amplitude declines exponentially toward zero: 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. m kx b dt dt x t Ae t bt m ( ) cos( ) (a) Underdamped, (b) critically damped, and (c) overdamped oscillations. Slide 13-11

12 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 F 0 cos d t, where d is the driving frequency, then Newton s law is The solution is where d x dx 0 cos d m kx b F t dt dt x( t) Acos( t ) A( ) F m ( ) b / m d 0 d 0 d 0 k m : natural frequency Slide 13-1

13 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. Slide 13-13

14 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) Acos 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. Slide 13-14