Periodic Motion

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Chapter 13

• Periodic Motion

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Special Case

• Simple Harmonic Motion (SHM)

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Simple Harmonic Motion (SHM)

• Only valid for small oscillation amplitude
• But SHM approximates a wide class of periodic
  motion, from vibrating atoms to vibrating tuning
  forks...

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Starting Model for SHM mass m attached to a
spring
Demonstration
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Simple Harmonic Motion (SHM)

• x displacement of mass m from
  equilibrium
• Choose coordinate x so that x 0 is the
  equilibrium position
• If we displace the mass m, a restoring force F
  acts on m to return it to equilibrium (x0)

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Simple Harmonic Motion (SHM)

• By SHM we mean Hookes Law holdsfor small
  displacement x (from equilibrium), F
  k x ma k x
• negative sign F is a restoring force(a and x
  have opposite directions)

Demonstration spring with force meter
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(No Transcript)
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What is x(t) for SHM?

• Well explore this using two methods
• The reference circlex(t) projection of
  certain circular motion
• A little mathSolve Hookes Law

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The Reference Circle

• P mass on spring x(t)
• Q point on reference circle
• P projection of Q onto the screen

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The Reference Circle

• P mass on spring x(t)
• Q point on reference circle
• A amplitude of x(t)
• (motion of P)
• A radius of reference circle (motion of Q)

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The Reference Circle

• P mass on spring x(t)
• Q point on reference circle
• f oscillation frequency of P 1/T
  (cycles/sec)
• w angular speed of Q 2p /T
  (radians/sec)
• w 2p f

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What is x(t) for SHM?

• P projection of Q onto screen.
• We conclude the motion of P is

See additional notes or Fig. 13-4 for q
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Alternative A Little Math

• Solve Hookes Law
• Find a basic solution

Solve for x(t)
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See notes on x(t), v(t), a(t)

• v dx/dt v 0 at x A v max
  at x 0
• a dv/dta max at x A a 0
  at x 0

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Show expression for f
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• going from 1 to 3,increase one of A, m, k
• (a) change A same T
• (b) larger m larger T
• (c) larger k shorter T

Do demonstrations illustrating (a), (b), (c)
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Summary of SHMfor an oscillator of mass m

• A amplitude of motion, f phase angle
• A, f can be found from the values of x and dx/dt
  at (say) t 0

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Energy in SHM

• As the body oscillates, E is continuously
  transformed from K to U and back again

See notes on vmax
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E K U constant
Do Exercise 13-17
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Summary of SHM

• x displacement from equilibrium (x 0)
• T period of oscillation
• definitions of x and w depend on the SHM

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Different Types of SHM

• horizontal (have been discussing so far)
• vertical (will see acts like horizontal)
• swinging (pendulum)
• twisting (torsion pendulum)
• radial (example atomic vibrations)

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Horizontal SHM
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Horizontal SHM

• Now show a vertical spring acts the same,if we
  define x properly.

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Vertical SHM
Show SHM occurs with x defined as shown
Do Exercise 13-25
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Swinging SHM Simple Pendulum
Derive w for small x
Do Pendulum Demonstrations
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Swinging SHM Physical Pendulum
Derive w for small q
Do Exercises 13-39, 13-38
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Angular SHMTorsion Pendulum (fiber-disk)
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Application Cavendish experiment (measures
gravitational constant G). The fiber twists when
blue masses gravitate toward red masses
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Angular SHMTorsion Pendulum (coil-wheel)
Derive w for small q
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Radial SHMAtomic Vibrations
Show SHM results for small x (where r R0x)
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Announcements

• Homework Sets 1 and 2 (Ch. 10 and 11) returned
  at front
• Homework Set 5 (Ch. 14)available at front, or
  on course webpages
• Recent changes to classweb accesssee HW 5 sheet
  at front, or course webpages

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Damped Simple Harmonic Motion
See transparency on damped block-spring
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SHM Ideal vs. Damped

