Motion of a particle on a spring

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Oscillatory motion (chapter twelve)

• Motion of a particle on a spring
• Simple harmonic motion
• Energy in SHM
• Simple pendulum
• Physical pendulum
• Damped oscillations
• Forced oscillations

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Particle attached to a spring
We can model oscillatory motion as a mass
attached to a spring (linear restoring force)
Causes displaced mass to to be restored to the
equilibrium position. Potential energy ? Kinetic
energy. At equilibrium large KE but force is
now zero. Newtons first law - keeps moving.
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Particle attached to a spring

• We can use Newtons 2nd law to quantitatively
  describe the motion

Acceleration proportional to displacement. Opposit
e direction.
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Simple Harmonic Motion
Defining the ratio k/m??2, the equation of motion
becomes (in one dimension)
This equation has the solution
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SHM
A amplitude of the motion (maximum
displacement) ? (k/m)½ angular frequency of
the motion ? phase where the motion starts A
and ? are set by the initial conditions, ? is
fixed by the mass and spring constant
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SHM

• Period of one full cycle of motion

Maximum velocity and acceleration
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Energy in SHM

• Kinetic energy
• Potential energy
• Total energy of the system
• Total energy is constant!

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Energy in SHM

• Oscillation is repeated conversion of kinetic to
  potential energy and back.
• Using the expression for the total energy, we can
  find the velocity as a function of position

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The simple pendulum
T
Fg
Small angle approximation - sin???
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The simple pendulum
This equation has the same form as that for the
motion of the mass attached to a spring. If we
define
we get the exact same differential equation, and
so the system will undergo the same oscillatory
motion as we saw earlier. Note the frequency
(and period) of the pendulum are independent of
the mass!
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The Physical Pendulum
An object hanging from a point other than its
COM
d
?
COM
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Damped Oscillations
If we add in a velocity dependent resistive
force
The solution to this DE when the resistive force
is weak
This describes an underdamped oscillator
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Damped Oscillations
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Damped Oscillations
The frequency of oscillation is
In other words, some natural frequency plus a
change due to the damping When b2m?, the system
is critically damped (returns to equilibrium) For
bgt2m?, the system is overdamped also returns to
equilibrium (slower rate).
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Forced Oscillations
If we try to drive an oscillator with a
sinusoidally varying force
The steady-state solution is
where ?0(k/m)½ is the natural frequency of the
system. The amplitude has a large increase near
?0 - resonance
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Forced Oscillations