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Many physical phenomena can be modeled with simple harmonic motion.

Many physical phenomena can be modeled with simple harmonic motion. the swinging of a pendulum. a spring-mass system. radio and television waves. light and sound waves. water waves. The amplitude is | a |.

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Many physical phenomena can be modeled with simple harmonic motion.

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  1. Many physical phenomena can be modeled with simple harmonic motion. the swinging of a pendulum a spring-mass system radio and television waves light and sound waves water waves

  2. The amplitude is |a|. The frequency is the number of oscillations per unit time. It is the reciprocal of the period.

  3. An object attached to a coiled spring is pushed up a distance 4 units from its rest position and then released. Assuming that the motion is simple harmonic with period 2 seconds, write an equation that relates the displacement d of the object from its rest potion after t seconds. Assume that the positive direction of the motion is up. Simple harmonic motion can be expressed with either sine or cosine but since this object starts at 4 when t = 0, cosine makes sense. If you used sine, you'd have to use a phase shift to have a different value than 0 at t = 0.

  4. This tells us that the motion will repeat every seconds. It is the reciprocal of the period so If we knew the equation of motion for an object, we could determine some things about it. The equation modeling the motion of an object is: Since we know , we can find the period This negative tells us that after t = 0, it will be stretched down first 1.6 What is the frequency? .64 This means in one second, it has completed .64 oscillations.

  5. Often the motion of an object is affected by friction or other resistive forces. These slow the object or damp its motion. The equation modeling an oscillating object with damping is: damping factor mass of the object

  6. What does the graph of look like? here is here is b and m are positive so this is a graph of e to the negative something t So as time increases, this function decreases and approaches zero

  7. What does the graph of look like? This is a basic cosine graph with amplitude a and period

  8. but this part which multiplies the amplitude is going to zero so this part wants to oscillate Notice the amplitude decreasing exponentially --- and this is what happens!

  9. When you push a button on your phone, the tone made is actually two tones of different frequencies added together. To obtain the graph of the sum of two or more functions, we can simply use the function property (f + g)(x) = f(x) + g(x) So we'll find the value of each function at x and then add the values together

  10. Let's find the graph of f(x) = 2 sin x + sin 2x Plot these points x 0 The purple is 2 sin x and the red is sin 2x. See how adding the values makes the blue graph.

  11. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

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