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SHM for IB and SL-DS Lebanese program

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  1. 1. Key points OSCILLATIONS
  2. 5. Oscillations <ul><li>Periodic Motion :- Any motion that repeats itself in equal intervals of time. </li></ul><ul><li>Oscillatory Motion :- If a particle in periodic motion moves back and forth over the same path. </li></ul><ul><li>Harmonic Motion :- The displacement of a particle in periodic motion can always be expressed in terms of sine and cosine functions therefore oscillatory motions are called Harmonic motions. </li></ul>
  3. 6. Oscillations <ul><li>The world is full of oscillatory motions </li></ul><ul><ul><li>A child on a swing </li></ul></ul><ul><ul><li>A mass attached to a spring </li></ul></ul><ul><ul><li>A guitar string being played </li></ul></ul><ul><ul><li>Swinging pendulum </li></ul></ul><ul><ul><li>Atoms in molecules or in solid lattice </li></ul></ul><ul><ul><li>Air molecules as a sound wave passes by </li></ul></ul><ul><ul><li>Radio waves, microwaves and visible light are oscillating magnetic and electric field vectors </li></ul></ul>
  4. 7. Oscillations <ul><li>Simple harmonic motion </li></ul><ul><li>Damped simple harmonic motion </li></ul><ul><li>- more realistic motion of oscillating systems </li></ul><ul><li>Forced oscillations and resonance </li></ul>
  5. 8. t = 0 t = T/4 t = T/2 t = 3T/4 t = T
  6. 9. <ul><li>Period T of a harmonic motion : </li></ul><ul><li>-The time required to complete one round trip of </li></ul><ul><li>the motion i.e. one complete oscillation </li></ul><ul><li>Frequency f : Number of oscillations in 1 second </li></ul><ul><li>Unit : 1 hertz (Hz) = one oscillation per second </li></ul><ul><li>= 1s -1 </li></ul><ul><li>It is inverse of the period </li></ul>
  7. 10. <ul><li>When the displacement of a particle at any time “t” can be given by this formula: </li></ul>Its motion is called simple harmonic motion
  8. 11. Simple Harmonic Motion <ul><li>The factors x m , ω and  are all constants </li></ul><ul><ul><li>x m is the amplitude of the oscillation </li></ul></ul><ul><ul><li>ω is angular frequency of the oscillation </li></ul></ul><ul><ul><li>The time varying quantity ( ωt +  ) is called the phase of the motion and  is called the phase constant or phase angle </li></ul></ul>
  9. 12. Simple Harmonic Motion <ul><li>and cosine function repeats itself when its argument (the phase) has increased by the value 2π </li></ul>
  10. 13. Simple Harmonic Motion
  11. 14. Simple Harmonic Motion <ul><li>We can see that the curves look identical except that one is ‘taller’ than the other </li></ul><ul><li>These two curves have a different maximum displacement – or amplitude x m </li></ul>Keeping ω & Φ constant varying x m …
  12. 15. Simple Harmonic Motion <ul><li>Keeping Φ & x m constant </li></ul><ul><li>Varying ω =2 π /T … </li></ul><ul><li>let T ´ = T /2 </li></ul>
  13. 16. Simple Harmonic Motion <ul><li>Keeping x m & ω constant </li></ul><ul><li>varying phase angle  </li></ul><ul><li>The 2nd curve has ‘slid over’ by a constant amount relative to the 1st curve… </li></ul>
  14. 17. The Velocity of SHM <ul><li>To get the velocity of the particle </li></ul><ul><li>differentiate the displacement function with respect to time </li></ul>
  15. 18. The Velocity of SHM
  16. 19. The Velocity of SHM <ul><li>velocity is a sine function </li></ul><ul><li>It is T/4 period (or π /2) out of phase with the displacement </li></ul>
  17. 20. The Acceleration of SHM <ul><li>lets differentiate once again to get the acceleration function: </li></ul>
  18. 21. The Acceleration of SHM <ul><li>We can combine our original equation for the displacement function and our equation for the acceleration function to get: </li></ul>
  19. 22. The Acceleration of SHM <ul><li>This relationship of the acceleration being proportional but opposite in sign to the displacement is the hallmark of SHM </li></ul><ul><li>Specifically, the constant of proportionality is the square of the angular frequency </li></ul>
