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SIMPLE HARMONIC MOTION
PREPARED BY:NAZIHAH BINTI MOHD NOOR
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2.1 Concept of simple !"monic motion#$HM%
2.1.1 De&ne te te"minolo'ies in $HM
2.1.2 $HM () *sin' ! s*it!(le +i!'"!m
2.1., C!lc*l!te time pe"io+- !mplit*+e-f"e*enc)- /elocit)- m!0im*m /elocit)-!ccele"!tion !n+ m!0im*m
!ccele"!tion of te $HM
2.2 ine!" motion of !n el!stic motion
2.2.1 El!stic s)stem#sp"in' !n+ m!ss%
CONTEN
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TERMINOLOGIES OF SHM
Motion in ic ! (o+) mo/es (!c3
!n+ fo"t o/e" ! &0e+ p!t- "et*"nin'to e!c position !n+ /elocit) !fte" !+e&nite inte"/!l of time.
E*il("i*mPosition
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4ORM5A
SHM
)(
cos
max
2
max
max
22
2
massmma Inertia
r A
Aa
AV
x Av
A x
xa
→=
=
=
=
−=
=
=
ω
ω
ω
θ
ω
ω
π
π
ω
π ω
2
1
2
2
=
=
=
=
T
f
T
f
f
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45NDAMENTADE4INITION$
Displacement that measured from theequilibrium point
x
Time T
Velocity or speedv
Acceleration a
Mass m
Force F
anular !elocity ω
Maximum Velocity"#peed V max
Maximum acceleration amax
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EXERCISE 1
Points moving with simple harmonic motion
have acceleration 9m/s2
and velocity 0.92m/swhen it was in 65mm from the centre position.Find
i. Amplitudeii. Periodic time the movementSOLUTIONS
i. Amplitude
ii. Pe"io+ic time te mo/eme
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EXERCISE 2 (i)
A particle moving with simple harmonic motion has a
periodic time of 0.s and it was !ac" and forth!etween two points is #.22m. $etermine
i. Fre%uency and amplitude of the oscillation
ii. &elocity and acceleration of the particle when it is00mm from the center of oscillationiii.'he ma(imum velocity and acceleration of the
movementSOLUTIONS
i. Frequency
Amplitude
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ii. &elocity and acceleration of the particle when
it is 00mm from the center of oscillation
EXERCISE 2 (ii)
EXERCISE 2 (iii)
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iii. 'he ma(imum velocity and acceleration of
the movement
EXERCISE 2 (iii)
EXERCISE 3 (i)
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A !ody of mass #.5"g moving with simple harmonic
motion is towards one end of the swing. At the time itwas in A) *60mm from the center of oscillation)velocity and acceleration is 9m/s and ##0m/s2 respectively.
i. Fre%uency and amplitude of the oscillationii. +a(imum acceleration and the inertia of the !ody
when it to the edge of swing
EXERCISE 3 (i)
SOLUTIONS
i. 4"e*enc)
=
Amplit*+e
EXERCISE 3 (ii)
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ii. +a(imum acceleration and the inertia of the
!ody when it to the edge of swing
EXERCISE 3 (ii)
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'he piston of a steam engine moves with simple
harmonic motion. 'he cran" rotates at #20r.p.m with astro"e of 2 metres. Find the velocity and accelerationof the piston when it is at a distance of 0.*5metre fromthe centre.
SOLUTIONS
i. Velocity of the piston
ii. Acceleration of the piston
SHM !A"#A
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SHM !A"#A
M!0. Displ!cement 6 A
M!0. 7elocit) 6 Vmax = ωA
M!0. Accele"!tion 6 amax =
EXERCISE 4
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$ Figures ,a- and ,!- are the displacementtime graphand accelerationtime graph respectively of a !ody in
simple harmonic motion. hat is the fre%uency of themotion
EXERCISE 4
SOLUTIONS
Frequency
EXERCISE 5
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'he following graphs show the variation of
displacement) x and velocity) v with time) t for a !odyin simple harmonic motion. hat is the value of '
EXERCISE 5
SOLUTIONS
EXERCISE 6
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A particle moves in simple harmonic motion along a
straight line a!out point x=0.40cm and the velocity is1ero. 'he fre%uency of the motion is 2.51. 3alculatethe4i. Periodii. Angular velocityiii. Amplitudeiv. $isplacement at time t v. +a(imum velocityvi. +a(imum acceleration
EXERCISE 6
SOLUTIONS
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i. Pe"io+
i. An'*l!" /elocit)
iii. Amplit*+e
i/. Displ!cement !t time t
/. M!0im*m /elocit)
/i. M!0im*m !ccele"!tion
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A sprin resists bein stretched or
compressed.
