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Physics 102 Waves

Physics 102 Waves. Lecture 1 The Harmonic Oscillator. Moza m. Al-Rabban Associate Professor of Physics mmr@qu.edu.qa. Useful links. http://www.physics.uoguelph.ca/tutorials/shm/Q.shm.html. Periodic Motion. A motion that repeat itself over and over is referred to as periodic motion.

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Physics 102 Waves

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  1. Physics 102 Waves Lecture 1The Harmonic Oscillator Moza m. Al-Rabban Associate Professor of Physics mmr@qu.edu.qa

  2. Useful links http://www.physics.uoguelph.ca/tutorials/shm/Q.shm.html Physics 102 - Lecture 1

  3. Periodic Motion A motion that repeat itself over and over is referred to as periodic motion. • Examples: • The beating of your heart, • The ticking of a clock, • The movement of a child on a swing…. The time required for the completion of one cycle of a periodic system’s repetitive motion called the period, T, of that system. Physics 102 - Lecture 1

  4. Period, T • Definition of Period: T = time required for one cycle of a periodic motion. SI unit: second/cycle = s Note that a cycle (that is, an oscillation)is dimensionless. Physics 102 - Lecture 1

  5. Frequency, f • The frequency of an oscillation is the number of oscillations per unit time. • It tells us how frequently, or rapidly, an oscillation takes place _ the higher frequency, the more rapid the oscillations. • Definition of Frequency, f • SI unit: cycle/second = 1/s = • 1 Hz = 1 cycle/second Physics 102 - Lecture 1

  6. Example 1: If the processing speed of a personal computer is 1.80 GHz, how much time is required for one processing cycle? The high frequency of the computer corresponds to a very small period to complete one operation. Physics 102 - Lecture 1

  7. Example 2: A tennis ball is hit back and forth between two players warming up for a match.If it takes 2.31 s for the ball to go from one player to the other, what are the period and frequency of the ball’s motion? The period of this motion is the time for the ball to complete one round trip. Therefore, Physics 102 - Lecture 1

  8. Example: Frequency and Period of a Radio Station What is the oscillation period for the broadcast of a 100 MHz FM radio station? Physics 102 - Lecture 1

  9. Simple Harmonic Motion • An oscillator is an object or system of objects that undergoes periodic oscillatory motion or behavior. Example: a mass and spring system. • Characteristics: • Oscillatory motion is about some (energy=0) equilibrium position; • Oscillatory motion is periodic, with a definite period or cycle time. Physics 102 - Lecture 1

  10. SHM Prototype Experiment Consider Fig. (a). An air-track glider attached to a spring. The glider is pulled a distance A from its rest position and released. Fig. (b) shows a graph of the motion of the glider, as measured each 1/20 of a second. The object’s maximum displacement from equilibrium is called the amplitude A of the motion. The object’s position oscillates between x=-A and x=+A. The graph shows the position of the glider from the same measurements. We see that A=0.17 m and T=1.60 s. Therefore the oscillation frequency of the system is f = 0.625 Hz Physics 102 - Lecture 1

  11. The Figure down is a velocity-versus-time graph. • The velocity graph is also sinusoidal, oscillating between –vmax ( maximum speed to the left) and +vmax (maximum speed to the right). As the figure shows, • The instantaneous velocity is zero at the points where x = ±A. These are the turning points in the motion. • The maximum speed vmax is reached as the object passes through the equilibrium position at x = 0 m. The velocity is positive as the object moves to the right but negative as it moves to the left. Physics 102 - Lecture 1

  12. We can ask three important questions about this oscillating system: • How is the maximum speed vmax related to the amplitude A? • How are the period and frequency related to the object’s mass m, the spring constant k, and the amplitude A? • Is the sinusoidal oscillation a consequence of Newton’s laws? Physics 102 - Lecture 1

  13. Kinematics of SHM Physics 102 - Lecture 1

  14. It’s especially important to remember to set your calculator to radian mode before working oscillation problems and use these equations. Physics 102 - Lecture 1

