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

2. Useful links http://www.physics.uoguelph.ca/tutorials/shm/Q.shm.html
Physics 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 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.
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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
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6. Example 1: If the processing speed of a personal computer is 1
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.
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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,
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8. Example: Frequency and Period of a Radio Station
What is the oscillation period for the broadcast of a 100 MHz FM radio station?
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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.
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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 = Hz
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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.
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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?
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13. Kinematics of SHM
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14. It’s especially important to remember to set your calculator to radian mode before working oscillation problems and use these equations.
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15. Example: A System in SHM
An air-track glider is attached to a spring, pulled 20 cm to the right, and released at 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?
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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
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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 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 for ƒ but not for  .
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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 Oscillations1 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).
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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? -
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23. Clicker Question 2 These figures show four oscillator systems at t=0.
Which one has the phase constant f0 = p/4 rad?
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24. The Energy of SHM
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25. Velocity and Amplitude of SHM
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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?
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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?
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28. The Dynamics of SHM
The acceleration is proportional to the negative of the displacement at any time.
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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.
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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!
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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 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.
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33. End of Lecture 1
Before the next lecture, read Knight, Chapters 14.6 through 14.8