University Physics: Waves and Electricity
Ch15. Simple Harmonic Motion
Lecture 1
Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com
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Textbook and Syllabus
Textbook:“Fundamentals of Physics”, Halliday, Resnick, Walker, John Wiley & Sons, 8th Extended, 2008.
Syllabus: (tentative)Chapter 15: Simple Harmonic MotionChapter 16: Transverse WavesChapter 17: Longitudinal WavesChapter 21: Coulomb’s LawChapter 22: Finding the Electric Field – IChapter 23: Finding the Electric Field – IIChapter 24: Finding the Electric PotentialChapter 26: Ohm’s LawChapter 27: Circuit Theory
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Grade Policy
Grade Policy:Final Grade = 5% Homework + 30% Quizzes +
30% Midterm Exam + 40% Final Exam + Extra Points
Homeworks will be given in fairly regular basis. The average of homework grades contributes 5% of final grade.
Homeworks are to be written on A4 papers, otherwise they will not be graded.
Homeworks must be submitted on time. If you submit late,< 10 min. No penalty10 – 60 min. –40 points> 60 min. –60 points
There will be 3 quizzes. Only the best 2 will be counted. The average of quiz grades contributes 30% of final grade.
Midterm and final exam schedule will be announced in time. Make up of quizzes and exams will be held one week after
the schedule of the respective quizzes and exams.
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Lecture Activities The lectures will be held every Tuesday and Wednesday:
17:30 – 18:30 : Class 17:15 – 18:1518:30 – 19:00 : Break 18:15 – 18:4519:00 – 20:45 : Class 18:45 – 20:30
Lectures will be held in the form of PowerPoint presentations. You are expected to write a note along the lectures to record
your own conclusions or materials which are not covered by the lecture slides.
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Lecture Material New lecture slides will be available on internet every Thursday afternoon. Please check the course homepage regularly. The course homepage is :
http://zitompul.wordpress.com You are responsible to read and understand the lecture slides. If there is any problem, you may ask me. Quizzes, midterm exam, and final exam will be open-book. Be sure to have your own copy of lecture slides. Extra points will be given if you solve a problem in front of the
class. You will earn 1, 2, or 3 points.
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Simple Harmonic Motion
The following figure shows a sequence of “snapshots” of a simple oscillating system.
A particle is moving repeatedly back and forth about the origin of an x axis.
One important property of oscillatory motion is its frequency, or number of oscillations that are completed each second.
The symbol for frequency is f, and its SI unit is the hertz (abbreviated Hz).
1 hertz = 1 Hz = 1 oscillation per second = 1 s–1
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Simple Harmonic Motion
Related to the frequency is the period T of the motion, which is the time for one complete oscillation (or cycle).
1T
f
Any motion that repeats itself at regular intervals is called periodic motion or harmonic motion.
We are interested here only in motion that repeats itself in a particular way, namely in a sinusoidal way.
For such motion, the displacement x of the particle from the origin is given as a function of time by:
( ) cos( )mx t x t
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Simple Harmonic Motion
This motion is called simple harmonic motion (SHM).
Means, the periodic motion is a sinusoidal function of time.
The quantity xm is called the amplitude of the motion. It is a positive constant.
The subscript m stands for maximum, because the amplitude is the magnitude of the maximum displacement of the particle in either direction.
The cosine function varies between ±1; so the displacement x(t) varies between ±xm.
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Simple Harmonic Motion
The constant ω is called the angular frequency of the motion.
The SI unit of angular frequency is the radian per second. To be consistent, the phase constant Φ must be in radians.
22 f
T
2 f radians radians cycles
second cycle second
2 radians 1 cycle 360
radian 1
cycle2
180
radian2
1 cycle
4 90
radian6
1 cycle
12 30
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Simple Harmonic Motion
'( ) cos( )mx t x t( ) cos( )mx t x t
( ) cos(2 )mx t x t
( ) cos( )4mx t x t
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A particle undergoing simple harmonic oscillation of period T is at xm at time t = 0. Is it at –xm, at +xm, at 0, between –xm and 0, or between 0 and +xm when:(a) t = 2T(b) t = 3.5T(c) t = 5.25T (d) t = 2.8T ?
Checkpoint
T1.5T0.5T
At +xm At –xm
At 0 Between 0 and +xm
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Velocity and Acceleration of SHM
By differentiating the equation of displacement x(t), we can find an expression for the velocity of a particle moving with simple harmonic motion:
( )( ) cos( )m
dx t dv t x t
dt dt
( ) sin( )mv t x t Knowing the velocity v(t) for simple
harmonic motion, we can find an expression for the acceleration of the oscillating particle by differentiating once more:
( )( ) sin( )m
dv t da t x t
dt dt
2( ) cos( )ma t x t 2( ) ( )a t x t
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Plotting The Motion
Plot the following simple harmonic motions:(a) x1(t) = xmcosωt (b) x2(t) = xmcos(ωt+π) (c) x3(t) = (xm/2)cosωt (d) x4(t) = xmcos2ωt
x1(t)
T0.5T
xm
–xm
0x2(t)
x1(t)
T0.5T
xm
–xm
0x3(t)
x1(t)
T0.5T
xm
–xm
0
x4(t)
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Homework 1: Plotting the Motions
Plot the following simple harmonic motions in three different plots:(a) xa(t) = xmcosωt (b) xb(t) = xmcos(ωt–π/2) (c) xc(t) = xm/2cos(ωt+π/2)(d) xd(t) = 2xmcos(2ωt+π)
xa(t)
T0.5T
xm
–xm
0xb(t)?
xa(t)
T0.5T
xm
–xm
0xc(t)?
xa(t)
T0.5T
xm
–xm
0
xd(t)?