Oscillations
Oscillations
Oscillations
metronome pendulum
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Condition for SHM
The equation defining SHM is:
a = –ω2x pendulum
ω = 2πf
equilibrium position
restoring force
So, for a body to perform simple harmonic motion, the
restoring force must be proportional to and in the opposite
direction to its displacement from the equilibrium position.
The maximum and minimum values of sin and cos are 1 and
–1, and these are achieved for some values of t.
v2 = (ωA)2 – (ωx)2
v = ±ω √ (A2 − x2)
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Velocity calculations
applied force = kx
–kx
a = F/m, therefore: a = m
mg
k and m are constant, so this is SHM.
then ω = 2πf =
√ k .
m
f = 1/T, therefore 2π =
T √ k
m T
x
√
m
T = 2π
k
The time period for a vertical mass on a mg
spring is only dependent on k and m.
a ≈ –gθ
mg θ
Therefore, at small angles of displacement,
a pendulum is a harmonic oscillator.
√
g
so ω = 2πf =
l
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Time period of a pendulum 2
√
g
ω = 2πf =
l
θ
f = 1/T, therefore l
T
2π
√
g
=
T l
x
√ l mg
T = 2π g
amplitude / m
the amplitude with
which a simple
harmonic oscillator
will vibrate, against
the driving frequency
applied to it.
f0
driving frequency / Hz
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Examples of resonance