  20. 23. <ul><li>The acceleration function is one-half period (or π radians) out of phase with the displacement and the maximum magnitude of the acceleration is ω 2 x m </li></ul>
  21. 24. <ul><li>When the displacement is at a maximum, the acceleration is also at a maximum (but opposite in sign) </li></ul><ul><li>And when the displacement and acceleration are at a maximum, the velocity is zero </li></ul>
  22. 25. <ul><li>Similarly, when the displacement is zero, the velocity is at a maximum </li></ul><ul><li>Does this remind you of anything you have seen previously? </li></ul>
  23. 26. The Force Law for SHM <ul><li>Knowing the acceleration of a particle in SHM, lets now apply Newton’s 2 nd law to get the equation of motion of SHM </li></ul>This force is called restoring force k is called force constant
  24. 27. The Force Law for SHM <ul><li>SHM is defined as when a particle of mass m experiences a force that is proportional to the displacement of the particle, but opposite in sign </li></ul><ul><li>Equation of SHM </li></ul>Where ω is natural frequency
  25. 28. Simple Harmonic Oscillator <ul><li>The block-spring system shown here is a classic linear simple harmonic oscillator where ‘linear’ means that the force F is proportional to the displacement x rather than to some other power of x </li></ul>
  26. 30. Mass-spring system <ul><li>The angular frequency ω and period T are therefore related to the spring constant k by the formulas: </li></ul>
  27. 31. Problem 1 <ul><li>Which of these relationships implies SHM? </li></ul><ul><ul><li>F = -5 x </li></ul></ul><ul><ul><li>F = -400 x 2 </li></ul></ul><ul><ul><li>F = 10 x </li></ul></ul><ul><ul><li>F = 3 x 2 </li></ul></ul>
  28. 32. Problem 2 <ul><li>At t = 0 the displacement x (0) of the block is -8.50 cm, the velocity is v (0) = -0.920 m/s and it’s acceleration is a (0) = +47.0 m/s </li></ul><ul><li>What are the values for ω, x m and  ? </li></ul>
  29. 33. Solution of problem 2
  30. 35. P 17.3 To be on the verge of slipping means that the force exerted on the smaller block is :-
  31. 36. P 17.5
  32. 37. Continued ……. P 17.6
  33. 39. P 17.7
  34. 40. P 17.7
  35. 41. E 17.17
  36. 42. Energy in SHM <ul><li>We already know from theChapters 11 &12 on Work,Kinetic Energy, Potential Energy and Conservation of Energy that kinetic and potential energy get transferred back and forth as a linear oscillator moves </li></ul><ul><li>Remember also that the total mechanical energy E remains constant </li></ul>
  37. 43. Energy in SHM <ul><li>We know from Chapter 12 that the potential energy stored in the spring is: </li></ul><ul><li>Substituting x(t), we get: </li></ul>
  38. 44. Energy in SHM <ul><li>The Kinetic energy of the block is: </li></ul>Using the velocity function and also making a substitution of ω 2 = k/m we get:
  39. 45. Energy in SHM <ul><li>the total mechanical energy is: </li></ul>
  40. 46. Energy in SHM So we finally get The total energy of a linear oscillator is constant and independent of time
  41. 47. Energy in SHM <ul><li>So an oscillating system contains an element of ‘springiness’ (which stores the potential energy) and an element of inertia (which stores the kinetic energy) – even if the system is not mechanical in nature </li></ul><ul><li>In an electrical system, capacitor is the element of ‘springiness’ and a choke (a coil i.e. inductor) is the element of ‘inertia’. </li></ul>
  42. 50. Energy Conservation in Oscillatory Motion
  43. 51. At any time t E = U (t) + K (t) = constant Differentiating w. r. t. t
  44. 52. Energy Law in SHM
  45. 53. Problem 3 <ul><li>In the figure the block has a kinetic energy of 3 J and the spring has an elastic potential energy of 2 J when the block is at x = +2.0 cm </li></ul><ul><ul><li>(a) What is the KE when the block is at x = 0? </li></ul></ul><ul><ul><li>What are the elastic potential energies when the block is at (b) x = -2.0 cm and (c) x = - x m ? </li></ul></ul>
  46. 54. <ul><li>(a) KE = K max = E = 5J </li></ul>Solution problem 3 (b) PE ( at x = - 2.0 cm) = PE ( at x = + 2.0 cm) = 2J (c) PE ( x = - x m ) = U max = E = 5J
  47. 55. P 17.13
  48. 56. P 17. 8 Continued …. .