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Chapter 16

Chapter 16. Oscillations. Oscillations. Motion that repeats itself at regular intervals: Spring Pendulum Rocking horse Diatomic molecule Kid on a trampoline. Special case: Simple harmonic motion (SHM). oscillation. Simple Harmonic Motion.

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Chapter 16

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  1. Chapter 16 Oscillations

  2. Oscillations Motion that repeats itself at regular intervals: Spring Pendulum Rocking horse Diatomic molecule Kid on a trampoline Special case: Simple harmonic motion (SHM)

  3. oscillation Simple Harmonic Motion Classic case of SHM: Mass on a massless spring with no friction M x = 0 Defines equilibrium length

  4. T = 1/f = 2p/w Simple Harmonic Motion Defined x(t) = xmcos(wt + f) xm = amplitude of oscillation w = angular frequency (radians/second) f = w/2p = “frequency”  # of cycles/second T = 1/f = “period” f = “phase” of motion (radians) (tells you position and velocity at t = 0) x xm t -xm

  5. What is Phase? x(t) = xmcos(t + ) • determines position and velocity at t = 0 (and shifts oscillation at any later time) x f = 0 f < 0 f > 0 t

  6. w x y r t x Visualizing SHM Equivalent to the 1-dimensional projection of an object executing uniform circular motion http://www.phy.ntnu.edu.tw/java/shm/shm.html

  7. a x v t SHM: Acceleration and Velocity

  8. k What Sort of Force Gives SHM? a(t) = -w2x(t) Ftot = ma = -mw2 x Force is proportional to displacement with a negative constant of proportionality Hooke’s law! F = -kx w = (k/m)½ m w is the frequency of oscillation of the mass w does not depend on amplitude of motion

  9. Fspring mg x0 m Example: Hanging Mass What about a hanging spring? Does it still obey SHM Ftot = mg-kx = ma Equilibrium when: mg = kx0 x0 = mg/k

  10. x0 m Example: Hanging Mass What is the frequency of oscillation? Need a solution to: Try: Same as before with x shifted by x0 = mg/k

  11. 100 80 60 Energy 40 20 0 -10 -5 0 5 10 x (m) Energy in a Spring Total energy is constant, but sloshes between kinetic and potential Kinetic energy: Potential energy:

  12. m Energy in a Spring Two special situations where calculating total energy is easy: All potential energy All kinetic energy

  13. Simple Pendulum For small angles, θ L m h = L(1-cosθ)

  14. Simple Pendulum Follows SHM Looks like a spring Solution by analogy Spring Pendulum

  15. Simple Pendulum: Questions Q1. If we double θm, the period: a) is half as large d) is 4 times greater b) is twice as large e) stays the same c) is √2 times greater Q2. If we double L, the period: a) is half as large d) is 4 times greater b) is twice as large e) stays the same c) is √2 times greater

  16. Physical Pendulum An object with physical extent: d θ COM We know the solution from before: mg Any system with a minimum in energy looks like a SHO near equilibrium

  17. Amplitude Time Damped Oscillations SHM is an idealization Energy is constant Motion never decays In real life the motion eventually stops Energy  0 Need to add a damping force in the equation of motion: Fd = -bv Direction opposite to motion Magnitude proportional to velocity

  18. Damped Oscillations SHM equation of motion (no damping) Adding the damping term: How do we solve this?

  19. What Happens in Real Life? 120 100 80 60 Energy 40 20 0 -20 0 5 10 15 20 Time 120 100 80 Energy 60 40 20 0 0 20 40 60 80 100 Time The Ideal Case: In real life the system loses energy Rate of energy loss proportional to energy

  20. 15 10 5 0 Amplitude -5 -10 -15 -20 0 20 40 60 80 100 120 Time The Solution Back to the damped oscillator Guess a solution of the form: You can fill in the gaps after you learn differential equations! It works if:

  21. Damped Oscillator 1) 2) 3) There are three types of solutions Underdamped Overdamped No oscillations! Critically damped

  22. 1) Underdamped

  23. 2) Overdamped

  24. 3) Critical Damping

  25. Forced Oscillations Two ways to apply a force Static – a constant push Dynamic – periodic push Consider a Dynamic Force

  26. 15 10 5 Force or Amplitude 0 -5 -10 -15 0 20 40 60 80 100 Time 20 15 10 Amplitude 5 0 -5 -10 -15 0 20 40 60 80 100 Time The Effects of a Force Examine limiting cases Tf >> T Slowly varying force Fast Oscillations F ~ constant during each oscillation Slow Force A constant force just shifts the equilibrium position

  27. 15 10 5 Force or Amplitude 0 -5 -10 -15 0 20 40 60 80 100 Time Rapidly Varying Forces Tf << T Force oscillates rapidly with respect to free oscillations The force oscillates many times during one cycle No effect Average value of F = 0

  28. 6 5 4 Amplitude 3 2 1 0 0 20 40 60 80 100 Frequency Resonance The equation of motion is The exact solution is Big response when d= !

  29. Resonance Resonance can lead to spectacular consequences!

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