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Transcript
Chapter 14 Outline
Periodic Motion
• Oscillations
• Amplitude, period, frequency
• Simple harmonic motion
• Displacement, velocity, and acceleration
• Energy in simple harmonic motion
• Pendulums
• Simple
• Physical
• Damped oscillations
• Resonance
Periodic Motion
• Many types of motion repeat again and again.
• Plucked string on a guitar, a child in a swing, sound waves in a
flute…
• This is called periodic motion, or oscillation.
• Stable equilibrium point
• Displacement from equilibrium leads to a force (or torque) to
return it to equilibrium.
• Kinetic energy leads to overshoot, causing repeat
Periodic Motion Notation
• Amplitude – The maximum
magnitude of displacement from
equilibrium
• Period, 𝑇 – The time for one cycle (s)
• Frequency, 𝑓 – Number of cycle per
unit time (Hz = 1/s)
• Angular frequency, 𝜔 – Equal to 2𝜋𝑓
(rad/s)
• From these definitions,
𝑓=
1
,
𝑇
and 𝑇 =
1
𝑓
Simple Harmonic Motion
• If the restoring force is directly proportional to the
displacement, the resulting oscillation is simple harmonic
motion.
• One example is a mass on spring that obeys Hooke’s law.
𝐹 = −𝑘𝑥
Mass on a Spring
• Using Newton’s second law, in one dimension,
𝐹 = −𝑘𝑥 = 𝑚𝑎
𝑘
𝑎=− 𝑥
𝑚
• Since acceleration is the second time derivative of
position,
𝑑2𝑥
𝑘
=− 𝑥
2
𝑑𝑡
𝑚
Mass on a Spring
• Sinusoidal functions satisfy this differential equation.
𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
• Plugging this into the equation, we can solve for 𝜔,
𝑑𝑥
= −𝜔𝐴 sin(𝜔𝑡 + 𝜙)
𝑑𝑡
𝑑2𝑥
2 𝐴 cos 𝜔𝑡 + 𝜙 = −𝜔 2 𝑥
=
−𝜔
𝑑𝑡 2
• Comparing to
𝑑2𝑥
𝑑𝑡 2
=−
𝑘
𝑥,
𝑚
𝜔=
𝑘
𝑚
Simple Harmonic Motion of a
Mass on a Spring
• Angular frequency
𝜔=
𝑘
𝑚
• Frequency
1 𝑘
𝑓=
2𝜋 𝑚
• Period
𝑚
𝑇 = 2𝜋
𝑘
• Amplitude does not enter the equations as long as the
spring obeys Hooke’s law. (Generally for small
amplitudes.)
Displacement, Velocity and Acceleration
• As we saw earlier, sinusoidal functions satisfy the
differential equation for simple harmonic motion.
𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
• 𝐴 is the amplitude, and 𝜙 is the phase angle (starting position)
• We find the velocity at any time by taking the time
derivative,
𝑑𝑥
𝑣=
= −𝜔𝐴 sin(𝜔𝑡 + 𝜙)
𝑑𝑡
• Likewise, for the acceleration,
𝑑2𝑥
𝑎 = 2 = −𝜔2 𝐴 cos 𝜔𝑡 + 𝜙
𝑑𝑡
SHM Example
Energy in Periodic Motion
• If there are no dissipative forces (friction), the total energy
is constant.
1
1 2 1 2
2
𝐸 = 𝑚𝑣 + 𝑘𝑥 = 𝑘𝐴 = constant
2
2
2
Energy Plots in Periodic Motion
1
1 2
2
𝐸 = 𝐾 + 𝑈 = 𝑚𝑣 + 𝑘𝑥
2
2
Vertical SHM
• We looked at a mass oscillating horizontally on a
frictionless table.
• How does this change if the mass is hanging vertically?
• The equilibrium point will not be at the same extension of the
spring. (∆𝑙 instead of 0)
𝑘∆𝑙 = 𝑚𝑔
𝐹net = 𝑘 ∆𝑙 − 𝑥 − 𝑚𝑔 = 𝑘∆𝑙 − 𝑚𝑔 − 𝑘𝑥
𝐹net = −𝑘𝑥
• This the same as before, but 𝑥 is measured from the new equilibrium
position.
Molecular Vibrations
• Interactions between neutral atoms can be described by the
Lennard-Jones potential.
𝑅0 12
𝑅0 6
𝑈 = 𝑈0
−2
𝑟
𝑟
• The positive term is due to the Pauli repulsion.
• The negative, is due to long range attractive forces (LDF).
