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Chapter 14

# PHYA11H3 Chapter Notes - Chapter 14: Moving Block, Propagation Constant, Unit Circle

Department
Physics and Astrophysics
Course Code
PHYA11H3
Professor
all
Chapter
14

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of 9
Ch 14 Oscillations
Oscillatory motion – a repetitive motion back and forth about an equilibrium position
oSwinging motions and vibrations of all kinds
oAll oscillatory motion is periodic
Oscillators – objects or systems that undergo oscillatory motion
o2 characteristics:
Oscillation takes place about an equilibrium position
Motion is periodic
Oscillations – An object oscillates if the position as a function of time is a periodic function, i.e. it repeats
Ex. Of d-t graphs for oscillating systems [see diagram below]
oOscillations takes place around an equilibrium position – the positive x-axis
oThe motion is periodic. One cycle takes time T
oThe 3rd oscillation is sinusoidal
Period (T) – the time to complete one complete cycle
oThe time in seconds required to complete one full cycle or oscillation
Frequency (f) – number of cycles per second
oNumber of cycles or oscillations completed in one second
oUnits: Hertz (Hz)
o1Hz = 1 cycle per second = 1s-1
oMeasured in s-1 or Hz (Hertz) and read as “cycles per second”.
Frequency Period
103 Hz = 1 kHz 1ms
106 Hz = 1 MHz 1μs
109 Hz = 1 GHz 1ns
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Simple harmonic motion SHM
Simple harmonic motion – sinusoidal oscillation that is the most basic type of all oscillatory motions
Harmonic motion is described by a trigonometric function ... cos or sin
Amplitude (A) – maximum displacement from the equilibrium position
oThe objects position oscillates from x= – A and x=A
oThe distance from the axis to the maximum, not the distance from the minimum to the maximum
The instantaneous velocity is zero at the points where x= ± A
oThese are the turning points in the motion
The maximum speed vmax is reached as the object passes through the equilibrium position at x=0 m
oThe velocity is positive as the object moves to the right but negative as the object moves toward the left
oA mass oscillating on a spring is the prototype of simple harmonic motion
Can we “solve” for the motion of the mass?
What is the position x of the mass at any given time t? What is x(t)?
Kinematics of simple harmonic motion
***The equations mentioned in this section are for , implying that ***
***The particle starts from rest at the point of maximum displacement***
The positions versus time graph is clearly a cosine function with period T
An object’s position can be written as
Where x(t) indicates that the position x is a function of time t
The cosine function must be used in radians mode
Angular frequency
oIf the oscillation is harmonic, we can “convert” the time to angles (radians)
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oThis defines the angular frequency ω
x=R cos ωt y=R sin ωt
The velocity versus time graph is clearly an “upside-down” sine function with the same period T
An object’s velocity can be written as
is the maximum speed and thus a positive number
The minus sign is needed to turn the sine function upside down
Velocity is the time derivative of position
oThe object moves faster if you stretch the spring farther and give the oscillation a larger amplitude
14.2 simple harmonic motion and circular motion
The above equations are for an oscillation in which the object just happened to be at xo=A at t=0
For different initial conditions:
oThe very close connection between simple harmonic motion and circular motion can be used
oUniform circular motion projected onto one dimension is simple harmonic motion
oThe x-component of a particle in uniform circular motion is simple harmonic motion
As φ increases, the particle’s x-component is
When used to describe oscillatory motion, ω is called the angular frequency rather than the angular velocity
The angular frequency of an oscillator has the same numerical value, in rad/s, as the angular velocity of the
corresponding particle in circular motion
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