• Ideal SHM
• We have only treated the restoring force
• Frestoring kx
• More realistic SHM
• We should add some damping force
• Fdamping bv

Demonstration of damped block-spring
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Damping Force

• this is the simplest model
• damping force proportional to velocity
• b damping constant (characterizes
  strength of damping)

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SHM Ideal vs. Damped

• In ideal SHM, oscillator energy is constant
  E K U , dE/dt
  0
• In damped SHM, the oscillators energy decreases
  with time E(t) K
  U , dE/dt lt 0

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Energy Dissipation in Damped SHM

• Rate of energy loss due to damping

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What is x(t) for damped SHM?

• We get a new equation of motion for x(t)
• We wont solve it, just present the solutions.

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Three Classes of Damping, b

• small (underdamping)
• intermediate (critical damping)
• large (overdamping)

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underdamped SHM
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underdamped SHMdamped oscillation, frequency
w
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underdamping vs. no damping

• underdamping
• no damping (b0)

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critical dampingdecay to x 0, no oscillation

• can also view this critical value of b as
  resulting from oscillation disappearing

See sketch of x(t) for critical damping
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overdamping slower decay to x 0, no
oscillation
See sketch of x(t) for overdamping
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Application

• Shock absorbers
• want critically damped (no oscillations)
• not overdamped(would have aslow response time)

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Forced Oscillations

• (Forced SHM)

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Forced SHM

• We have considered the presence of a damping
  force acting on an oscillator
  Fdamping bv
• Now consider applying an external force
  Fdriving Fmax coswdt

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Forced SHM

• Every simple harmonic oscillator has a natural
  oscillation frequency
• (w if undamped, w if underdamped)
• By appling Fdriving Fmax coswdt we force the
  oscillator to oscillate at the frequency wd (can
  be anything, not necessarily w or w)

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What is x(t) for forced SHM?

• We get a new equation of motion for x(t)
• We wont solve it, just present the solution.

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x(t) for Forced SHM

• If you solve the differential equation, you find
  the solution (at late times, t gtgt 2m/b)

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Amplitude A(wd)

• Shown (for f 0)A(wd) for different b
• larger b smaller Amax
• ResonanceAmax occurs at wR, near the natural
  frequency,w (k/m)1/2

Do Resonance Demonstrations
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Resonance Frequency (wR)

• Amax occurs at wdwR (where dA/dwd0)

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natural, underdamped, forcedw gt w gt wR

• natural frequency
• underdamped frequency
• resonance frequency

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Introduction toLRC Circuits

• (Electromagnetic Oscillations)

See transparency on LRC circuit
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Electric Quantity Counterpart

• charge Q(t) x(t)
• current I dQ/dt v dx/dt(moving
  charge)(generates a magnetic field, B)

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Electrical Concepts

• electric charge Q
• current (moving charge) I dQ/dt
• resistance (Q collides with atoms) R
• voltage (pushes Q through wire) V RI

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Voltage (moves charges)

• resistance R causes charge Q to lose energy
  V RI
• (voltage potential energy per unit charge)
• C and L also cause energy (voltage) changes

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Circuit Element D(Voltage)

• R resistance VR RI(Q collides with atoms)
  
• C capacitance VC Q/C(capacity to store Q
  on plate)
• L inductance VL L(dI/dt)(inertia towards
  changes in I)

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Change in Voltage Change in Energy

• voltage potential energy per unit charge
• recall, around a closed loop

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• Which looks like

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Circuit Element Counterpart

• 1/C 1/capacitance k
• L inductance m
• R resistance b
• (Extra Credit Exercise 31-35)
• Use this table to write our damped SHM as damped
  electromagnetic oscillations

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In the LRC circuit, Q(t) acts just like
x(t)!underdamped, critically damped, overdamped
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Driven (and resonance) Vdriving Vmax coswdt