• This does not look like the simple parabolic potential well from a mass
on a spring, but we can still look at the oscillations.
Molecular Vibrations
• The restoring force is:
𝑑𝑈
𝑈0
𝐹𝑟 = −
= 12
𝑑𝑟
𝑅0
𝑅0
𝑟
13
𝑅0
−
𝑟
7
• Using, the binomial theorem,
𝑛 𝑛−1 2
𝑛
1 + 𝑢 = 1 + 𝑛𝑢 +
𝑢 +⋯
2!
• For small oscillations, the restoring force reduces to:
72𝑈0
𝐹𝑟 = −
𝑥
2
𝑅0
• Where 𝑥 = 𝑟 − 𝑅0
• This is now Hooke’s law.
Simple Pendulum
• The simplest pendulum is a
point mass on a massless
string.
• What provides the restoring
force?
• How can we describe the
motion?
• Mass 𝑚, length 𝐿.
Simple Pendulum
• For small amplitudes,
𝜔=
𝑔
𝐿
𝜔
1 𝑔
𝑓=
=
2𝜋 2𝜋 𝐿
1
𝐿
𝑇 = = 2π
𝑓
𝑔
Physical Pendulum
• A real, or physical, pendulum
is an extended object.
• Now, we need to know the
moment of inertia, 𝐼, as well as
the mass 𝑚, and distance from
the pivot point to the center of
mass, 𝑑.
Physical Pendulum
• Again, for small amplitudes,
𝜔=
𝑚𝑔𝑑
𝐼
𝐼
𝑇 = 2π
𝑚𝑔𝑑
Physical Pendulum Example
Damped Oscillations
• We have assumed there was no friction, and therefore the
amplitude never decreases, but in reality, energy will be
lost to friction.
• This decrease in amplitude is damping.
• Consider the simplest case where the frictional damping is
proportional to the velocity of the object.
Σ𝐹 = −𝑘𝑥 − 𝑏𝑣
• This force always opposes the motion.
−𝑘𝑥 − 𝑏𝑣 = 𝑚𝑎
𝑑𝑥
𝑑2𝑥
−𝑘𝑥 − 𝑏
=𝑚 2
𝑑𝑡
𝑑𝑡
Underdamped Oscillations
• Solving this differential
equation for small 𝑏,
𝑥 = 𝐴𝑒 − 𝑏/2𝑚 𝑡 cos(𝜔′ 𝑡 + 𝜙)
• This is very similar to simple
harmonic motion but with an
exponentially decreasing
amplitude.
• Also, the angular frequency
is decreased.
𝜔′ =
𝑘
𝑏2
−
𝑚 4𝑚2
Critical and Over-damped Oscillations
• When the equation for 𝜔′ = 0,
𝑏 = 2 𝑘𝑚
• The system is critically
damped, and it no longer
oscillates, but returns
exponentially to equilibrium.
• If 𝑏 > 2 𝑘𝑚, the system is
overdamped, and it decays to
zero with a double exponential.
Forced Oscillations and Resonance
• If the system is driven at some arbitrary frequency, there
will not be much of a response.
• But, if it is driven near the natural resonance frequency,
the response can be quite large.
Chapter 14 Summary
Periodic Motion
• Oscillations
• Amplitude, 𝐴
• Period, 𝑇 =
1
𝑓
• Frequency, 𝑓, and angular frequency, 𝜔 = 2𝜋𝑓
• Simple harmonic motion – Spring: 𝜔 =
𝑘
𝑚
• 𝑥 = 𝐴 cos(𝜔𝑡 + 𝜙)
• 𝑣=
𝑑𝑥
𝑑𝑡
• 𝑎=
𝑑2 𝑥
𝑑𝑡 2
= −𝜔𝐴 sin(𝜔𝑡 + 𝜙)
= −𝜔2 𝐴 cos 𝜔𝑡 + 𝜙
• Energy in simple harmonic motion
1
1 2 1 2
2
𝐸 = 𝑚𝑣 + 𝑘𝑥 = 𝑘𝐴 = constant
2
2
2
Chapter 14 Summary
Periodic Motion
• Pendulums
• Simple: 𝜔 =
𝑔
𝐿
• Physical: 𝜔 =
𝑚𝑔𝑑
𝐼
• Damped oscillations
• 𝑥 = 𝐴𝑒 − 𝑏/2𝑚 𝑡 cos(𝜔′ 𝑡 + 𝜙)
•
𝜔′
=
𝑘
𝑚
−
𝑏2
4𝑚2
• Resonance – driven near 𝜔′