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ALL INVOLVE
SIMPLE HARMONIC MOTION
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A body will undergo SIMPLE HARMONIC MOTION when the
force that tries to restore the object to its REST POSITION is
PROPORTIONAL TO the DISPLACEMENT of the object.
A pendulum and a mass on a spring both undergo this type of
motion which can be described by a SINE WAVE or a COSINE
WAVE depending upon the start position.
Displacement x
+A
Time t
-A
x  A cos 2ft
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SHM is a particle motion with an acceleration (a)
that is directly proportional to the particle’s
displacement (x) from a fixed point (rest point), and
this acceleration always points towards the fixed
point.
Rest point
a
a
x
x
F  kx is Hooke' s Law
and as
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F  ma
then
a x
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Displacement x
T
+A
time
-A
Amplitude ( A ): The maximum distance that an object moves
from its rest position. x = A and x = - A .
Period ( T ): The time that it takes to execute one complete
cycle of its motion.
Unit: seconds (s)
Frequency ( f ): The number of oscillations the object
1
completes per unit time.
f

-1
©JParkinson Units:Hertz (Hz) = s .
T
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Displacement x
x  A cos 2ft
t
Velocity v
Velocity = slope of displacementtime graph
Maximum velocity vo in the middle
t
of the motion
ZERO velocity at the end of the
motion
Acceleration a
t
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Acceleration = slope of velocity time graph
Maximum acceleration at the
end of the motion – where the
restoring force is greatest!
ZERO acceleration in the middle
of the motion!
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THE PENDULUM
The period, T, is the time for one complete cycle.
l
T  2
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l
g
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MASS ON A SPRING
x
M
A
F = Mg = kx
Stretch &
Release
k = the spring constant in N m-1
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m
T  2
k
T  2
x
g
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The link below enables you to
look at the factors that influence
the period of a pendulum and
the period of a mass on a spring
http://www.explorelearning.com/index.cfm?method
=cResource.dspView&ResourceID=44
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DAMPING
DISPLACEMENT
INITIAL AMPLITUDE
time
THE AMPLITUDE DECAYS EXPONENTIALLY WITH TIME
If damping is negligible, the total energy will be constant
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ENERGY IN SHM
SPRING
PENDULUM
potential
M
M
potential
Eg
Kinetic
EK
Potential
Eg
M
kinetic
potential
ETOTAL = EP + EK
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Energy in SHM
= kinetic
= potential
= TOTAL ENERGY, E
energy
Energy Change with POSITION
-A
Energy Change with TIME
E
+A
0
energy
x
E
N.B. Both the kinetic and the
potential energies reach a
maximum TWICE in one cycle.
time
T/2
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T
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Energy in SHM
Maximum Kinetic Energy, EK (max) = ½ m (vo)2
where vo is the velocity at the rest position (origin) = max velocity.
TOTAL ENERGY = Eel + Ek
For a spring, energy stored = ½ Fx = ½ kx2, [as F=kx]
m
F m
x=A
x=0
MAXIMUM POTENTIAL ENERGY = TOTAL ENERGY = ½ kA2
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Formula Summary and Consolidation:
ET = (½)mv2 + (½)kx2
ET=Eel+Ek
ET=(½)mvo2
ET=(½)kA2
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