■ Often a physical system will repeat the same motion over and over. We call this periodic motion, or an oscillation.
■ The time it takes for the motion to complete one cycle is called the period, T.
■ The number of times the motion repeats in unit time is called the frequency, f.
1 . f T
■ Periodic motion refers to any type of repeated motion.
■ Simple harmonic motion refers to a specific type of periodic motion where the restoring force is proportional to the displacement.
Chap 3. Oscillations
A function can be approximated by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error introduced by the use of such an approximation..
) ...
(
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1 )
(
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(
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becomes series
Taylor the
0 a case special For the
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(
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(
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(
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) ( 1
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if is that , at x ) (expansion tion
representa series
power a
has f(x) If : heorem
3 0 3
3 2
0 2
2
0 0
3 3 0
3 2
2 0 2 1
0 0
0
0 0
0
0
0
0 0 0
0
0
dx x x f x d
dx x f x d
dx x f df
n! x ) ( f(x) f
x dx x
x f x d
dx x x f x d
dx x x x df
f
x dx x
x f d f(x) n!
dx x f d R, then c n!
x-x and x
x c f(x)
T
x x x
n
n (n)
x x x
x x x
n
n x x
n n
x x n n n
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Examples of Taylor series
1
1 5
4 3
2
0
1 2 9
7 5
3
0
2 8
6 4
2
0 4
3 2
0 4
3 2
) 1 ( ...
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! 4
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! ) 2
1 ln(
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1 2
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9
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cos
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! 1 2
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n
n n
n
n n
n
n n
n
n x
n
n
n x x
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: series s
Taylor'
following the
is it Then m.
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far not to nts
desplaceme for
nt x displaceme the
of function polynomial
a by ely approximat described
be can F force restoring the
general, In
0
0 3 3 3 0
2 2 2 0
0
x F F
dx F x d
dx F x d
dx x dF F
x F
by given
2. Simple harmonic motion
■ A very special form of periodic motion where the restoring force is proportional to the displacement: F
■ One way to construct such an oscillator is to use a spring obeying Hooke’s law: F=-kx.
x .
. 0
2 0 2 0
m where k
x x
-kx x
m F
particle).
a of position initial
(the angle phase The
:
particle.
a of mass The : m
constant.
spring The
: k
).
k/m ( frequency angular
The :
A).
x (-A amplitude The
: A
sin cos
or ) sin(
) (
0
0 0
0
A t x(t) A ω t B ω t
t x
!
frequency ,
2
1 1
2
0 0
0 0
0 0
s isochronou
period m
k
frequency angular
m k
. 0
'
2 0 2 0
m where k
x x
-kx x
m F
low s Hooke
particle).
a of position initial
(the angle phase The
:
particle.
a of mass The : m
constant.
spring The
: k
).
k/m ( frequency angular
The :
A).
x (-A amplitude The
: A
) cos(
or ) sin(
) (
0
0 0
A t A t
t x
2 2 2
2 0 2 2
2 0
2 2
0 2 2
0 2 2
0 2
0
0 0
0 0
0
) (
1 sin ) (
1 cos
) (
2 sin 1
2 1 by
calculated is
energy potential
The
m ) k
( 2 cos
1
) (
2 cos 1 2
T 1
by given is
energy kinetic
Then the
) sin(
) (
) cos(
) (
sin cos
or x(t) )
sin(
) (
t kA
t kA
U T E
t kA
U
kx kxdx
dW kxdx & U
-Fdx dW
t kA
t mA
mv
t A
t x
t A
t x
t B
t A
t A
t x
cm k v
x m
kx mv
a
cm erg
mv c
erg cm
dyne kA
E b
gram Hz cm
cm gram gram
cm dyne m
a k
1 1
10 1 1 10 10 2 1 2
)1 (
energy.
potential maximum
the (b) and nt displaceme maximum
the (a) calculate
position.
m equilibriu its
at cm/s 1 of velocity initial
an it giving by motion into
set is oscillator The
3) - 3 Prob
sec / 100 30
10 5 . 4 v 2
10 5 . 2 4
)1 (
10 5 . 4 3
2 10 1 2
) 1 (
sec 63 . 0 10 sec
2 1
6 . 1 2 sec
sec 10 10 2 10 10
/ 10
2 1 2
) 1 (
speed.
maximum the
(c) and energy, total
the (b) , period the
and frequency natural
the (a) Calculate rest.
from released and
cm 3 displaced is
mass The dyne/cm.
10
is constant force
whose spring
a to attached mass
g - 100 a of consists oscillator
harmonic smiple
A 1) - 3 Prob
4 2 0
0
2 0 0
2 0
4 max
4 max
2
4 2
4 2
0 0
2 1 4
2 4 0
0
0 4
l g
l g
g l mg
x m
0
2 0 2
0 0 ,
0
0 sin
. 0
'
2 0 2 0