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Oscillations I-Classical Physics-Handouts, Lecture notes of Classical Physics

University of AllahabadClassical Physics

This course includes alternating current, collisions, electric potential energy, electromagnetic induction and waves, momentum, electrostatics, gravity, kinematic, light, oscillation and wave motion. Physics of fluids, sun, materials, sound, thermal, atom are also included. This lecture includes: Oscillation, Motion, Period, Cycle, Frequency, Amplitude, Displacement, Equilibrium, Central, System, Differential, Equation

Typology: Lecture notes

2011/2012

Uploaded on 08/12/2012

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PHYSICS –PHY101 VU
© Copyright Virtual University of Pakistan
41
Summary of Lecture 15 – OSCILLATIONS: I
1. An oscillation is any self-repeating motion. This motion is characterized by:
a) The period , which is the time for completing one full cycle.
b) The frequency 1/ , which is the number of
T
fT=cycles per second. (Another
frequently used symbol is ).
c) The amplitude A, which is the maximum displacement from equilibrium (or the
size of the oscillation).
2. Why does a syste
ν
m oscillate? It does so because a force is always directed towards
a central equilibrium position. In other words, the force always acts to return the
object to its equilibrium position. So th
restore
e object will oscillate around the equilibrium
position. The restoring force depends on the displacement , where
is the distance away from the equilibrium point, the negative si
Fkx x=− Δ Δ
gn shows that the
force acts towards the equilibrium point, and is a constant that gives the strength of
the restoring force.
k
3. Let us understand these matters in the context of a spring tied
to a mass that can move freely over a frictionless surface. The
force ( ) (or - because the extension will be
ca
Fx kx k x x=− Δ Δ
2
lled for short). In the first diagram is positive, is
negative in the second, and zero in the middle one. The energy
1
stored in the spring, ( ) , is positive in the first and
2
th
xxx
Ux kx=
2
ird diagrams and zero in the middle one. Now we will use Newton's second law to
derive a differential euation that describes the motion of the mass: From ( )
and it follows that
Fx kx
d
ma F m
=−
=
2
22
22
, or 0 where . This is
the equation of motion of a simple harmonic oscillator (SHO) and is seen widely in
many different branches of physics. Although we have derived it
xdx k
kx x
dt dt m
ωω
=− + = ≡
for the case of a
mass and spring, it occurs again and again. The only difference is that , which is
called the oscillator frequency, is defined differently depending on the situation.
4. In
ω
order to solve the SHO equation, we shall first learn how to differentiate some
d
elementary trignometric functions. So let us first learn how to calculate cos
dt
starting from the basic defini
t
ω
tion of a derivative:
Compressed
Stretched
Relaxed
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Summary of Lecture 15 – OSCILLATIONS: I

  1. An oscillation is any self-repeating motion. This motion is characterized by: a) The period , which is the time for completing one full cycle. b) The frequency 1/ , which is the number of

T

f = T cycles per second. (Another frequently used symbol is ). c) The amplitude A, which is the maximum displacement from equilibrium (or the size of the oscillation).

  1. Why does a syste

m oscillate? It does so because a force is always directed towards a central equilibrium position. In other words, the force always acts to return the object to its equilibrium position. So th restore

e object will oscillate around the equilibrium position. The restoring force depends on the displacement , where is the distance away from the equilibrium point, the negative si

F = − k Δ x Δ x gn shows that the force acts towards the equilibrium point, and is a constant that gives the strength of the restoring force.

k

  1. Let us understand these matters in the context of a spring tied to a mass that can move freely over a frictionless surface. The force ( ) (or - because the extension will be ca

F x = − kx k Δ x Δ x

2

lled for short). In the first diagram is positive, is negative in the second, and zero in the middle one. The energy stored in the spring, ( ) 1 , is positive in the first and 2 th

x x x

U x = kx

2

ird diagrams and zero in the middle one. Now we will use Newton's second law to derive a differential euation that describes the motion of the mass: From ( ) and it follows that

F x kx ma F md

(^22 ) 2 , or^2 0 where^. This is the equation of motion of a simple harmonic oscillator (SHO) and is seen widely in many different branches of physics. Although we have derived it

x (^) kx d x (^) x k dt dt m

for the case of a mass and spring, it occurs again and again. The only difference is that , which is called the oscillator frequency, is defined differently depending on the situation.

  1. In

order to solve the SHO equation, we shall first learn how to differentiate some elementary trignometric functions. So let us first learn how to calculate dcos dt starting from the basic defini

ω t

tion of a derivative:

Compressed

Stretched

Relaxed

( ) ( ) ( ) ( ) ( )

cos and cos. Take the difference: ( ) cos cos sin sin( / 2)

Start : x t t x t t t t x t t x t t t t t t t

sin as becomes very small. cos sin. (Here you should know that sin for small , easily proved by drawing trian

t t t d (^) t t dt

≈ gles.) You should also derive and remember a second important result: sin cos. (Here you should know that cos 1 for small .)

  1. What

d (^) t t dt

( )

( )

(^22) 2 (^22) 2

happens if you differentiate twice? sin cos sin

cos sin cos. So twice differentiating either sin

d (^) t d t t dt dt d (^) t d t t dt dt t

(^22) 2

or cos gives the same function back!

  1. Having done all the work above, now you can easily see that any function of the form ( ) cos sin satisfies. But what do , a, b represe

t

x t a t b t d x x dt

( )

nt?

a) The significance of becomes clear if your replace by 2 in either sin

or cos. You can see that cos 2 cos 2 cos. That is, the function merely repeats it

t t t

t t t t

ω^ π ω

self after a time 2 /. So 2 / is really the period of the

motion , 2 2. The frequency of the oscillator is the number of

complete vibrations per unit time: 1 1 so 2 2

T T m k k T m

[ ] 1

Sometimes is also called the angular frequency. Note that dim , from it is clear that the unit of is radian/second. b) To understand what and mean let us note that fro

k T m T

a b

−

m ( ) cos sin it follows that (0) and that d ( ) sin cos (at 0). Thus, dt is the initial position, and is the initial velocity divided by.

x t a t b t x a x t a t b t b t a b