PHY 1023: Waves and Op+cs (Pete Vukusic)
PHY 1023 Waves and Optics
05/11/2014
Prof. Pete Vukusic School of Physics, University of Exeter 1
1
Simple harmonic motion (undamped) .
Mass m
Increasing x
Equilibrium position (x=0)
spring constant k
(Pete Vukusic)
PHY 1023 Waves and Optics
In 1D, a mass on a spring is displaced from an equilibrium position.
Hooke’s Law
F = −kx
Displacement
Force used to
stretch spring
Spring constant
Restoring force exerted
by the spring
2
Torsional
Ball in
parabolic
dish
Simple
Torsional
SHM analysis (undamped)
We set up the equation of motion by balancing forces;
Fnet = ma
Any net force on an object of mass m will produce an
acceleration a
F = −kx
The restoring force F of the system, causes the object of
mass m to accelerate at a.
Balancing forces;
(Pete Vukusic)
PHY 1023 Waves and Optics
i.e. Σ (all forces) = F1 + F2 + F3 = Fnet = mass × accl’n
rearranging;
ma = −kx
!x! = −
k
x
m
•  Accl’n of an object undergoing SHM is
proportional to its displacement
3
•  Accl’n is in the opposite direction to
the displacement
(-ve sign).
And re-writing;
m!x! = −kx
This is the equation of motion
for undamped SHM
Definition of
SHM
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05/11/2014
Solution to the equation of SHM.
is a periodic equation that predicts the variation of
displacement x with time t
(ω is the angular frequency of the oscillation)
Is it a solution to the equation of motion?
x! = − Aω sin ωt + φ
Differentiate once to get velocity:
Differentiate a second time to get
acceleration:
(Pete Vukusic)
PHY 1023 Waves and Optics
x = A cos(ωt + φ )
!x! = − Aω 2 cos ωt + φ
Substitute into the SHM equation:
Recall
!x! = −ω 2 x
Therefore; x =A cos (ωt + φ) is a solution, but only when:
!x! = −
ω=
k
x
m
k
m
4
Tie these equations together with ω = 2πf :
k
m
ω=
f =
1
2π
k
m
This is the resonant frequency of the spring / mass system;
(Pete Vukusic)
PHY 1023 Waves and Optics
Oscillation frequency is a function of the m and k.
i.e. when it is set oscillating, it will automatically oscillate
with this frequency f.
Note;
•
It only depends on k and m.
•
It does not depend on the amplitude of the oscillation.
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Question.
Answer.
First find the spring constant using Hooke’s Law.
(F
(Pete Vukusic)
PHY 1023 Waves and Optics
A spring with mass 400g, is stretched by 75cm. If it is
then set oscillating, at what frequency does it resonate?
(
= kx ) ⇒ k = F = 4 N = 5.3N / m
x
0.75m
)
Then use the frequency equation for SHM,
f =
1
2π
5.3
= 0.58 Hz
0.4
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05/11/2014
SHM Phase relations: displacement, velocity, and acceleration.
Max. value = A
x! = Aω cos(ωt + φ )
Max. value = Aω
(Velocity amplitude)
(Pete Vukusic)
PHY 1023 Waves and Optics
x = A sin(ωt + φ )
!x! = − Aω 2 sin (ωt + φ )
Max. value = Aω2
(Accl’n amplitude)
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Note on SHM phase relations.
Velocity is out
of phase with displacement by 90 degrees
(π/2 radians)
Velocity leads the displacement by π/2 radians
When velocity is maximum, displacement is zero (and vice
versa)
2.
(Pete Vukusic)
PHY 1023 Waves and Optics
1.
Acceleration is out
of phase with displacement by 180
degrees (π radians).
Accl’n leads the velocity by π/2 radians
When displacement is maximum (positive), the acceleration is
maximum (negative)
8
Energy of Simple Harmonic Motion
Total energy = KE + PE
Since the velocity at certain points becomes zero,
there must be an exchange between KE and PE.
This transfer is only perfect if no energy is lost
during the motion (undamped motion).
For undamped motion, this total energy E is constant.
(Pete Vukusic)
PHY 1023 Waves and Optics
(We continue with the spring /
mass model of SHM)
Kinetic Energy term
for SHM:
E = 12 mv2 = 1 mx! 2
2
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