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Chapter 14 Oscillations

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Chapter 14 Oscillations To understand the physics and mathematics of oscillation. To draw and interpret oscillatory graphs. To learn the concepts of phase and phase ... – PowerPoint PPT presentation

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Title: Chapter 14 Oscillations


1
Chapter 14 Oscillations
  1. To understand the physics and mathematics of
    oscillation.
  2. To draw and interpret oscillatory graphs.
  3. To learn the concepts of phase and phase constant
  4. To understand and use energy conservation in
    oscillatory system
  5. To understand the basic ideas of damping and
    resonance.

2
Stop to think 14.1 page 414 Stop to think
14.2 page 417 Stop to think 14.3 page 419 Stop to
think 14.4 page 423 Stop to think 14.5 page 428
  • Example 14.2 page 413
  • Example 14.4 page 417
  • Example 14.6 page 422
  • Example 14.7 page 424
  • Example 14.9 page 426
  • Example 14.10 page 428

3
Simple Harmonic Motion
  • Object or systems of objects that undergo
    oscillatory motion are called oscillators. All
    these oscillators have two things in common
  • 1. The oscillation takes place about an
    equilibrium position, and
  • 2. The motion is periodic.

Sinusoidal oscillation Is called simple harmonic
motion.
4
Period and frequency
  • Period T time per cycle, units second
  • Frequency f the number of cycles per second.
    units 1/s Hz (Hertz)

5
Graph of simple harmonic motion
  • The amplitude A the maximum displacement from
    equilibrium.
  • Measured A 0.17 cm
  • Measured T 1.60s
  • How to describe the displacement x using A, T,
    and t.

6
Position vs time graph and velocity vs time graph
  • Position-vs time graph Velocity vs
    time graph

7
Angular frequency ?
We define ?2p/T 2pf, is called angular
frequency
V(max) ?A
8
Simple Harmonic Motion and Circular Motion.
  • Uniform circular motion projected onto one
    dimension is simple harmonic motion
  • The figure shows the x-component, when the
    particle does uniform circular motion
  • With
  • So

9
The Phase constant
  • In more general case, particle start phase Fo is
    not zero. then,
  • The harmonic motion function is
  • Fo is called the phase constant or initial phase.
  • is called
    phase.
  • When t 0, initial condition

10
Show phase constant
  • The following show the oscillations by different
    phase constant

Notice Fop/3 and Fo-p/3 have the same starting
x, but different Vo
11
P14.2
  • From the Figure, how we get
  • Amplitude
  • Frequency
  • Phase constant.
  • First, you write general Harmonic
  • Wave function
  • Then you compare this trigonometric
  • Function and the figure, you can get
  • A 10 cm
  • T 2 s, frequency f ½ 0.5 Hz.
  • When t 0 x(0) 5cm 10cos(Fo)
  • cos(Fo)0.5, Fop/3.
  • But at t 0, the slope of curve is negative
  • So V0 is negative, from
  • Sin(Fo) is positive, that makes Fop/3.

12
Energy in simple Harmonic Motion
  • The mechanical energy of an object oscillating on
    a spring is
  • When x A, E ½ kA2 0
  • When x 0 E 0 1/2mV2 max
  • From conservation of energy

13
The Dynamics of Simple Harmonic Motion
  • The spring force is
  • From Newtons second Law
  • The dynamics equation
  • This is second derivative equation, the solution
    is

14
Vertical oscillations
  • The equilibrium position, ?L.
  • The harmonic oscillation equation should be the
    same on a horizontal spring.
  • In right figure
  • K 10N/m, The spring stretch at equilibrium
    is given by ?Lmg / K 19.6 cm
  • That is the amplitude of oscillation
  • A 30cm-19.6cm 10.4 cm
  • The initial condition y0-A AcosFo
  • Fop. So the oscillator function is

15
The Pendulum
  • Lets look another oscillator a pendulum

16
Small-angle Approximation
  • The Dynamical equation is
  • Using
  • We can write
  • If ? is very small sin (?) ? (? in radians)
  • Then
  • Solution is
  • Or

or
17
The Physical pendulum
18
Period vs Meff in log plot
19
V-10p sin(ptp/3)
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