university physics: waves and electricity

University Physics: Waves and Electricity

Ch15. Simple Harmonic Motion

Lecture 1

Dr.-Ing. Erwin Sitompulhttp://zitompul.wordpress.com

1/2Erwin Sitompul University Physics: Waves and Electricity

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


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.


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.


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.


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



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



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.



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



'( ) cos( )mx t x t( ) cos( )mx t x t

( ) cos(2 )mx t x t

( ) cos( )4mx t x t


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


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


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)


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)?

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