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13. Oscillatory Motion

13. Oscillatory Motion. Oscillatory Motion. If one displaces a system from a position of stable equilibrium the system will move back and forth, that is, it will oscillate about the equilibrium position. The maximum displacement from the equilibrium is called the amplitude , A.

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13. Oscillatory Motion

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  1. 13. Oscillatory Motion

  2. Oscillatory Motion If one displaces a system from a position of stable equilibrium the system will move back and forth, that is, it will oscillate about the equilibrium position. The maximum displacement from the equilibrium is called the amplitude, A.

  3. Oscillatory Motion The time, T, to go through one complete cycle is called the period. Its inverse is called frequency and is measured in hertz (Hz). 1 Hz is one cycle per second.

  4. Simple Harmonic Motion For many systems, if the amplitude is small enough, the restoring force F satisfies Hook’s law. The motion of such a system is called simple harmonic motion (SHM) Hook’s law

  5. Simple Harmonic Motion As usual, we can compute the motion of the object using Newton’s 2nd law of motion, Fnet = ma: The solution of this differential equation gives x as function of t.

  6. Simple Harmonic Motion Suppose we start the system displaced from equilibrium and then release it. How will the displacement x depend on time, t ? Let’s try a solution of the form

  7. Simple Harmonic Motion Note that at t = 0, x = A. A is also the amplitude. Why? To find the value of ω we need to verify that our tentative solution is in fact a solution of the equation of motion.

  8. Simple Harmonic Motion . therefore

  9. Simple Harmonic MotionFrequency and Period We can get a solution if we set k = mω 2, that is, By definition, after a period T later the motion repeats, therefore:

  10. Simple Harmonic MotionFrequency and Period The equation can be solved if we set ωT = 2π, that is, if we set ω is called the angular frequency

  11. Simple Harmonic MotionFrequency and Period For simple harmonic motion of the mass-spring system, we can write

  12. Simple Harmonic MotionPhase It is easy to show that is a more general solution of the equation of motion. The symbol ϕ is called the phase. It defines the initial displacement x = Acosϕ

  13. Simple Harmonic MotionPosition, Velocity, Acceleration Position Velocity Acceleration

  14. Applications of SHM

  15. Vertical Mass-Spring System At equilibrium upward force of spring = weight of block Gravity changes only the equilibrium position

  16. The Pendulum A simple pendulum consists of a point mass suspended from a massless string! Newton’s 2nd law for such a system is The motion is not simple harmonic. Why?

  17. The Pendulum If the amplitude of a pendulum is small enough, then we can write sinθ≈θ, in which case the motion becomes simple harmonic This yields

  18. The Pendulum For a point mass, m, a distance L from a pivot, the rotational inertia is I = mL2. Therefore, and

  19. Energy in SHM

  20. Energy in Simple Harmonic Motion Kinetic Energy Potential Energy

  21. Energy in Simple Harmonic Motion Total Energy = Kinetic + Potential For a spring: In the absence of non-conservative forces the total mechanical energy is constant

  22. Damped Harmonic Motion & Driven Oscillations

  23. Damped Harmonic Motion Non-conservative forces, such as friction, cause the amplitude of oscillation to decrease.

  24. Driven Oscillations When an oscillatory system is acted upon by an external force we say that the system is driven. Consider an external oscillatory force F = F0 cos(ωdt), we find that the amplitude exhibits the resonance.

  25. Example – Resonance November 7, 1940 – Tacoma Narrows Bridge Disaster. At about 11:00 am the Tacoma Narrows Bridge, near Tacoma, Washington collapsed after hitting its resonant frequency. The external driving force was the wind. http://www.enm.bris.ac.uk/anm/tacoma/tacnarr.mpg

  26. Summary • Systems that move in a periodic fashion are said to oscillate. If the restoring force on the system is proportional to the displacement, the motion will be simple harmonic. • The mass-spring system is a simple model that undergoes simple harmonic motion. • If the presence of non-conservative forces the system will undergo damped harmonic motion. • If driven, the system can exhibit resonant motion.

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