  15. Example: A System in SHM • An air-track glider is attached to a spring,pulled 20 cm to the right, and releasedat t=0. It makes 15 oscillations in 10 s. • What is the period T of oscillation? • What is the object’s maximum speed? • What is the position and velocity at t=80 s? Physics 102 - Lecture 1

  16. Example: Finding the Time A mass on a spring oscillating in simple harmonic motion (SHM) starts at x=A at t=0 and has period T. At what time, as a fraction of T, does the object first pass through x = ½A? 2 √3 600 =p/3 1 Physics 102 - Lecture 1

  17. Clicker Question 1 An object moves with simple harmonic motion. If the amplitude and period of the oscillation are both doubled, the object’s maximum speed is: • quadrupled; • doubled; • unchanged • halved; • quartered. Physics 102 - Lecture 1

  18. SHM and Circular Motion Uniform circular motion projected into one dimension is simple harmonic motion (SHM). Consider a particle rotating ccw, with the angle f increasing linearly with time:

  19. Note » when used to describe oscillatory motion,  is called the angular frequency rather than the angular velocity. Note » Hertz is specifically “ cycle per second” or “ oscillations per second” . It is used forƒ but not for . Physics 102 - Lecture 1

  20. The Phase Constant But what if f is not zero at t = 0 ? The position xo and velocity vox at t = 0 are the initial conditions. Different values of the phase constant correspond to different starting points on circle and thus different initial conditions.

  21. Phases and Oscillations 1 2 3 • Here are three examples of differing initial conditions: • f0 = p/3, implying x0 = A/2 and moving to the left (v<0); • f0 = -p/3, implying x0 = A/2 and moving to the right (v>0); • f0 = p, implying x0 = -A and momentarily at rest (v=0). Physics 102 - Lecture 1

  22. Example: Using Initial Conditions A mass on a spring oscillates with a period of 0.80 s and an amplitude of 10 cm. At t=0, it is 5.0 cm to the left of equilibrium and moving to the left. What is the position and direction of motion at t=2.0 s? - Physics 102 - Lecture 1

  23. Clicker Question 2 These figures show four oscillator systems at t=0. Which one has the phase constant f0 = p/4 rad? Physics 102 - Lecture 1

  24. The Energy of SHM Physics 102 - Lecture 1

  25. Velocity and Amplitude of SHM Physics 102 - Lecture 1

  26. Example:Using Conservation of Energy A 500 g block on a spring is pulled a distance of 20 cm and released. The subsequent oscillation is measured to have a period of 0.80 s. At what position(s) is the block’s speed 1.0 m/s? Physics 102 - Lecture 1

  27. Clicker Question 3 These figures show four springs that have been compressed from their equilibrium positions at x=0. When released, they will begin to oscillate. Which one will have the largest maximum speed of oscillation? Physics 102 - Lecture 1

  28. The Dynamics of SHM The acceleration is proportional to the negative of the displacement at any time. Physics 102 - Lecture 1

  29. The Equation of Motion in SHM This is the equation of motion for the system. It is a homogeneous linear 2nd order differential equation. Physics 102 - Lecture 1

  30. Solving the Equation of Motion Q: How do you solve this differential equation? A: Our method - guess the solution and see if it works. It works! Physics 102 - Lecture 1

  31. Example:Analyzing an Oscillator At t=0, a 500 g block oscillating on a spring is observed to be moving to the right at x=15 cm. At t=0.30 s, it reaches its maximum displacement of 25 cm. • Draw a graph of the motion for one cycle. • At what time in the first cycle is x=20 cm? Physics 102 - Lecture 1

  32. Clicker Question 4 The position vs. time of a mass-and-spring oscillator is shown. What can you say about the velocity and force at the time indicated by the dashed line? • Velocity is positive; force is to the right. • Velocity is zero; force is to the right. • Velocity is negative; force is to the right. • Velocity is positive; force is to the left. • Velocity is zero; force is to the left. Physics 102 - Lecture 1

  33. End of Lecture 1 • Before the next lecture, read Knight,Chapters 14.6 through 14.8

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