1. 13. Oscillatory Motion

2. Oscillatory Motion

3. 3 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. 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. 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. 6 Simple Harmonic Motion As usual, we can compute the motion of the object using Newton’s 2 nd law of motion, F = m a: The solution of this differential equation gives x as function of t.

7. 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. 8 Simple Harmonic Motion Note that at t = 0, x = A. A is also the amplitude. Why? To find the value of  we need to verify that our tentative solution is in fact a solution of the equation of motion.

9. 9 Simple Harmonic Motion. therefore

10. 10 We can get a solution if we set k = m  2, that is, By definition, after a period T later the motion repeats, therefore: Simple Harmonic Motion Frequency and Period.

11. 11 The equation can be solved if we set  T = 2 , that is, if we set Simple Harmonic Motion Frequency and Period.  is called the angular frequency

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

13. 13 It is easy to show that is a more general solution of the equation of motion. The symbol  is called the phase. It defines the initial displacement x = A cos  Simple Harmonic Motion Phase.

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

15. 15 Simple Harmonic Motion Position, Velocity, Acceleration

16. Applications of SHM

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

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

19. 19 The Pendulum A simple pendulum consists of a point mass suspended from a massless string! Newton’s 2 nd law for such a system is The motion is not simple harmonic. Why?

20. 20 The Pendulum If the amplitude of a pendulum is small enough, then we can write sin  ≈ , in which case the motion becomes simple harmonic This yields

21. 21 The Pendulum For a point mass, m, a distance L from a pivot, the rotational inertia is I = mL 2. Therefore, and

22. Energy in SHM

23. 23 Energy in Simple Harmonic Motion Position Velocity Acceleration

24. 24 Energy in Simple Harmonic Motion Kinetic Energy

25. 25 Energy in Simple Harmonic Motion Potential Energy

26. 26 Energy in Simple Harmonic Motion Total Energy = Kinetic + Potential In the absence of non-conservative forces the total mechanical energy is constant For a spring:

27. 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. 29 Damped Harmonic Motion Non-conservative forces, such as friction, cause the amplitude of oscillation to decrease.

30. 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. 31 The solution of the differential equation is of the form For simplicity, we take x = A at t = 0, then  = 0. Damped Harmonic Motion

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

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

34. 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. 35 Example – Bad Shocks First find out how long it takes for the amplitude to drop to half its initial value:  = 2m/b = 10.43 s exp(-t/  ) = ½ → t =  ln 2 = 7.23 s http://static.howstuffworks.com/gif/car-suspension-1.gif

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

37. Driven Oscillations

38. 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 = F 0 cos(  d t). Newton’s 2 nd law for the system becomes

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

40. 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. 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.