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13. Oscillatory Motion

13. Oscillatory Motion. 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

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13. Oscillatory Motion

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  1. 13. Oscillatory Motion

  2. Oscillatory Motion

  3. 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.

  4. 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.

  5. 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

  6. 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.

  7. 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

  8. 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.

  9. Simple Harmonic Motion . therefore

  10. 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:

  11. 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

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

  13. 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

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

  15. Simple Harmonic MotionPosition, Velocity, Acceleration

  16. Applications of SHM

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

  18. The Torsional Oscillator A fiber with torsional constant κ provides a restoring torque: The angular frequency depends on κ and the rotational inertia I: Newton’s 2nd law for this device is

  19. 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?

  20. 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

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

  22. Energy in SHM

  23. Energy in Simple Harmonic Motion Position Velocity Acceleration

  24. Energy in Simple Harmonic Motion Kinetic Energy

  25. Energy in Simple Harmonic Motion Potential Energy

  26. 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

  27. Energy in Simple Harmonic Motion In a simple harmonic oscillator the energy oscillates back and forth between kinetic and potential energy, in such a way that the sum remains constant. In reality, however, most systems are affected by non-conservative forces.

  28. Damped Harmonic Motion

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

  30. Damped Harmonic Motion In many systems, the non-conservative force (called the damping force) is approximately equal to where b is a constant giving the damping strength and v is the velocity. The motion of such a mass-spring system is described by

  31. Damped Harmonic Motion The solution of the differential equation is of the form For simplicity, we take x = A at t = 0, then ϕ = 0.

  32. Damped Harmonic Motion If one plugs the solution into Newton’s 2nd law, one will find the damping time and the angular frequency, where is the un-damped angular frequency

  33. Damped Harmonic Motion The larger the damping constant b the shorter the damping timeτ. There are 3 damping regimes: (a) Underdamped (b) Critically damped (c) Overdamped

  34. Example – Bad Shocks A car’s suspension can be modeled as a damped mass-spring system with m = 1200 kg, k = 58 kN/m and b = 230 kg/s. How many oscillations does it take for the amplitude of the suspension to drop to half its initial value? http://static.howstuffworks.com/gif/car-suspension-1.gif

  35. Example – Bad Shocks First find out how long it takes for the amplitude to drop to half its initial value: τ = 2m/b = 10.43 s exp(-t/τ) = ½ → t =τ ln 2 = 7.23 s http://static.howstuffworks.com/gif/car-suspension-1.gif

  36. Example – Bad Shocks The period of oscillation is T = 2π/ω = 2π/√(k/m – 1/τ 2) = 0.904 s Therefore, in 7.23 s, the shocks oscillate 7.23/0.904 ~ 8 times! These are really shocking shocks! http://static.howstuffworks.com/gif/car-suspension-1.gif

  37. Driven Oscillations

  38. 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). Newton’s 2nd law for the system becomes

  39. Driven Oscillations Again, we try a solution of the form x(t) = A cos(ωdt). When this is plugged into the 2nd law, we find that the amplitude has the resonance form

  40. 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

  41. 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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