Medradsc 1c03

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Medradsc 1c03
Dr Fiona McNeill Waves and Vibrations
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Simple Harmonic Motion
x
Fapplied
Frestoring
Frestoring -kx
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Simple Harmonic Motion
x
Fapplied
Frestoring
Frestoring kx
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Forces in System
m
x
Fapplied
Frestoring
m is displaced from equilibrium at constant
velocity Since the acceleration is zero, the net
force must be zero (Newtons Second Law
Fma) Fapplied must equal Frestoring but in the
opposite direction
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Elastic Potential Energy
As the mass moves, work is done by the applied
force If the force were constant, then the work W
Fapplied.?x However F is proportional to x which
is variable The work done in stretching the
spring from 0 to x is W ? Fapplieddx? kx.dx½kx2
x
x
0
0
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Elastic Potential Energy
m
Fapplied
W ½kx2 The work is positive for both elongation
and compression This can be considered elastic
potential energy, U, stored in the spring, U½kx2
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Simple Harmonic Motion
If the mass is pulled aside from equilibrium and
released, how will its position vary with time?
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Energy from Springs
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Energy from Springs
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Simple Harmonic Motion
xA
x-A
In a frictionless system, m will oscillate
indefinitely between the positions xA and
x-A The position x is a sinusoidal function of
time, t, XA sin(?t)
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Simple Harmonic Motion
xA
x-A
XA sin(?t) ? is called the angular frequency of
the oscillation Its SI unit is radians per second
(rad/s) 2p radians 360 , p radians 180