INEAR MOTION O4 AN
EA$TIC MOTION
comp"esse+
st"etce
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Hoo3e8s !
When a spring is stretched, there is arestorin$ force that is proportional to thedisplacement.
F % &'(
F
x
m
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HOOKE'S LAW
The restorin force of an ideal sprin is i!en by%
&here k is the sprin constant and x is the
displacement of the sprin from its
unstrained lenth' The minus sin indicates
that the restorin force al&ays points in a
direction opposite to the displacement of
the sprin'
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Te fo"ce +esc"i(e+ () Hoo3e8s! is te net fo"ce in Neton8s$econ+ !
k stiffness of the sprin ("m) sprin constant ("m)
xm
k a
makx
F F Newton Hooke
−==−
=
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xa 2ω =
xmk a
−=
m
k
m
k
=
=
ω
ω 2
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SHM Mass & SpringSyst!
Si!p" Pn#$"$!
ω
π
π
ω
π ω
2
1
2
2
=
=
==
T
f T
f
f
k
mT
f T
m
k
f
f
m
k
f m
k
π
π
π
π ω ω
2
1
2
2
2*
=
=
=
=
==
g
l T
f T
l
g
f
f
l
g
f l g
π
π
π
π ω ω
2
1
2
2
2*
=
=
=
=
==
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e
x
estatic deflectiongra!ity
kemg
keT mg T
=
==11 *
xm
k a
makx
makxkeke
maT mg
ma F
xek T
−=
=−=−−
=−
=
+=
+
1
+
1 )(
g
eT
k
mT
g
e
k
m
kemg
π
π
2
2
=
=
=
=
%ISPLACEMENT IN SHM
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%ISPLACEMENT IN SHM
m
x = 0 x = +A x = -A
x
$ $isplacement is positive when theposition is to the right of the e%uili!riumposition ,( 0- and negative whenlocated to the left.
$ 'he m!0im*m displacement is calledthe amplitude A.
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ELOCIT IN SHM
m
x = 0 x = +A x = -A
v (+)
$ 7elocit) is positi/e en mo/in' tote "i't !n+ ne'!ti/e en mo/in'
to te left.$ It is 9e"o !t te en+ points !n+ !m!0im*m !t te mi+point in eite"+i"ection # o" ;%.
v (-)
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Acceleration in SHM
m
x = 0 x = +A x = -A
$ Accele"!tion is in te +i"ection of te"esto"in' fo"ce. #a is positi/e en 0 is ne'!ti/e- !n+ ne'!ti/e en 0 ispositi/e.%
$ Accele"!tion is ! m!0im*m !t te en+points !n+ it is 9e"o !t te cente" of
oscill!tion.
+x -a
-x +a
F ma kx= = −
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EXERCISE 7
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A !ody of mass #"g !eing hung with springs straightfrom one end attached to a rigid support. 'he !odyproduced 25mm static deection. 7t was pulled down28mm and then released. Find
i. 'he acceleration !egan to the !ody
ii. Periodic timeiii. 'he spring ma(imum forceiv. &elocity and acceleration the !ody when it is #2mm
from the e%uili!rium positionSOLUTIONS
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Te !ccele"!tion (e'!n to te (o+)
ii. Pe"io+ic time
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Te sp"in' m!0im*m fo"ce
i/. 7elocit) !n+ !ccele"!tion te (o+) en it is 12mm f"om tee*ili("i*m position
THE SIMPLE PEN%ULUM
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,n order to be in #-M% the restorinforce must be proportional to the
neati!e of the displacement' -ere
&e ha!e.
&hich is proportional to sin θ and
not to θ itself'
-o&e!er% if the anle is small%sin θ / θ '
THE SIMPLE PEN%ULUM
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Therefore% for small anles% &e ha!e.
&here
The period and frequency are.
EXERCISE 7
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A mass is suspended from a string 60mm long. 7t isnudged so that it ma"es a small swinging oscillation.
$etermine the fre%uency and periodic time.
SOLUTIONS