Modelling Pendulums

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Modelling Pendulums

• Pendulum - A body suspended from a fixed support
  so that it swings freely back and forth under the
  influence of gravity. Commonly used to regulate
  various devices, especially clocks.
• From the Latin pendulus meaning hanging.

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Aims

• Model a simple pendulum.
• Investigate whether the model is realistic, and
  make necessary improvements.
• Find a solution to the equation of motion.
• Look at modelling other systems related to the
  simple pendulum.
• Use techniques from senior school, A level, and
  degree.

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PERIOD - time taken to complete one full
swing. What affects the period?

• length
• not mass
• gravity

Equation of motion must include g and l, but not
m.
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Galileos findings

• Pendulums nearly return to their release heights.
• All pendulums eventually come to rest, with the
  lighter ones coming to rest faster.
• The period is independent of the bob weight.
• The period is independent of the amplitude.
• The square of the period varies directly with the
  length.

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Assumptions

• Weight of string can be ignored
• String is inextensible
• Pendulum bob can be modelled as point mass
• No air resistance
• No friction
• Constant gravity, g9.80665

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Identify the forces and acceleration, parallel
and perpendicular to the string.
Apply Newtons second law, Fma, perpendicular to
string
d2? -g sin? dt2 l
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For small oscillations d2? ? -g ? dt2 l
Simple harmonic motion d2x -n2x dt2
Period of SHM 2?
n Period of pendulum 2? ?l
?g
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Other pendulum systems
Use Lagrangian mechanics -only need to find
kinetic and potential energies of the system to
find the equation of motion. NORMAL MODES-
particular solutions which oscillate with a
single frequency.

• Double pendulum
• Coupled oscillator

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Back to simple pendulum... Equation of
motion d2? -g sin? dt2 l Analyse as a
dynamical system and draw phase portrait.
Pendulum will never come to rest!
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Implies our assumptions are wrong. Introduce
damping factor d2? -g sin? - ?d? dt2 l
dt
Better model, but still not perfect.
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Finding a solution
Find ?(t) that satisfies our equation of
motion ? 2 sin-1 (sin½? sn(??t, sin½?)) where
sn is a Jacobi elliptic function. Also use
elliptic functions to find an expression for the
period (not just for small oscillations).
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Foucault Pendulum
1851 - Foucault uses a pendulum to show that the
earth rotates. 1954 - Maurice Allais reports that
a Foucault pendulum exhibits peculiar movement
during a solar eclipse.
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• Found equation of motion for the simple pendulum.
• Newtons second law.
• Dynamical Systems.
• Model wasnt realistic - add damping term.
• Solution to equation of motion, and expression
  for the period.
• Elliptic Functions.
• Looked at double pendulum and coupled oscillator.
• Lagrangian Mechanics.